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# POL S 1 POL S 1

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Ls Full text provided by wwwsciencedirectcom SCIENCEltampDIFIECT Enzymatic transition states and transition state analogues Vern L Schramm Transition states are the balance point of catalysis Bonds are partially made andor broken at the transition state and the energy of the extended system provides near equal probability that the system forms products or reverts to reactants Enzymatic catalytic sites provide dynamic electronic environments that increase the probability that the transition state will be formed Alignment of reactants in the Michaelis complex and motion of the catalytic site architecture are necessary to achieve the transition state Transition state lifetimes are a fraction of a picosecond preventing chemical equilibrium in extended covalent systems Thus dynamic descriptions of enzymatic transition states are required Stable analogues similar to the transition state capture dynamic excursions that generate the transition state and convert them into thermodynamic binding energy These analogues bind with extraordinary affinity relative to reactants Addresses Department of Biochemistry Albert Einstein College of Medicine 1300 Morris Park Avenue Bronx NY 10461 USA Corresponding author Schramm Vern L vernaecomyuedu Current Opinion in Structural Biology 2005 15604 613 This review comes from a themed issue on Catalysis and regulation Edited by William N Hunter and Ylva Lindqvist Available online 4th November 2005 0959 440X see front matter 2005 Elsevier Ltd All rights reserved DOI 101016j3bi200510017 Introduction The idea of the enzymatic transition state developed from the chemical rate theory of Eyring who formulated the transition state as a stable state to enable mathema tical treatment using the thermodynamic and activated state concepts of the time 1 The approach of Eyring dominated thinking about enzymatic transition states for decades and led to the early concept that enzymes achieve their catalytic potential by binding tightly to the activated complex or transition state in the same way that antibodies were known to bind with high af nity to their haptens 23 Propagation of these ideas contin ued with the lock and key and transition state binding descriptions of enzyme function 45 Wolfenden quan titated the concept of transition state stabilization by creating a thermodynamic box with the hypothetical construct of the binding energy between the free transi tion state and the enzyme Figure 1 56 These ideas have been experimentally supported by X ray crystal lography which shows loose complexes between enzymes and reactants Michaelis complexes and tigh ter complexes between enzymes and analogues of their transition states 78 Equilibrium treatment of transition states Despite its practical utility and broad acceptance the Eyring treatment which assumes thermodynamically stable transition states is physically impossible A recent review of current developments in reaction rate theory summarizes these problems in the provocative state ments a de nite quantum transition state theory has not been formulated to date and transition state theory is no longer valid and cannot even serve as a conceptual guide for understanding the critical factors which determine rates away from equilibrium 9quot The opposing view with regard to modern classical transition state theory is alternative descriptions proposed as the source of enzyme catalysis are encompassed in mod ern transition state theory and do not require the intro duction of new concepts 10quot With this degree of uncertainty regarding the nature of transition states how can we interpret the general principles of catalysis which appear empirically correct or are experimentally useful and attractive These include enzymes bind tightly to their transition states limiting values of transi tion state binding energy are given by enzyme chemical gtlt binding energy of the Michaelis complex chemically stable analogues of the transition state bind tightly to their cognate enzymes with a limiting value of enzyme chemical gtlt binding energy of the Michaelis complex 11quot Additionally how can we resolve these principles with the growing evidence that dynamics play an impor tant role in reaching the transition state in enzymatic catalysis 12quot13 that dynamic excursions promote hydride tunneling 14quot and that near attack con gura tions NACs overcome most of the barrier to achieve the transition state 1539 This opinion article will provide an intuitive approach to examining conflicting issues concerning the nature of enzymatic transition states tight binding at the transition state rapid release of products and enzymatic dynamics The exploration of well known timescales of chemical events pertaining to transition states in chemistry and in enzymatic catalytic sites will be useful for this exercise Some of these concepts have been discussed 16quot and other reviews and this article expands these analyses with a discussion of recent literature and adds some speculation Current Opinion in Structural Biology 2005 15604 613 wwwsciencedirectcom Enzymatic transition states and transition state analogues Schramm 605 Figure 1 Uncati kchemlcal 1 I Products I39x 4 gt E 1 E L 39 A EIi i Kdiz Kd kchemicai 7 A E Products kenzyme u A AAGli blndil ig I k i Stable El M N 10 15 10 10 v kchemicai Reaction or binding coordinate Currenl Opinion in Strtlclurai Biology Reaction coordinate diagram and energy relationships for uncatalyzed and enzyme catalyzed reactions and transition state analogue binding to an enzyme E Uncat and EA are transition state barriers for the uncatalyzed and enzymatic reactions respectively A EA and El are y substrate reactant transition state analogue and their complexes with the enzyme EI39 is the complex of a transition state analogue that has captured transition state dynam uncatalyzed chemical reaction kenzymek Ic motion into a stable complex as described in the text AAG is the enzymatic efficiency compared to the mammal expressed in Gibbs free energy AAG I is the energetic difference in binding energy for a perfect transition state analogue and substrate The upper right hand scheme is a thermodynamic box for the relative reaction rates of an uncatalyzed reaction k icai or enzyme catalyzed reaction k me Dissoc iation constants for the substrate Kd and the substrate i1 nzy molecule at the transition state configuration Kd predict tight binding of the transition state complex Adapted from 3 Timescales in chemistry and enzymology The conversion of a bond vibrational mode stretching and restoring forces to a bonddissociating translational mode an imaginary frequency de nes the transition state Along the reaction coordinate the transition state forms within the time span of a bond vibration typically 10 13 s for bonds of biochemical interest Figure 2 This timescale contrasts sharply with that of typical enzyme catalytic rates ofl to s For the purpose of compar ison the catalytic rate of purine nucleosidase PNP Z is near the middle of this range The capitalistic scaling of the time constants of the transition state life time and catalysis by PNP is the equivalent of spending a 5 billion fortune one cent at a time Why is catalysis so slow when bond excursions and transition state lifetimes are so fast The answer comes from the limits of diffu sional capture between a substrate and enz me 10 9 M 1 s 1 and from the relatively long times needed for enzymes to rearrange protein structure and position the substrates for productive chemistry Recent and insightful studies have measured multiple conforma tional changes in lactate dehydrogenase following the formation of encounter complexes of enzyme NADH 1739 A microsecond protein conformational adjustment and a millisecond catalytic site closing event are required before the enzyme NADH complex reaches equilibrium Therefore chemistry is delayed by a relatively long time simply because of conformational adjustments that align reactants in the catalytic site Similar rates have been seen for loop opening and closing events in orotate phosphor ibosyltransferase OPRTase and triose phosphate iso merase TIM Figure 2 181939 Even after these adjustments Michaelis complexes are bound weakly and the reactants require additional time to adjust flexible bond angles within the catalytic site These pretransition state steps are usually reversible and substrates often diffuse in and out of the catalytic site many times before catalysis This happens when loop motions that capture release the substrates in the catalytic site are faster than the time to form the transition state Tight binding at the transition state The fleeting lifetime of an enzymatic transition state prevents a true thermodynamic equilibrium from orm ing a tightly bound transition state complex Researchers have recognized this by calling the transition state a quasi wwwsciencedirectcom Current Opinion in Structural Biology 2005 15604 613 606 Catalysis and regulation Figure 2 e l B 74gt Age of the universe 0 O m 0 39 Common km range 8 O 1 H kca PNP r 7 7 7 7 7 7 7 7 7 7 7 7 7 gt O O F 8 E gt Flap opening TIM 7c gt Rate 02 HD to deoxy Hb gt Flap closing OPRTase 2 l e g gt Domain motions in proteins in 7 Rotationtranslation lor NACs V g a Catalytic site capture limit a To 7 H20 diffuses two diameters Collisionless H ion transfer m a To 7 Light travels 03 mm gt Bond vibrationTS lifetime gt 3 a b 2 Current Opinion in Struclural Biology TIme constants of molecular dynamics protein motion and catalysis equilibrium state which we can interpret to mean not in equilibrium7 10quot A thermodynamic description of the transition state complex requires bond equilibrium to propagate through both the enzyme and reactants during the lifetime of the transition state As protein conforma tional changes are slow this is impossible during the transition state lifetime Clearly thermodynamic equili brium cannot occur at the transition state However enzymes invariably close loops flaps or domains over the catalytic site before reaching the transition state Con formational changes are required to productively position reactants and to shield reactive species from solvent The catalytic site remains closed during the lifetime of the transition state therefore we can say with certainty that reactants cannot diffuse from the catalytic site during this period a de nition of tight binding at the transition state This is a hollow argument for tight binding because the lifetime of the transition state is shorter than the time required for a water molecule to diffuse in or out of the catalytic site or even over distances equivalent to its own diameter Figure 2 Below I attempt to reconcile the dynamic nature of the transition state with the extraor dinary binding af nity of transition state analogues We propose that transition state analogues capture a rare con formational event that generates the transition state Protein dynamic motion Computational advances have led to insights into the atomic excursions that result from thermal motion in enzyme reactant complexes 10quot20 Dynamic motion of linked regions of the protein has been proposed to transfer vibrational energy into the catalytic site 12quot212223 Domain and loop motions are responsi ble for the appropriate alignment of groups for catalysis in the Michaelis complex Reaching the transition state from this complex involves a stochastic search of enzyme substrate contact distances to achieve the transition state 16quot Nature of enzymatic transition states Dynamic excursions of enzymebound reactants together with motions of enzyme sidechains loops and domains all change the reactant enzyme distances and dynamic interaction energies on the timescales summarized in Figure 2 Rotation about covalent bonds and ap and domain motions 10 8 to 10 3s form an alignment poised to achieve the transition state Bruice has called this the NAC or the turnstile to the transition state77 2425 In X ray crystal studies of highresolution struc tures of Michaelis complexes it is clear that reactant protein interactions are too weak to account for the formation of the transition state and the enzymeimposed rate accelerations Flap opening and ap closing dynamics 10 3 to 10 s position speci c amino acids with respect to the reactants Superimposed on slow domain move ments are higher frequency bond vibration modes from amino acid sidechains and from the reactants Typical atomic excursions of approximately 05A occur on the subpicosecond timescale Figure 3 Therefore during the time between loop closure and opening typically s there is time for approximately 0 0 ynamic excursions of 05 A magnitude of each atom at the cata lytic site thus interrogating the bound reactants Mole cular dynamics MD or quantum mechanicalmolecular mechanical QMMM calculations are stymied by long timescales because computational dynamics calculations Figure 3 Legend Dynamic motion in PNP a The left panel shows distances from the crystal structure of bovine PNP in complex with immucillin H a transition state analogue and phosphate The panel on the right indicates the catalytic red and anti catalytic blue vibrational modes of the Michaelis complex To generate these panels the structure of the bovm replaced with phosphate inosine was replaced PNP guano e PNP inosln with guanosine and dynamic excursions were calculated by MD simulation for the human sine P04 complex 28quot b The distances between 05 04 and 04 0 e 804 complex was taken from 36 the sulfate was pmsz are shown during a 100 ps MD simulation From the 05 A distance of the atomic excursions we see fluctuating hydrogen bond lengths and electrostatic interactions on the subpicosecond timescale as discussed in the text Adapted from 16 Current Opinion in Structural Biology 2005 15604 613 wwwsciencedirectcom Figure 3 Enzymatic transition states and transition state analogues Sohramm a N243 NM219 27 03220 2 277 OW 393 39 HOY192 7 4S03 229 O116 5 1 x x A H64 N v 30 23 29 NE2 N 833 RB4 N NH86 NH NE2 b I l l 45 ll 1 l l r li 1 1 ll l I l i l l y y l r 1 I I I 4 E 8 c 35 9 0 3 25 9 I I I l l 25 50 75 1 0 Time ps Current Opinion in suumural Biology www50iencedirectcom Current Opi on in Structural Biology 2005 15604 613 608 Catalysis and regulation Figure 4 a PNP inosine 804 PNP ImmH PO4 INSN N243 We N243 1 o 41 O NT 45 W 9 34 sycw gijo 52m 33 o 29quot 7 O vegoquot 2 6 5330 quot H647N N NE R84 N NH WES Km17uM Kd23pM MTAP MTImmA POA o 023 gtD222 MTA P MTA SO4 EI74 E174 114 MM Kd 2 pM Current Opinion in Stvuclural Biology Comparative structures of reactant state Michaelis or substrate analogue complexes and transition state analogue complexes of a bovine P b human MTAP and c Escherichia coli MTAN Reactant complexes L L are on the ri ht quot e l a 39 MIchaelis are shown on the left and transition state analogue complexes I I dllu I I my Major differences in interatomic contact distances are emphasized in color red is tighter in the transition state analogue complex and blue is Current Opinion in Structural Biology 2005 15604 613 wwwsciencedirectcom Enzymatic transition states and transition state analogues Schramm 609 can simulate on the nanosecond timescale but important loop and domain motions are up to a million times slower 20 What does this mean in terms of reaching the transition state Hydrogen bonds are the most common chemistry promoting force between enzyme and reactants held in the NAC con guration A 05 A nuclear excursion between a hydrogen bond acceptor donor pair canucon vertuthe hydrogen bond distance from 31 to 26 A At 31 A the hydrogen bond energy is weak typically lt1 kcalmol However at 26 A the bond is highly ener getic and can be considered a low barrier7 or partial covalent hydrogen bond well suited to electron with drawal or contribution Short hydrogen bonds can con tribute 4 6 kcalmol or more toward catalysis 2627 Electronic rearrangements necessary for catalysis are typically made up of many such interactions all changing distance on the subpicosecond timescale An example of the distance fluctuations between atoms that play elec tronic roles in catalysis is seen in PNP Figure 3 82839 Because each atomic nucleus can in principle move in all directions motions that make a hydrogen bond longer or adopt less favorable angles are anticatalytic blue vectors in Figure 3 and those that shorten bonds and favor electron migration red vectors in Figure 3 are catalytic contributing to the reaction coordinate These excursions provide fluctuations in the instantaneous electronic envir onment at the enzyme reactant interface The transition state is achieved when each of the cata lytically critical bonds moves in a coordinated excursion that addswithdraws electrons simultaneously as needed to reach the transition state Computational and experi mental dynamics approaches to the study of enzyme function indicate that the protein architecture is designed to make catalytic motions more frequently coordinated than random anticatalytic motions 12quot21 In the PNP case when all ve of the catalysispromoting dis tances shorten simultaneously the transition state is reached 16quot Note that it takes a long time millise conds to accomplish this state because the probability of ve or more dynamic excursions coinciding is small The instantaneous geometry that forms the transition state will not generate a conformational change transmitted through the protein or cause clamping down7 global protein conformational compression speci c to the transi tion state nor will it achieve a tightly bound7 transition state except in the dynamic sense of the fraction of a picosecond in which the transition state occurs However as discussed below if we could freeze time at the instant of the excursion that forms the transition state keep the reactants and their instantaneous local contacts frozen at the transition state and let the rest of protein equilibrate we would have a tightly bound transition state complex with transition state binding energy equivalent to enzyme chemical gtlt binding energy of the Michaelis complex e term transition state stabilization7 has fostered the idea that an energyequilibrated transition state exists Realistic enzymatic transition states might more appro priatel be ex ressed in terms of statistical dynamic probabilities rather than thermodynamic expressions Dynamics of hydride transfer and quantum mechanical tunneling In hydride transfer reactions the dynamic motions are relatively simple acceptor and donor are pushed together by the vibrational mode of a promoting amino acid7 near either or both the acceptor or donor 14quot29 In this dynamic mode the distance between the acceptor and donor determines the extent of quantum mechanical tunneling The dynamic principles outlined above also hold for hydride transfer atomic excursions that move the hydride donor acceptor pair apart are anticatalytic and only when acceptor and donor move simultaneously toward each other does hydride transfer occur Computa tional analysis ofhydride transfer reactions has shown that the hydride has the ability to cross the transition state barrier and with nite probability recross or return to reactants before forming products 30 The recrossing phenomenon also occurs in nonenzymatic systems 3139 Enzymes that limit transition state recrossing are said to aid catalysis by increasing the transition coef cient the probability that reactants at or past the transition state will form products rather than returning to reactants 10quot Studies by the Klinman group have investigated this phenomenon as a function of temperature by com paring thermophilic and psychrophilic enzymes and mutagenesis of the promoting groups with convincing results 3233 The coupled protein networks elegantly described by the Benkovic and Wright groups in dihy drofolate reductase are proposed to be responsible for increasing the probability of these speci c transition statepromoting protein vibrational modes 343 Weak binding of reactants Compared to forces imposed at the instantaneous transi tion state reactants are bound weakly The relatively weak binding of substrate to the catalytic site of bovine PNP Michaelis constant of 28 uM is reflected in the long hydrogen bond distances seen in crystal structures of PNP Michaelis complexes Figures 3 and 4 7836 Figure 4 Legend Continued weaker in the transition state analogue complex Bovine PNP shows several shortened hydrogen bonds that form interactions with leaving groups Human MTAP shows sh ortened hydrogen bonds to the ribosyl that are presumed to be interactions that form the ribooxacarbenium ion In E coli MTAN there are some hydrogen bond losses in the MT DADMe lmmA complex and minimal changes to the inhibitor H owever from capturing a dynamic excursion that forms the ribooxacarbenium hydro 39 39iun pair Wllll uie water u come xide ion pair at the transition state MTA MT lmmA M39l39l39 and MT DADMe lmmA are 5 methylthioadenosine methylthio immuncillin A 5 methylthiotubercidin and methylthio DADMe immucillin A respectively 42quot46391 wwwsciencedirectcom Current Opinion in Structural Biology 2005 15604 613 610 Catalysis and regulation However when immucillinH a transition state analo gue is crystallized in the PNP catalytic site six new hydrogen bonds are formed and an ion pair moves closer together If one treats these interactions as static pro blems in thermodynamics and estimates the sum of the hydrogen bond and ion pair energies it can be estimated that up to 30 kcalmol more binding energy is present in the transition state analogue complex than in the Michae lis complex 37 This is far more than the 101 kcalmol experimentally observed for the binding of immucillinH relative to the binding of substrate inosine This ener getic difference is a result of both cooperativity between interaction sites on the inhibitor and protein reorganiza tion to hold the protein statically in the con guration that it acquires only i 39 and i in the actual transition state The transition state is therefore not a thermodynamic entity but a dynamic mode that is improbable on the timescale of bond vibrations taking millions to billions of excursions to nd the saddle point that de nes the transition state spending the 5 billion fortune 1 cent at a time This explains why catalysis is so slow relative to the lifetime of the transition state and to the timescales of bond vibrations and domain motion Transition state analogues If the enzymatic transition state is a dynamic mode then what are transition state analogues and why do they bind so tightly Transition state analogues capture the simul taneous molecular excursions that generate the dynamic transition state This state is linked to the dynamic path of protein conformational change that the enzyme has evolved to favor In the presence of a transition state analogue the protein structure collapses around the ana logue into a static thermodynamically equilibrated potential energy well Release of the analogue requires the s ow and energetically unfavorable expansion of the protein Release rates of transition state analogue inhibi tors are very slow from minutes to days resulting in tight binding Dissociation constants in the picomolar range 10 12 M are not uncommon 3738 39 40 and analo gues in the femtomolar range 10 5 M have been described 41quot Studies of transition state analogue inhibitors with picomolar and femtomolar dissociation constants bound to their cognate enzymes reveal that the difference between substrate binding and transition state analogue binding is subtle and transitionstate promoting conformational changes are frequently within the range of dynamic excursions of enzymatic sidechains at the catalytic site Figure 4 839 4Zquot When presented with a chemically stable mimic of the transition state thermodynamically stable tight binding occurs All of the improbable statistical dynamic motions that lead to the transition state including protonation hydrogen bond formation and ion pair formation now become energetically favorable thermodynamic interac tions This causes the classical conversion of AAG from barrier lowering at the transition state to AAGIi binding of the transition state analogue Figure 1 16quot43 The protein conformation present at the transition state is closely related to that of the Michaelis complex after all adjustments have been made to reach the NAC or EAI of Figure 1 In catalysis the conversion of EAI into the transition state is a dynamic search for coincident inter actions for the fraction of a picosecond that leads to transition state formation But when a transition state analogue is bound the favorable fleeting and dynamic interactions of the transition state are converted into stable interactions and the protein undergoes a major conformational change driven by the conversion of dynamic interactions around the catalytic site into stable i i 4 his gives rise to the stable compressed state of enzyme complexes with transition state analogues Hydrogendeuterium exchange into the protein peptide bonds is decreased in this state relative to any other complex 44 This condensed state of the enzyme is not formed during catalysis and is a stable state only in the presence of the transition state analogue Bond breaking following normal transition state forma tion causes a rapid return to an open enzyme conforma tion facilitating product release Enzymes have evolved to form exquisite vibrationally linked structures that exhibit dynamic excursions leading to the transition state followed by loopdomain opening to complete the cata lytic cycle Thus an electrostatic mimic of the transition state draws the protein into a conformation that is related to but is off the normal reaction path This is illustrated in Fig 1 by the blue potential energy well for the complex with transition state analogues Limits of transition state analogue binding It has been proposed that a 10 11 M dissociation constant is the practical limit of binding energies for ligand pro tein interactions 45 However transition state analogues commonly violate this limit with numerous femtomolar dissociation constants 10 15 M having been described eg 46quot These are easily within the limits of the transition state thermodynamic box Wolfenden Figure 1 and remain valid estimates of the af nity of analogues of the transition state Other analogues made by click chemistry have also reached the femtomolar range and are not necessarily related to the transition states of these enzymes 47 The 10 11M limit of binding energy has led to proposals that any ligand including transition states or transition state analogues that exceeds 10 11 M binding energy must have covalent character between protein and ligand 48 49 Cova lency is neither necessary nor desirable for most enzy matic transition states as it necessarily involves multiple transition states and slows catalysis by requiring multiple statistical dynamic searches as described above In addi tion transition state analogues with picomolar and fem tomolar af nity have been rigorously explored and show no evidence of covalency Three examples of enzyme Current Opinion in Structural Biology 2005 15604 613 www5ciencedirectcom Enzymatic transition states and transition state analogues Schramm 611 transition state analogue complexes and their contacts are shown or NP 5 methylthioadenosine phosphorylase MTAP and 5 methylthioadenosineS adenosylhomo cysteine nucleosidase MTAN Figure 4 Conclusions Enzymes reach their transition states through dynamic stochastic searches Binding equilibrium does not exist at the transition state The transition state complex exists only for the duration of bond breaking Thermodynamic binding forces between enzyme and reactant are instan taneously altered as all groups required for catalysis are synchronized in a dynamic excursion towards the reac tant This dynamic excursion involves local motion at the catalytic site linked to larger scale dynamic modes of the protein that favor transition state formation Transition state analogues form stable complexes by capturing and stabilizing the local dynamic modes at the catalytic site these trigger conformational collapse of the lin ed dynamic modes throughout the protein Transition state analogues capture the dynamic networks ofprotein struc ture that have evolved to produce dynamic transition states and convert the dynamic modes into thermodyna mically stable protein complexes References and recommended reading Papers of particular interest published within the annual period of review have been highlighted as o of special interest so of outstanding interest Eyring H The activated complex in chemical reactions J Chem Phys 1935 3107 115 2 Pauling L Chemical achievement and hope for the future Am Sci 1948 3650 58 3 Eyring H Johnson FH The elastomeric rack in biology Proc Natl Acad Sci USA 1971 682341 2344 4 Koshland DE Jr Correlation of structure and function in enzyme action Science 1963 1421533 1541 5 Wolfenden R Transition state analogues for enzyme catalysis Nature 1969 223704 705 6 Wolfenden R Transition state analog inhibitors and enzyme catalysis Annu Rev Biophys Bioeng 1976 5271 306 7 Lolis E Petsko GA Transition state analogues in protein crystallography probes of the structural source of enzyme catalysis Annu Rev Biochem 1990 59597 630 8 Fedorov A Shi W Ncska G Fedorov E Tyler PC Furneaux RH Hanson JC Gainsford GJ Larese JZ Schramm VL Almo SC 394 and principles of atomic mo39tion in enzymatic catalysis Biochemistry 2001 40853 860 9 Pollak E Talkner P Reaction rate theory where it was where is n It to ay an w e e is it going Chaos 2005 1526116 A trip through the history of rate theory includin the proposals of Arrhenius 1889 Wigner 1932 Eyring 1935 Kramers 1940 Pechukas 1976 Chandler 1978 and Pollock 1986 Current problems are posed iugeu lei 10 Garcia Viloca M Gao J Karplus M Truhlar DC How enzymes on work analysis by modern rate theory and computer simulations Science 2004 303186 195 quot 39 mpu llllUldllUll isexplored in the context of enzymatic catalytic power Using examples it is proposed ALA u k g meury lowering the free energy of activation and increasing the tran m39ss39on coefficient The quasi thermodynamic free energy of activation is used to indicate that equilibrium is not established at the transition s a e 11 Wolfenden R Thermodynamic and extrathermodynamic u requirements of enzyme catalysis Biophys Chem 2003 105559 572 A review of the proposal that the transition state binding energy of enzymatic catalysis is kcajKm rate for the uncatalyzed reaction in water Of specific interest here is a thorough discussion of t e remote binding interactions that contribute to catalysis and transition state stabilization f 39 39 39 39 39 12 Benkovic SJ Hammes Schiffer S A perspective on enzyme u catalysis Science 2003 301 1 1 96 1202 Dihydrofolate reductase is exemplified as an enzymatic scaffold for coupled d namic networks of conserved residues that foster the forma tion of the transition state Protein dynamic motion is s own to be central to catalysis and the need for additional experimental and theoretical ed studies is emphasiz 13 Wong KF Selzer T Benkovic SJ Hammes Schiffer S Impact of 1 to hydride transfer in dihydrofolate reductase Proc Natl Acad Sci USA 2005 1026807 6812 14 Liang ZX Klinman JP Structural bases of hydrogen tunneling in u enzymes progress and puzzles Curr Opin Struct Biol 2004 5 H drogen tunneling is highlighted by three enzyme systems including soybean lipoxygenase 1 thermophilic alcohol dehydrogenase and dihy drofolate reductase icular atten ion been devote o the issues of whether protein dynamics modulate hydrogen tunneling probability and whet er the tunneling process contributes to the catalytic power 0 15 Zhang X Zhang X Bruice TC A defini 39ve mechanism for o chorismate mutase Biochemistry 2005 4410443 10448 Computational chemistry is used to conclude that the formation of the N e enz me is major contributor to catalysis omputed transition state structures are similar in vacuum in water or in the enzyme 16 Schramm VL Enzymatic transition states thermodynamics mics and analogue design Arch Biochem Biophys 2005 43313 26 The short lifetime of the transition state is emphasized to demonstrate that the dynamic nature of enzymatic transition states cannot be a e quater described in thermodynamic terms including transition state binding energy I is proposed that transition states are reached stochastic dynamic excursions of protein groups that participate in transition state formation Transition state analogues capture dynamic energy and convert it into thermodynamic energy 17 McClendon S Zhadin N Callender R The approach to the o ichae is complex in lacta e dehydrogenase the substrate binding pathway Biophys J 2005 892024 Binding of NADH to lactate dehydro enase occurs via an encounter complex followed by two unimolecular conformational adjustments on the microsecond to millisecond timescale inclu ing the closure of the catalytic site loop amic analysis suggests that transient melting of 10 15 of the protein is involved in this structural alteration to align NADH in the catalytic site 18 Wang GP Cahill SM Liu X Girvin ME Grubmeyer C Motional dynamics of the catalytic loop in 0MP synthase Biochemistry 1999 38284 295 Desamero R Rozovsky S Zhadin N McDermott A Callender R A r 612 Catalysis and regulation 22 Wolf Watz M Thai V Henzler Wildman K Hadjipavlou G Eisenmesser ern ink ynamics and catalysis in a thermophilic mesophpilic enzyme pair Nat Struct Mol Biol 2004 11 945 949 23 Antoniou D Abolfath MR SchwartzSD Transition path sampling 0 of class39cal rate promoting vibrations J Chem Phys 2004 121 6442 6447 Attempts were made to explain enzymatic proton transfer reactions that a evaluated and dynamic atomic motion was found to contain slow and fast oscillations optimal for catalysis but outside current concepts of transition state theory 24 BruiceTC 39 39 39 39 39 catalysis Acc Chem Res 2002 35139 148 25 Luo J Bruice TC Anticorrelated motions as a driving force in enzyme catalysis the dehy rogenase reaction Proc NatAcad Sci USA 2004 10113152 13156 26 Gerlt JA Kreevoy MM Cleland W Frey PA Understanding enzymic catalysis t e importance of short strong hydrogen bonds Chem Biol 1997 4259 267 27 Neidhart D Wei Y Cassidy C Lin J Cleland WW Frey PA Correlation of low barrier hydrogen bonding and oxyanion bin ing in transition state analogue complexes of chymotrypsin Biochemistry 2001 402439 2447 28 Nu ez S Antoniou D Schramm VL Schwartz SD Promoting o vibrations in hum n purine nucleoside phos horylase ular dynamics and hybri quantum mechanical molecular mechanical study J Am Chem Soc 2004 1261572015729 Classical and mixed quantumclassical MD simulations were performed to assess the existence of protein facilitated promoting vibrations The hybrid QMMM methods were used to cons39 er whether an enzymatic vibration that pushes oxygens of PNP together is coupled to the reaction r inate e calculations revealed the existence of promoting vibra tions coupled to the reaction coordinate 29 Soudackov A Hatcher E Hammes Schiffer S Quantum and mical effects of proton donor acceptor vibrational motion in nonadiabatic proton coupled electron transfer reactions J Chem Phys 2005 12214505 30 Hammes Schiffer S Quantum classical simulation methodsfor rogen transfer in en es a case study of dihydrofolate reductase Curr Opin Struct Biol 2004 14192 201 31 Pritchard HO Recrossings and transition state theory 0 J Phys Chem 2005 1091400 1404 Computational chemistry using simple chemical isomerization demon strates transition state recrossings Some reacting molecules cross the transition state ba ance oint but return to reactant before going to product making the transmission coefficient near 05 32 Liang ZX Tsigos Lee T Bouriotis V Resing KA Ahn NG Klinman JP39 Evidence for increased local flexi n psyc rop c alcohol de ydrogenase relative to I s thermophilic homologue Biochemistry 2004 4314676 14683 Liang ZX Lee T Resing KA Ahn NG Klinman JP Thermal activated protein mobility and its correlation with catalysis in thermophilic alcohol dehydrogenase Proc Natl Acad Sci USA 2004 101 9556 9561 w w w 4 Venkitakrishnan RP Zaborowski E McEIheny D Benkovic SJ yson HJ ig 39 L 39 L quot quot loops of dihydrofolate reductase during the catalytic cycle Biochemistry 2004 4316046 16055 w 01 Sikorski RS Wang L Markham KA Rajagopalan PT Benkovic SJ Kohen A dihydrofolate reductase catalysis J Am Chem Soc 2004 1264778 4779 Mao C Cook WJ Zhou M Federov AA Almo SC Ealick SE Calf spleen purine nucleoside phosphorylase complexed with substrates d substrate analogues Biochemistry 1998 377135 7146 w 7 w i Kicska GA Tyler PC Evans GB Fumeaux RH Shi W Fedorov A Atomic dissection of the hydrogen bond network for transition state analogue binding to purine nucleoside phosphorylase Biochemistry 2002 41 14489 14498 to 9 Lewandowicz A Taylor Ringia EA Ting LM Kim K Tyler PC Evans GB Zubkova OV Mee S Painter GF Lenz et al Energetic mapping of transition state analogue interactions with hum n and Plasma ium fa ciparum purine nuc 39 hosphorylases J Biol Chem 2005 28030320 30328 The binding of different transition state analogues derived from specific atomic alterations of immucillin measured at the catalytic sites of human and malarial PNP This study demonstrates remarkable differ ences in the binding of transition state analogues among P isozymes Energetic mappin PNP immucillin H binding energy gives 30 kcal mol whereas the catalytic rate enhancement of PNP is 18 kcalmol Binding interactions are strongly cooperative 9 Singh V Shi W Evans GB Tyler PC Furneaux RH Almo SC Schramm VL Picomo rtr n 39tion state analogue inhibitors of uman 5 methylthioadenoslne hos o ase an structure with MT immucillin A Biochemistry 2004 439 18 39 racter of N ribosyltransferase transition states transition state analogues were designed synthesized as describe in and co crystallized with human i erences in catalytic site contacts between substrate bound and transition state complexes are modest Evans GB Furneaux RH Lenz DH Painter GF Schramm VL Singh V T ler PC Second generation transition state analogue inhibitors of human met ylt ioa enosine phosphorylase J Med Chem 2005 484679 4689 41 Singh V Evans GB Lenz DH Mason JM Clinch K Mee S u Painter GF Tyler PC Furneaux RH Lee JE et al Femtomolar transition state analogue inhibitors of 5 t ioadenosineS adenosylhomocysteine rom Escherichia coli J Biol Chem 2005 w 40 nucleosidase 5 The transition state structure described in 46quot was used to desi n transition state analogue inhibitors This study reports ten transition state analogue inhibitors with equilibrium dissociation constants between 47 and 900 fM 10 42 Lee JE Singh V Evans GB Tyler PC Furneaux RH Cornell KA u Riscoe MK Schramm VL Howell PL Structural rationale for the affinity of pico and emtomolar transition s ate analogues of Escherichia coli 5 methylthioadenosineS adenosylhomocysteine nucleosidase J Biol Chem 2005 28018274 18282 Crystal structures of MTAN complexed with transition state analogues were determine to 22A resolution and comp 39 h other MTAN inhibitor complexes These e amon the tightest binding enzyme ligand complexes ever described anal sis of t eir mode of binding provides extraordinary insight into the structural basis of their affinity The inhi itor complex revealst e presence 0 a new ion pair between the 4 iminoribitol atom and the nucleophilic water AT3 that captur Piracy and Music The Recording Industry Association of America According to its Web site wwwriaacom The Recording Industry Association of America RIIA is the trade group that represents the US recording industry The RIAA works to protect intellectual property rights worldwide and the F irstAmendment rights of artists conduct consumer industry and technical research and monitor and reviewistate and federal laws regulations and policies The following article is from the RIAA s site A Piracy generally refers to the illegal duplication and distribution of sound recordings There are four speci c categories of music piracy Pirate recordings are the unauthorized duplication of only the sound of legitimate recordings as opposed to all the packaging ie the original art label title sequencing combination of titles etc This includes mixed tapes and compilation CDs featuring one or more artists Counterfeit recordings are unauthorized recordings of the prerecorded sound as well as the unauthorized duplication of original artwork label trademark and packaging Bootleg recordings or underground recordings are the unauthorized recordings of live concerts or musical broadcasts on radio or television Online piracy is the unauthorized uploading of a copyrighted sound recording and making it available to the public or downloading a sound recording from an Internet site even if the recording isn t resold Online piracy may now also include certain uses of streaming technologies from the Internet Many do not understand the signi cant negative impact of piracy on the music industry Though it would appear that record companies are still making their money and that artists are still getting rich these impressions are mere fallacies Each sale by a pirate represents a lost legitimate sale thereby depriving not only the record company of profits but also the artist producer songwriter publisher retailer and the list goes on The consumer is the ultimate victim as pirated product is generally poorly manufactured and does not include the superior sound quality art work and insert information included in legitimate product Each year the industry loses about 42 billion to piracy worldwideiwe estimate we lose millions of dollars a day to all forms of piracy Review of Linear Algebra amp Linear Systems I 1050 P Hespanha 1 Linear Algebra The material reviewed here can be found in 2 Chapter 1 or 1 Chapter 22 l 72 7 2 l 4 l 1 What does it mean for aA to be nonsingular Consider the matrix A 19 Is A nonsingular L How would you compute the inverse of A What is the image rank of a matrix V He What is the kernel nullity of a matrix Compute bases for the image and kernel of A Q How would you do this in MATLAB 2 LTI Systems Most of the material reviewed here is summarized in 2 Chapter 2 A more detailed treatment can be found in 3 Lectures 177 or 1 Chapter 2 Consider the following system 36Axbu7 ycxu7 l where 72 0 0 l 01 107 A 0 0 0 7 1 0 i 0 0 71 71 1 What is the transfer function of this system Q How does the transfer function relate system inputs and outputs L How can one nd the solution to the system for nonzero initial conditions xlo x0 4 Compute the state transition matrix for this system How would you compute it for more complicated systems Lquot What is the de nition of e and how would you compute it in MATLAB 3 Stability Most of the material reviewed here is summarized in 2 Chapter 2 A more detailed treatment can be found in 3 Lectures 8710 or 1 Chapter 6 Consider again the system 1 1 Is the system BIBO stable BIBO unstable asymptotically stable marginally stable unstable Give examples of all cases 2 Give examples of a Jordan blockJ with the following properties Choose a 2 X 2 block whenever possible a The system it Jx is asymptotically stable b The system it Jx is unstable c The system it Jx is stable but not asymptotically stable 3 Classify the following systems with respect to BIBO and Lyapunov stability parallel cascade 4 Controllability and Observability Most of the material reviewed here is summarized in 2 Chapter 2 and Table l A more detailed treatment can be found in 3 Lectures 11717 or 1 Chapter 3 1 Consider the following system 0 0 72 3 3 x 0 71 73 x 0 72 u7 yl 0 2x7 0 0 2 0 0 Classify it in terms of controllability observability stabilizability and detectability 2 Consider the following singleoutput system in Jordan form 12 7 J1 0 7 21 1 7 x7 0 J2 x7 J170 ll J2 7 8 ycl 02 C3 C4 C5x 1 0 712 1 0 M with 21 y lg When is this system observable detectable Will your answer change if 21 12 References l P J Antsaklis and A N Michel Linear Systems McGrawHill Series in Electrical and Computer Engineering McGrawHill New York 1997 2 G E Dullerud and F Paganini A Course in Robust Control Theory Number 36 in Texts in Applied Mathematics Springer New York 1999 3 J P Hespanha Topics in Undergraduate Control Systems Design Univ California at Santa Barbara Apr 2006 Available at http www ece ucsb eduhespanhapublished troduc n Revealing the world of RNA interference Craig c Mottoquot2 amp Darryl Conte Jr2 01505 USA emcol 57mg mella ummxmed edu The recent discoveries of RNAinterferelice and related RNA silencing pathways have revolutionized our understanding of gene regulation RNA us experime EH08 has been used as a research tool to GOHUOI the mat organism and has potential as a therapeutic strategy to reduce the expression of problem genes At the heart of RNA interference lies a remarkable RNA processing mechanism tha is now known to llnderlie many distinct biological phenomena he term RNA world was rst coined to describe r p re on Earth This original RNA world if it ever existed on Earth is long gone But this Insight deals with a within our cells 7 RNA silencing or RNA interference RNAi quot 39 39 39 DNA many organisms mount highly speci c counter attacks to silence the invading nucleiceacid sequences r c L translation in mammalian cells Second dsRNA isenere 39 39 39 39 pecinc A t 39 39 L So a model in which dsRNA ofcellular mechanisms for unwinding the dsRNA and pro moting the search for complementary baserpairing party A q Hypotheses that requireaparadigm shiftand depend on the rarelyappealing o w y was dsRNA proposed as a trigger for RNAi and directed immunity mechanisms is doubleestranded RNA dsRNA Interestingly dsRNA does more than help to defend cells against foreign nucleic acids 7 it also guides 4 4 H n ulati n and an n comml r c n r m chromatin In some organisms RNAi signals are trans mitted horizontally between cells and in certain s 39 L 39 n39 39 39 Tn199 Guoand Kemphues5 attempted to use RNA complementary to the C elegmls pine mRNA to block pare expression This technique is known as antisensermediated silencing n pi in nun ar I li into the cytoplasm ofa cell Base pairing between the sense mRNA sequence and the complementary antisense intere ne t ing or f W 39 39 J 39 39 39 439 t A A t t39 H To their sure I I 1 I A A t 39I t OT 0 nvi I j a r ran 39 39 and In M DKY 39 39 39 l r 39 39 abriefoverview ofthisrapidly growing field the mRNA to cause interference raising the question ofhow this RNA could induce silencing Was an active silencing Discovering the trigger Crucial to understanding a genersilencing mechanism such responsebeing triggered against the foreign RNA regardless of its polarity Or was the silencing apparently induced by W the theoretical rem arlmhle 39r n hl tool see review in this issue by Hannon and Rossi page 371 The observation by Fire emf that dsRNA is a potent trigger for RNAi in the nematode l W 39 7 Fig i 39 39 39 39 C elegans and other organisms and accelerated the dis covery ofa unifying mechanism that underlies a host of cellular and developmental pathways However there were substantial barriers to the acceptance of the idea that A W W I t t t th type ofin vitmtranr r n L n A r ued to beused to silence genes in c clegcns More surprises were in store While using this antisense L L quot be transmitted in the germ 1 u k A through the sperm or the eg for up to several generations Equally remarkable the silencing effect could also spread from tissue to tissue within the injected animala Taken together the apparentlack ofstrand specificity the remark First at the time dsRNA was thought to be a nonspecific silencing agent that triggers a general destruction ofmesr senger RNAs and the complete suppression of protein 338 inheritance properties of the silencing phenomenon prompted the creation ofa new term RNAi Importantly the properties of RNAi demanded the existence ofcellular NATURE VOLASI t 16 SEPTEMBER moo twwwnature comnature 2004 NaturePublishing Group ahhhals that are defective tor RNAl h Note that silencing occurs throughoutthe body w the e orthe alllm xceptlon orarew cellsln the tarlthat express some residual GFP The signal is lostlh intestinal cells hearthetarl arrowhead as well as hearthe harilarrlll W l t D the systemic spread and rhhehtahee otsrleherhg antH in responseto foreignRNAa Alth ough our initial models saw dsRNA as an intermediate in the ampli cation of the silencing signal Fire3 suggested that dsRNA m m introduction Tenammo These and other breakthroughs united previously disparate fields by identifying a common core mechanism that involves the processing ofdsRNA into small RNArsilencing guides Fig 2 lnshort dsRNA had taken thebiological world by storm other silencing ii39iggers trigger or as an intermediate in all the RNAirrelated silencing pathr tri r il n in 1 v amnl quot 39 39 V tran 7 itselfinto the genome in such a way that a nearby promoter or an inverted copy of the transgene itself leads to the production of sRNA which could in turn enter directly into the RNAi pathway genesexpressingeither strandalone But several quot f 394 that transgenes can trigger A key gene family involved in silencing pathways in plants quot fungi and C elegmlsu contains genes that encode putative aellularRNAr a rm A m bers of this family of proteins were identified in forward genetic it IF mm m d l 39 39 screens 7 whereby mutant genes are isolated from an organism th me n f showing abnormal phenotypic characteristics 7 as facto s l t l I I 39 39 required forcorsuppressionin plants and quelling in Neurospom l quot p 39 results from p 39 t39 I silencing of introduction into the animal this dsRNA could be recognized as L 39 quot foreign thereby activating cellular amplification and inheritance cellular gene Interestingly although cellular RdRP genes were mechani m 39 39 39 is vii is required for transgenermediated corsuppression in plantsmquot they 39 39 1 39 for virusinduced silencing ofa transgene pier need for transgenerdriven expression this theory was easily tested dsRNA roved to be an extremely potent activator ofRNAi 7 at least 107fold and perhaps 1007fold more effective than purified preparationsofsinglerstrandedRNAa sumably because the virus provides its own viral RNA polymerase ninth vm he i 4 A A pc ax original stimulus for corsuppression and quelling In this type of Taking ihe biological world by storm silencing the RdRP somehow recognizes transgene products s quot 39 139 39itb abnormalor aberrant ndsubsequentlyconvertsthisinitialsilencing r m o u u c o mals 39 t arm wilnthi a th ADM 39 39 139 39 were soaked in dsRNA or given food containing bacterially the silencing pathway rather than the trigger The Rdluxderived a a on lzla m m m A i a quot 39 quot rid nti al 39 139 Comparison of the C elegmls genes required for RNAi to genes required for gene silencing in Dmsophila m plants 7 and fungi a p 7 transcriptional gene silencing PTGS 9 corsuppressionm quelling and RNAi share a common underlying mechanism that retlects an ancient origin in a common ancestor offungi plants and animals T39 39 39 39 f 39 39 dsRNA i nmmme i organisms including organisms that were otherwise unsuited to genetic analysism Small RNAswere shown to beproduced in plants W 26723 page356 The dsRNArprocessing enzyme Dicerzgwas found to prom mr i iiii i 2 70 shown to in uce sequencerspeci c gene silencing in human without initiating the nonspeci c gene silencing pathways A class i is response to a dsRNA trigger But how might the transgene mRNA be recognized as foreign 39quot39 39 39 39 H n 39 aberrant tranr script model has perhaps undeservedly received little attention of late One possibility discussed by Baulcombe review in this issue page 356 is that high levels ofexpression ofthe transgene mRNA 3 IR It mati l L 39 39 be marked for silencing by the cell When in ally delivered to cells associated proteins During the rapid assembly of naked DNA into chromatin the host cell may in selfdefence somehow mark the transgene chromatin so that RdRP is recruited RdRP acting on quent silencing Consistent with this possibility fission yeast RdRP was found to physically associate with silent heterochromatin I L i r 39 39 39 39 no m HARD 39 39 r by Dice 5 and to function together with RDErl homologues l 1w 1 i i u Final m 1 v ntl A u a a quot 4 39 39 39 yeast 4 t 394 I exist in C elegmls Both RDErl and the dsRNArbinding protein 39 39 am A f NATURE lVOLASl l 16 SEPTEMBERZOOA l w nature comnature 339 2004 NaturePublishing Group introduction Multigene lamtly or Favelgn DNA lnnkedt whenth trnnsgene Foreign dsRNA Developmclllal or experimental transcnpmn n r ns 9 assembly SNHNA prermlRNA numnmmnm modificallon CH 39 hem TI39ITI39I39I39I39I ITI ITF l sense RNA 1 l y t lntenneniaie 5amp5 I Ulcer Ulcer 11111131111111 1 Ulcer Trrrn 39rrmr 39rrrnT HNA LL quotWM LLJJJL mm M 5 Silencer I JU J JJJLLL Trrrn 39rm39rr Tmn ullll w J JLLL 39 I f quot2 TGA Targel quot5 AAAA NSC RISC 915 inlsc Helemcmamalm larmmlon and mRNA deshucllon Dr nunscnpnonnl suenelng translational repression f r 2mm Transcription can l Ml h m on an n ll H 77 VSUBHOSWB h t tallTGAttanslatlon termination codon injecting feedingorexpressingdsRNA HoweverRDErlandRDEr4 Subsequently dsRNA targeting a promoter was shown to trigger arenotrequiredfortmnsposonsilencingorforcorsuppressio 5 dDM and initiate transcriptional silencing The silencing was Furthermore RDErl and RDEV4 are not required for the inheritance accompanied by the production ofsiRNAss pointing to an RNAir fw 39 39 39 39 39 39 39 39 like mechanism for the initiation oftranscriptional gene silencing during the initial exposure to dsRNA W P A39 39 A39 t that 39 39 39 39 quot 39 1 39 39 1 39 39 39 n the formation of silent heterochromatin can be guided by small 139 1 l 39 39 RNAs54 39 39 39 IansophiZc theRNAr a chromatin sigmture 39 39 1 39 f b H 39 39 39 39 mmar scripts and the formation of a novel species of dsRNA perhaps tion and for silencing multicopy transgenes and pericentric DNA undemooLo39 4 c edsRNAmt39w39t 39I 39 h quotM 39 39 39 meansofRDErl andRDEJ in 39 39 39 39 39 391 1 39 39 39 39 39 39 39 39 39 39 39 2 Per haps a similar RdRPrderived dsRNA functions in the RDErlr and quot 39 39 39 39 1 39 39 m A m ThepossibilityoffeedbackbetweenRNAiitspotential chromatin associated trigger and chromatinrmediate silencing maintenance anisms raises further questions about the ultimate causes o il n in 39 39 39 39 by means of an RDErlRDEArdependent dsRNA signal resulting generation to the next Ten years ago dc nova cytosine methylatio n ofgenomic DNA was were homologous to the methylated genomic sequences This a Ann 340 NATURElvoLAai 16 SEPTEMBER mm wwwnatureoomnature 2004 NaturePublishing Group from sense and ant39sense readthrough transcriptionfrom insertion points in the genom 57 Perha s ime this initial dsRN e triggered silencing signal was replaced and augmented by a chromatineassociated silencingemaintenance s1gnal Outlook for the RNA wm ld The numerous branching and converging silencing pathways that 39 139 39 39quot no doubt 39 introduction s RN 1m 1f 1 n on 12371320999 V hmvlt 7497750 1999 1 n 1 151 1 111m transposahle elements 1nthe D melamgnsmgermlrne 51111 Ewl 111017710272001 FaardMRHtt M1VVR 111111 1 n1 11mm seemto ex1s research to unravel It is already clear that different organisms have evolved distinct mechanisms or at least variations on a common is mode of regulation has been found in animals The diversity of RNA silencing phenomena suggests that other interesting findings await discovery For e amp e ofanm nrr39 39 39 39 V R ll1n1 11 e1 r111ll1n 1nf11n1 anRN ear 11 w A USA9111 11 1 V t l n 11 lm M 1m n1 N AM 2452000 1 11 1 1h 1 17 EMBO 11259572502 1992 16 N 11 l1 V m1 11 V v n n R 5359 D 7 72391990 transformatronwrth homologous sequencesMal Mmahal 533434353 1992 V ll V R 1th R W FA PM 101 40251993 h11rV1 11 V VTqu F F1 H AND C39 elegam Wmmamabmca Pm NatlALzd 5m 052195145377145920993 24Eosher1 M 1 h 11 M M 1111 raises the question of whether natural small RNAs are transmitted in germ cells or other developmental cell lineages in other animals including humans Extrachromosomal inheritance of silencing J B10121E317E362000 lt R 1l m m1 n 1 1 11 11 m 1H 1H 11 nt 7115 079520999 ummna MRvnthR hr Hnnn 1 patterns by 1 icated layers of gene regulation at both postetranscriptional and chromatin modifying levels These small RNAs maybe important 1n stemecell maintenance and development and differential localization of such RNAs may have a role in the generation of cellular diversity It will be interesting to discover if the phenomenon of lateral transport of RNA from cell to cell so far observed inplantsSquot62 and C elegans is more widespread As well as having a role in immunity could enetic RNA mor hogens allow cells to modulate the activity developmentally important genes or mRNAs in neighbouring cells This type of regulation might be particularly useful when cells such as neurons communicate at junctions that are far from the cell nucleus The past ten years have seen an explosion in the number of none coding RNAs found to orchestrate remarkably diverse functionssm These functions include sequenceespeci c modification of cellular RNAs guided by small nucleolar RNASSS 39 d t39 wide domains of chromatin condensation by the mammalian none coding RNA Xist Xeinactive specific transcript autosomal gene imprinting and silencing by noncoding mammalian Air antisense 1 transcrrptronal gene srlencrng 1n Emsaphmcells Nature4042937296 2000 m1 Dn39m hl39Vquot hvnD thlnDDH1H AA dependent cleavage omeNA at21 to 23 nucleotrde 1nter1als Cell1o125733 2000 D 111 V1 R11 1 Alarm A deg111 RNA7 139771402 2001 R m t 1n F 11d H mm nri n 1 1n1t1atronstep ofRNA 1nterference N122111e409 3537355 2001 0 1111 1111 M n 1 c 1 mammalran cells 111111112411 49H9s 2001 V R mh 1 RV 391 1 21231115 111111111211039017905 2000 V R R ml 1 11m R V mlquot 39V b wlth antrsense complementarrty to 1111714 Cell75 3437354 1993 111111 n 1 the lethmalltemporalRNA 551211522918347838 2001 AV quotm 12 1 developmental trmrng 1n 1 21231115 Genes Dev 152551y2559 2001 11 V1 k 1n 1 35 RNAl 52121122 297 133371337 2002 M h1r11k1k Rm N R1111 39Vquot m k M small RNAs 1n genome rearrangement 1n Tetrahymeml 41111104397699 2002 3BWaterhouse P M Graham M w ampWangM BV11us resrstanoe and gene srlencrng 1n plants canhe 1 M A USAQS 1PM lngrRNA and a 139 J repre ion 01 target J L 39nv 1 J 39nxv 1 J39 J 1 1 13959713964 199B 1 1 1 1 1 lm TV Vm1ltn 12111111 n 11 11111m1V n 1m at a luuto anRNA revolution BIut considering the potential role of RNA as a primordial biopolymer of life it is perhaps more apt to call it an RNA revelation r transgenehutnothyavrrus 5211101543453 2000 M7 M 1111 1n D srlencrng and naturalvrrus resrstanoe Cell 101 5337542 2000 1s not talllt1ng over the celli1thas been We Just M m M m dldnt real H mm HOW dependent RNA polymerase 111111112399 1557159 1999 11 n 39V In A 7475 dol10103BnatureOZB7Z 2001 1 1 111nm 12m 121 h39T R tk1nVFV49777 4 m39dquot quot4quot i 1 Cold SprmgHarborLaboratoryPressNewYorh 1999 development and RNAmterference 1n C deg111 51111 Ewl 10159717s2000 2 GllbextWTheRNAwoxldletu123196181986 441561116 3611 4 R M quotquotk 1 V nquot 39 n vquot V W i i 39 339 11 F1 m1 m 1 1 11 12 7511 11 7113571993V deg111 N111u1239180678111998 451Makeye aE R W M n H quot 39 W 4 Proud c G PKRanewname and newroles nendslawchem 5212024172451995 M quot 4 44272002 5 1111 m 11 12 1a 1 111 1 11 1 1 11m 45nougherty v 11 39T 1 Emcth Sac T111115 2150975130997 Cuquot Opm 521131017139974050995 6 11 V mnhu K V J L A R 11l mh 139 111 11520 1995 plants leMal 310132797880996 n 1111 n t1n 1 71 12 1x 4 N 391 quotV F NY 4121141231 41974230993 20572170937 8 lzantIGamp 1nt1111111 1m ATh111 1111 1m1V V M11 1 1 1 111 a molecularapproach 10 genencanalym521135100710151gg41 RDELDCRJ andaDExHrboxhellcase to d1rectRNAr 1n 1 21231111 521110986178712002 1D mhm R l V1 MD 1lln1 39T e A ta 911111 T111 14 M11 Sclence 237249Ay2497 2000 0 12 h 1 11 1 embryos 5211907077715 1997 11nbara1l 11 h k M 11 28243074310998 T1mm n 1 111 1111 r11 111 and potent genetrc 1nterference 1n 022110 111121711111 212311113 52112253 1037112 2001 NATURE lVOL 431 1 15 SEPTEMBER 2004 1 wwwmturecommture V 1 M corsupptesslonm the c elegamgermlme 521151221 14157371533 2000 1V111n RR Dl tkaVV 1 21231115 111111111211042957293 2000 n vMV V1m R1erT VHYDHJ 4 sequences 1n plants Cell 75557575 1994 53MetteM EvanderWrnden1MatzkeMA13eMatzkeAProductronofaherrant promoter EMBO 13241243 1999 2004 Nature Publishing Group PHYSICAL REVIEW B 79 205112 2009 3 Spin Bosemetal phase in a spin model with ring exchange on a twoleg triangular strip D N Sheng1 Olexei I Motrunich2 and Matthew P A Fisher3 1Department 0f Physics and Astranamy Califarnia State University Narthridge Califarnia 91330 USA 2Department 0f Physics Califarnia Institute 0f Technalagy Pasadena Califarnia 91125 USA 3Micr0s0ft Research Stati0n Q University 0f Califarnia Santa Barbara Califarnia 93106 USA Received 4 March 2009 published 20 May 2009 Recent experiments on triangular lattice organic Mott insulators have found evidence for a twodimensional 2D spin liquid in close proximity to the metalinsulator transition A Gutzwiller wave function study of the triangular lattice Heisenberg model with a fourspin ring exchange term appropriate in this regime has found that the projected spinon Fermi sea state has a low variational energy This wave function together with a slave particlegauge theory analysis suggests that this putative spin liquid possesses spin correlations that are singular along surfaces in momentum space ie Bose surfaces Signatures of this state which we will refer to as a spin Bose metal SBM are expected to manifest in quasionedimensional quasi1D ladder sys tems the discrete transverse momenta cut through the 2D Bose surface leading to a distinct pattern of 1D gapless modes Here we search for a quasi1D descendant of the triangular lattice SBM state by exploring the Heisenberg plus ring model on a twoleg triangular strip zigzag chain Using density matrix renormalization group DMRG supplemented by variational wave functions and a bosonization analysis we map out the full phase diagram In the absence of ring exchange the model is equivalent to the 1112 Heisenberg chain and we nd the expected Bethechain and dimerized phases Remarkably moderate ring exchange reveals a new gapless phase over a large swath of the phase diagram Spin and dimer correlations possess singular wave vectors at particular Bose points renmants of the 2D Bose surface and allow us to identify this phase as the hoped for quasi1D descendant of the triangular lattice SBM state We use bosonization to derive a lowenergy effective theory for the zigzag spin Bose metal and nd three gapless modes and one Luttinger parameter controlling all power law correlations Potential instabilities out of the zigzag SBM give rise to other interest ing phases such as a period3 valence bond solid or a period4 chirality order which we discover in the DMRG Another interesting instability is into a spin Bosemetal phase with partial ferromagnetism spin polarization of one spinon band which we also nd numerically using the DMRG DOI 101103PhysRevB79205112 I INTRODUCTION A promising regime to search for elusive twodimensional 2D spin liquids is in the proximity of the Mott metal insulator transition In such yveak Mott insulators signi cant local charge uctuations induce multispin ring exchange processes which tend to suppress magnetic or other types of ordering Indeed recent experiments1gt2 on the triangular lat tice based organic Mott insulator KET2C112CN3 reveal no indication of magnetic order or other m e breakin down to temperature several orders of magnitude smaller an the characteristic exchange interaction energy J W250 K Under pressure the KET2C112CN3 undergoes a weak rstorder transition into a metallic state while at am bient pressure it has a small charge gap of 200 K as ex pected in a weak Mott insulator Thermodynamic transport and spectroscopic experiments 4 all point to the presence of a plethora of lowenergy excitations in the KET2C112CN3 indicative of a gapless spin liquid phase Several authors have proposed7 that this putative spin liq uid can be described in terms of a Gutzwillerprojected Fermi sea of spinons Quantum chemistry calculations suggest that a oneband triangular lattice Hubbard model at half lling is an appro priate theoretical starting point to describe K ET2Cu2CN318 Variational studies of the triangular lat tice Hubbard model9 nd indications of a nonmagnetic spin 1098012l2009792020511229 2051121 PACS numbers 71 10Hf 7510Jm 7110Pm liquid phase just on the insulating side of the Mott transition Moreover exact diagonalization ED studies of the triangu lar lattice Heisenberg model show that the presence of a foursite ring exchange term appropriate near the Mott tran sition can readily destroy the 120 antiferromagnetic order10 One of us5 performed variational wave function studies on this spin model and found that the Gutzwillerprojected Fermi sea state6 has the lowest energy for suf ciently strong foursite ring exchange interactions appropriate for the K ET2C112CN3 Despite these encouraging him the theoretical evidence for a spin liquid phase in the triangular lattice Hubbard model or Heisenberg spin model with ring exchanges is at best suggestive Variational studies are biased by the choice of wave functions and can be notoriously misleading Exact diagonalization studies are restricted to very small sizes a act which is especially problematic for gapless spin liquids Quantum Monte Carlo fails due to the sign problem The density matrix renormalization group DMRG can reach the ground state of large onedimensional 1D systems but cap turing the highly entangled and nonlocal character of a 2D gapless spin liquid state is a formidab hallenge Thus with new candidate spin liquid materials and increasingly re ned experiments available the gap between theory and experiment becomes ever more dire Effective eld theory approaches such as slave particle gauge theories or vortex dualities while unable to solve any 2009 The American Physical Society SHENG MOTRUNICH AND FISHER x x1 x2 x3 FIG 1 Top Heisenberg plus ring exchange model on a twoleg triangular strip Bottom convenient representation of the model as a 1112 chain with additional foursite terms the Hamiltonian is writ ten out in Eq 1 particular Hamiltonian do indicate the possibility of stable gapless 2D spin liquid phases Such gapless 2D spin liquids generically exhibit spin correlations that decay as a power law in space perhaps with anomalous exponents and which can oscillate at particular wave vectors The location of these dominant singularities in momentum space provides a con venient characterization of gapless spin liquids In the alge braic or critical spin liquidsll 14 these wave vectors are limited to a nite discrete set often at high symmetry points in the Brillouin zone and their effective eld theories can often exhibit a relativistic structure But the singularities can also occur along sulfaces in momentum space as they do in the Gutzwillerprojected spinon Fermi sea state the 2D spin Bosemetal SBM phase It must be stressed that it is the spin ie bosonic correlation functions that possess such singular surfacesithere are no fermions in the systemiand the lowenergy excitations cannot be described in terms of weakly interacting quasiparticles It has been proposed recently15 that a 2D Bosonring model describing itinerant hardcore bosons hopping on a square lattice with a frustrat ing foursite term can have an analogous liquid ground state which we called a d wave Bose liquid DBL The DBL is also a Bosemetal phase possessing a singular Bose surface in momentum s ace Recently we have suggestedl v17 that it should be possible to access such Bose metals by systematically approaching two dimensions from a sequence of quasi1D ladder models On a ladder the quantized transverse momenta cut through the 2D surface leading to a quasi1D descendant state with a set of lowenergy modes whose number grows with the num ber of legs and whose momenta are inherited from the 2D Bose surfaces These quasi1D descendant states can be ac cessed in a controlled fashion by analyzing the 1D ladder models using numerical and analytical approaches These multimode quasi1D liquids constitute a different and previ ously unanticipated class of quantum states interesting in their own right But more importantly they carry a distinctive quasi1D ngerprint of the parent 2D state The power of this approach was demonstrated in Ref 16 where we studied the new Bosonring model on a twoleg ladder and mapped out the full phase diagram using the DMRG and ED supported by variational wave function and gauge theory analysis Remarkably even for a ladder with only two legs we found compelling evidence for the quasi1D descendant of the 2D DBL phase This new quasi1D quantum state possessed all of the expected signa tures re ecting the parent 2D Bose surface In this paper we tum our attention to the triangular lattice Heisenberg model with ring exchange appropriate for the PHYSICAL REVIEW B 79 205112 2009 o5 o 05 1 15 2 25 3 35 J2J1 FIG 2 Color online Phase diagram of the ring model Eq 1 determined in the DMRG using system sizes L48796 Filled squares red denote Bethechain phase Open squares with black outlines denote valence bond solid with period 2 Open circles blue denote spin Bose metal Open circles with crosses denote where the DMRG has dif culties converging to singlet for the larger sizes but where we still think this is spinsinglet SBM Star symbols denote points where the ground state appears to have true nonzero spin for all points here the magnetization is smaller than full polarization of the smaller Fermi sea in the SBM interpreta tion Filled diamonds magenta denote VBS with period 3 Our identi cations are ambiguous in the lower VBS3 region approach ing VBS2 Lines indicate phase boundaries determined in VMC using spinsinglet wave functions described in the text we also used appropriate dimerized wave functions for the VBS states A detailed study of a cut K Jl1 is presented in Sec III cf Fig 8 mg K ET2Cu2CN3 material In hopes of detecting the quasi1D descendant of the triangular lattice spin Bose metal Gutzwillerprojected spinon Fermi sea state we place this model on a triangular strip with only two legs shown in Fig l The allimportant ring exchange term acts around foursite plackets as illustrated we also allow different Heisenberg exchange couplings along and transverse to the ladder It is convenient to view the twoleg strip as a J 112 chain studied extensively beforelgvlg with additional fourspin ex changes The Hamiltonian reads H E 211555 55 1 21251 55 2 5 KringPgtcgtc2gtc3gtcl HCJI 1 The fourspin operators act as P1234 010203930394 gt 0394 0391 0392 0393 P1234P1234 1 We attack this model us ing a combination of numerical and analytical techniquesi DMRG ED variational Monte Carlo VMC as well as em ploying bosonization to obtain a lowenergy effective eld theory from the slave particlegauge formulation andor from an interacting electron Hubbardtype model Our key ndings are summarized in Fig 2 which shows the phase diagram for antiferromagnetic couplings J 1 12 and King For Kinggt0 the 1112 model has the familiar lD Bethechain phase for JZSO2411 and period2 valence bond solid VBS2 for larger Jz For ng202ll different physics opens up In fact Klironomos et 41120 considered such 2051122 SPIN BOSEMETAL PHASE IN A SPIN MODEL 3 2 t1 cosk39 2 t2 cos2k u 2 1 A 0 g kF2 39kF1 kF1 39sz v 1 39 w kF2kF1 n2 4 5 Tc 0 T k FIG 3 Color online Spinon dispersion for t2gt05t1 showing two occupied Fenni sea segments here and throughout we use the 1D chain language see bottom Fig 1 Gutzwiller projection of this is the zigzag SBM state at the focus of this work JlJzng model motivated by the study of Wigner crystals in a quantum wire21 Using ED of systems up to L224 they found an unusual phase in this intermediate regime called 4P in Fig 8 of Ref 20 but it had proven dif cult to clarify its nature We identify this region as a descendant of the triangular lattice spin Bosemetal phase or further deriva tives of the descendant as discussed below A caricature of the zigzag spin Bose metal is provided by considering a Gutzwiller trial wave function construction on the twoleg strip The 2D SBM is obtained by letting spinons hop on the triangular lattice with no uxes and then Gutzwiller projecting to get a trial spin wave function So here we also take spinons hopping on the ladder with no uxes which is tltz hopping in the 1D chain language that we mainly use For t2lt05t1 the mean eld state has one Fermi sea segment spanning 7727r2 spinons are at half lling and the Gutzwiller projection of this is known to be an excellent state for the Bethe chain On the other hand for t2gt05t1 the spinon band has two Fermi seas as shown in Fig 3 The Gutzwiller projection of this is a phase that we identify as a quasi1D descendant of the triangular lattice spin Bose metal The wave function has one variational pa rameter t2 II or equivalently the ratio of the two Fermi sea volumes Using this restricted family of states our VMC energetics study of the J lJzng model nds three regimes broadly delineated by solid lines in Fig 3 for larger ng i Bethechain regime the optimal state has one Fermi sea ii For suf ciently large Km and upon increasing Jz we enter a different regime where it is advantageous to start populating the second Fermi sea As we further increase Jz moving away from the Bethechain phase we gradually transfer more spinons from the rst to the second Fermi sea This whole region is the SBM iii Finally at still larger Jz the volumes of the two Fermi seas become equal which corresponds to tz tlaw ie decoupledlegs limit The DMRG is the crucial tool that allows us to answer how much of this trial state picture actually holds in the JlJzng model Figure 2 shows all points that were studied using the DMRG and their tentative phase identi cations by looking at various groundstate properties Remarkably in a PHYSICAL REVIEW B 79 205112 2009 broadbrush sense the three regimes found in VMC for K gt02 one spinon Fermi sea two generic Fermi seas and decoupled legs match quite closely different qualitative behaviors found by the DMRG study and marked as Bethe chain SBM and VBS2 regions Here we note that the decoupledlegs Gutzwiller wave function is gapless and does not have VBS2 order but it is likely unstable toward open ing a spin gap1819 still it is a good initial description for large Jz On the other hand away from the decoupledlegs limit we expect a stable gapless SBM phase The DMRG measures spin and dimer correlations and we identify the SBM by observing singularities at characteristic wave vec tors that evolve continuously as we move through this phaseithese are the quasi1D Bose points remnants of a 2D Bose surface The singular wave vectors are reproduced well by the VMC although the Gutzwiller wave functions apparently cannot capture the amplitudes and power law ex ponents An effective lowenergy eld theory for the zigzag SBM phase can be obtained by employing bosonization to analyze either a spinongauge theory formulation or an interacting model of electrons hopping on the zigzag chain In the latter case we identify an umklapp term which drives the twoband metal of interacting electrons through a Mott metalinsulator transition The lowenergy bosonized description of the Mott insulating state thereby obtained is identical to that obtained from the zigzag spinongauge theory In the interacting elec tron case there are physical electrons that exist above the charge gap On the other hand in the gauge theory the spinons are unphysical and linearly con ned The lowenergy xedpoint theory for the zigzag spin Bosemetal phase consists of three gapless free Boson modes two in the spin sector and one in the singlet sector the latter we identify with spin chirality uctuations Be cause of the SU2 spin invariance there is only one Lut tinger parameter in the theory and we can characterize all power laws using this single parameter The dominant corre lations occur at wave vectors 2kF1 and 2km connecting op posite Fermi points and the power law can vary between 32 and x 1 depending on the value of the Luttinger param eter We understand well the stability of this phase We also understand why the Gutzwillerprojected wave functions while capturing the singular wave vectors are not fully adequateiour trial wave functions appear to be described by a speci c value of the Luttinger parameter that gives 32 power law at 2kF1 and 2km The difference between the DMRG and VMC in the SBM phase is qualitatively captured by the lowenergy bosonized theory The full DMRG phase diagram ndings are in fact much richer Prominently present in Fig 2 is a new phase occur ring inside the SBM and labeled VBS3 This has period3 valence bond solid order dimerizing every third bond and also has coexisting effective Bethechainlike state formed by nondimerized spins see Sec V A and Fig 14 for more explanations A careful look at the SBM theory reveals that at a special commensuration where the volume of the rst Fermi sea is twice as large as that of the second Fermi sea the SBM phase can be unstable gapping out the rst Fermi sea and producing such VBS3 state Another observation in Fig 2 is the possibility of devel oping a partial ferromagnetic FM moment in the SBM re 2051123 SHENG MOTRUNICH AND FISHER gion labeled partial FM to the left of the VBS3 phase We do not understand all details in this region In the SBM fur ther to the left we think the ground state is spin singlet which is what we nd from the DMRG for smaller system sizes up to L248 However the DMRG already has dif cul ties converging to the spinsinglet state for the larger system sizes L96 In the partial FM region it seems that the ground state has a small magnetization Given our SBM pic ture it is conceivable that one or both spinon Fermi seas could develop some spin polarization The most likely sce nario is for the polarization to rst appear in the smaller Fermi sea since it is more narrow in energy more atband like The total spin that we measure in the partial FM region in Fig 2 is smaller than what would be expected from a full polarization of the second Fermi sea and it is hard for us to analyze such states To check our intuition we have also considered a modi ed model with an additional thirdneighbor coupling J3 which can be either antiferromagnetic or ferromagnetic tai lored to either suppress the ferromagnetic tendencies or to reveal them more fully We have studied this model at ng 2111 varying 12 With antiferromagnetic 13205 we have indeed increased the stability of spinsinglet states in the region to the left of the VBS3 Interestingly this study which is not polluted by the small moment dif culties also revealed a spingapped phase near the VBS3 The phase has a particular period4 order in the spin chirality and we can understand the occurrence as an instability at another com mensuration point hit by the singular wave vectors as they vary in the SBM phase see Sec IV E for details Tuniing now to ferromagnetic 132 05 we have found a more clear example of the partial ferromagnetism where the ground state is well described by Gutzwiller projecting a state with a fully polarized second Fermi sea and an unpolarized rst Fermi sea The paper is organized as follows To set the stage in Sec II we develop general theory of the zigzag SBM phase In Sec III we present the DMRG study of the ring model that leads to the phase diagram Fig 2 We consider carefully the cut at ng211 and provide detailed characterization of the new SBM phase In Sec IV we study analytically the stabil ity of the SBM We also consider possible phases that can arise as some instabilities of the SBM This is done in par ticular to address the DMRG ndings of the VBS3 and chirality4 states which we present in Sec V To clarify the regime to the left of the VBS3 where the DMRG runs into convergence dif culties or small moment development we also perturb the model with antiferromagnetic Sec V B or ferromagnetic Sec VI thirdneighbor interaction J 3 and dis cuss partially polarized SBM Finally in Sec VII we brie y summarize and suggest some future directions one might ex plore In Appendix A within our effective eld theory analy sis we summarize the bosonization expressions for physical observables that are measured in the DMRG and VMC In Appendix B we provide details of the Gutzwiller wave func tions that are used throughout in the VMC analysis In Ap pendix C we summarize the DMRG results for the conven tional Bethechain and VBS2 phases PHYSICAL REVIEW B 79 205112 2009 II SPIN BOSEMETAL THEORY ON THE ZIGZAG STRIP Since a wave function does not constitute a theory and can at best capture a caricature of the putative SBM phase it is highly desirable to have a eldtheoretic approach The goal here is to obtain an e ective lowenergy theory for the SBM on the zigzag chain In two dimensions the usual ap proach is to decompose the spin operators in terms of an SU2 spinorithe fermionic spinons 5 in rm fix 1 2 In the mean eld one assumes that the spinons do not interact with one another and are hopping freely on the 2D lattice For the present problem the mean eld Hamiltonian would have the spinons hopping in zero magnetic eld and the ground state would correspond to lling up a spinon Fermi sea In doing this one has arti cially enlarged the Hilbert space since the spinon hopping Hamiltonian allows unoccu pied and doublyoccupied sites which have no meaning in terms of the spin model of interest It is thus necessary to project back down into the physical Hilbert space for the spin model restricting the spinons to single occupancy If one is only interested in constructing a variational wave function this can be readily achieved by the Gutzwiller pro jection where one simply drops all terms in the wave func tion with unoccupied or doublyoccupied sites The alteniate approach to implement the single occupancy constraint is by introducing a gauge eld a U1 gauge eld in this instance which is minimally coupled to the spinons in the hopping Hamiltonian This then becomes an intrinsically strongly coupled lattice gauge eld theory To proceed it is necessary to resort to an approximation by assuming that the gauge eld uctuations are in some sense weak In two dimen sions one then analyzes the problem of a Fermi sea of spinons coupled to a weakly uctuating gauge eld This problem has a long history14v22 29 but all the authors have chosen to sum the same class of diagrams Within this un controlled approximation one can then compute physical spin correlation functions which are gauge invariant It is unclear however whether this is theoretically legitimate and even less clear whether or not the spin liquid phase thereby constructed captures correctly the universal properties of a physical spin liquid that can or does occur for some spin Hamiltonian Fortunately on the zigzag chain we are in much better shape Here it is possible to employ bosonization to analyze the quasi1D gauge theory as we detail below While this still does not give an exact solution for the ground state of any spin Hamiltonian with regard to capturing universal lowenergy properties it is controlled As we will see the lowenergy effective theory for the SBM phase is a Gaussian eld theory and perturbations about this can be analyzed in a systematic fashion to check for stability of the SBM and possible instabilities into other phases As we will also brie y show the lowenergy effective theory for the SBM can be obtained just as readily by start ing with a model of interacting electrons hopping on the zigzag chain ie a Hubbardtype Hamiltonian If one starts 2051124 SPIN BOSEMETAL PHASE IN A SPIN MODEL with interacting electrons it is in principle possible to con struct the gapped electron excitations in the SBM Mott insu lator Within the gauge theory approach the analogous gapped spinon excitations are unphysical being con ned to gether with a linear potential Moreover within the electron formulation one can access the metallic phase and also the Mott transition to the SBM insulator A SBM via bosonization of gauge theory We rst start by using bosonization30 32 to analyze the gauge theory 35 Motivated by the 2D triangular lattice with ring exchanges we assume a mean eld state in which the spinons are hopping in zero ux Here the spinons are hop ping on the zigzag strip with nearneighbor and second neighbor hopping strengths denoted t1 and t2 This is equiva lent to a strictly lD chain with likewise rst and second neighbor hoppings The dispersion is k 2t1 cosk 21 2 cos2k M 3 For t2gt05t1 there are two sets of Fermi crossings at wave vectors kFl and ikm as shown in Fig 3 Our convention is that fermions near km and km are moving to the right the corresponding group velocities are 01 02gt0 The spinons are at half lling which implies kF1kF2 7r2 mod 27739 The spinon operators are expanded in terms of continuum elds frx 2 e39Pme 4 1113 with a l 2 denoting the two Fermi seas 042T 1 denoting the spin and PR L i denoting the right and left moving fermions We now use bosonization30 32 reexpressing these lowenergy spinon operators with bosonic elds fPacz 7711a51aPMa with canonically conjugate boson elds MAX 235060 655500 bgx l 0 6 MAX bgx l iw5ab5agbc x 7 where x is the Heaviside step function Here we have introduced Klein factors the Majorana fermions 7711a 77 28ab8a5 which assure that the spinon elds with different avors anticommute with one another The slowly varying fermionic densities are simply figmfpmz IPltpm 6 2 77 A faithful formulation of the physical system in this slave particle approach Eq 2 is a compact Ul lattice gauge theory In llD continuum theory we work in the gauge eliminating spatial components of the gauge eld The imaginarytime bosonized Lagrangian density is then 1 1 L E ltz96m2valta6m2 A lt8 27739m 1 Here LA encodes the coupling to the slowly varying lD sca lar potential eld Ax PHYSICAL REVIEW B 79 205112 2009 l A 9cA7T2 iPAAa 9 m where pA denotes the total gauge charge density pA E Matw 10 Ila It is useful to de ne charge and spin boson elds 6W iw 6 11 E and even and odd avor combinations 1 6M 091M 1 92M 12 with Lp039 Similar de nitions hold for the p elds The commutation relations for the new 6 p elds are unchanged Integration over the gauge potential generates a mass term LAme 352 13 for the eld 692m6m2 In theegauge theory analysis we cannot determine the mean value 91 which is important for detailed properties of the SBM in Appendix A as well as for the discussion of nearby phases in Secs IV BiIV E But if we start with an interacting electron model one can readily argue that the correct value in the SBM phase satis es 46191 7739 mod 27739 14 B SBM by bosonizing interacting electrons Consider then a model of electrons hopping on the zigzag strip We assume that the electron hopping Hamiltonian is identical to the spinon mean eld Hamiltonian with rst and secondneighbor hopping strengths 1 2 H E tchxcax l tzcxcax 2 Hc Him 15 The electrons are taken to be at half lling The interactions between the electrons could be taken as a Hubbard repulsion perhaps augmented with furtherneighbor interactions but we do not need to specify the precise form for what follows For t2lt05t1 the electron Fermi sea has only one seg ment spanning 77 2 77 2 and at low energy the model is essentially the same as the 1D Hubbard model We know that in this case even an arbitrary weak repulsive interaction will induce an allowed fourfermion umklapp term that will be marginally relevant driving the system into a 1D Mott insu lator The residual spin sector will be described in terms of the Heisenberg chain and is expected to be in the gapless Bethechain phase On the other hand for t2gt05t1 the electron band has two Fermi seas as shown in Fig 3 This is the case of pri mary interest to us As in the oneband case umklapp terms are required to drive the system into a Mott insulator But in 2051125 SHENG MOTRUNICH AND FISHER this twoband case there are no allowed fourfermion um klapp terms While it is possible to study perturbatively the effects of the momentum conserving fourfermion interac tions and address whether or not the twoband metal is stable for some particular form of the lattice Coulomb repulsion we do not pursue this here Rather we focus on the allowed eightfermion umklapp term which takes the form H3 08Ci211Ci211Ci221Ci221CL11CL11CL21CL21 HCJ 16 where we have introduced slowly varying electron elds for the two bands at the right and left Fermi points For repulsive electron interactions we have 03gt0 This umklapp term is strongly irrelevant at weak coupling since its scaling dimen sion is A324 each electron eld has scaling dimension 12 much larger than the spacetime dimension D2 To make progress we can bosonize the electrons just as we did for the spinons che39 PMWM The eightfermion umklapp term becomes H3 2 203 cos4 p 17 where as before 6922m6m2 and pgx219k997r is now the physical slowly varying electron density The bosonized form of the noninteracting electron Hamiltonian is precisely the rst part of Eq 8 and one can readily con rm that A324 But now imagine adding a strong densitydensity re pulsion between the electrons The slowly varying contribu tions on scales somewhat larger than the lattice spacing will take the simple form HpVppix Vp19k992 These for ward scattering interactions will stiffen the 99 eld and will reduce the scaling dimension A8 If A3 drops below 2 then the umklapp term becomes relevant and will grow at long scales This destabilizes the twoband metallic state driving a Mott metalinsulator transition The 99 eld gets pinned in the minima of the H3 potential which gives Eq 14 Expanding to quadratic order about the minimum gives a mass term of form 13 For the lowenergy spin physics of primary interest this shows the equivalence between the di rect bosonization of the electron model and the spinongauge theory approach The difference between the spinongauge theory and the interacting electron theory is manifest in the charge sector In the latter case the electron excitations CI above the gap will correspond to instantons connecting adjacent minima of the cosine potential Eq 17 In the spinongauge theory there are no such fermionic excitations fl and the spinon excita tions are linearly con ned This is appropriate for the spin model which has no charge sector and no notion of spinons In the weak Mott insulating phase of the electron model the Fermi wave vectors kFl km denote the momenta of the minimum energy gapped electron excitations What is the meaning then of the spinon Fermi wave vectors if the spinon excitations are unphysical Within the spinongauge theory the only gaugeinvariant ie physical momenta are the sums and differences of km km which correspond to momenta of the lowenergy spin excitations In the electron model the spin excitations below the charge localization length of the Mott insulator will be similar to that of elec trons in the metal On longer scales the spin sector remains gapless and this is the regime described below by the low PHYSICAL REVIEW B 79 205112 2009 energy effective theory of the SBM Mott insulator It is these physical longer length scale spin excitations which are cor rectly captured by both the spinon gauge theory and interact ing electron approaches C Fixed point theory of the SBM phase The lowenergy spin physics in either formulation can be obtained by integrating out the massive 9 eld as we now demonstrate Performing this Gaussian integration leads to the effective xedpoint quadratic Lagrangian for the SBM spin liquid L BMLgLg 18 with the charge sector contribution LL i319 2 1919 2 19 0 2780 pomp Moot lt and the spin sector contribution 039 1 1 2 2 0 2 E a6w Mama 20 77 11 011 The velocity 0 in the charge sector depends on the product of the avor velocities 120 0102 while the dimensionless conductance depends on their ratio 2 21 go wlv2 xvzvl Notice that go S 1 with g0gt 0 upon approaching the limit of a single Fermi surface 121 at 0v2gt0 and goal in the limit of two equally sized Fermi surfaces 122 v1gt 1 that occurs when the two legs of the triangular strip decouple In Sec IV A we also consider all symmetry allowed re sidual shortrange interactions between the lowenergy de grees of freedom and conclude that the above xedpoint theory can indeed describe a stable phase with the only modi cation that g0gt g is now a general Luttinger param eter Stability requires glt 1 There are also three marginal interactions that need to have appropriate signs to be margin ally irrelevant The gapless excitations in the SBM lead to power law correlations in various physical quantities at wave vectors connecting the Fermi points Here and in the numerical study Sec III we focus on the following observables spin x bond energy Bx ie VBS order parameter and spin chirality xx 3x x x 1 22 Xx 5 1 x x x1 23 In Appendix A we give detailed expressions in the con tinuum theory The most straightforward contributions are obtained by writing out eg Sx flx 5fx in terms of the continuum fermion elds and then bosonizing see also Eqs A7 and A9 for Bx and xx We expect dominant power laws at wave vectors 2kFa and i7 2 1kF1 2051126 SPIN BOSEMETAL PHASE IN A SPIN MODEL PHYSICAL REVIEW B 79 205112 2009 TABLE I Spin Bosemetal xedpoint theory scaling dimensions of the spin S bond energy 8 and chirality X observables at various wave vectors Q in the top row Columns with one listed value have all scaling dimensions equal to this value Entries with subdominant power laws are listed as subd 1392th lm kn QO 2sz 772 3kF1kn 77 4kF1 s 12g4 1 Subd 8 1 I2g4 I2I4g I2I4gg4 1 X Subd Ig Subd km originating from fermion bilinears composed of a par ticle and a hole moving in opposite directions Such bilinears become enhanced upon projecting down into the spin sector ie upon integrating out the massive 99 in the bosonized eld theory and it is possible to compute the scaling dimen sion of any operator in terms of the single Luttinger param eter g It is also important to consider more general contri butions eg containing fourfermion elds this is best done using symmetry arguments and the corresponding expres sions can be found in Appendix A Table I summarizes such analysis of the observables by listing scaling dimensions at various wave vectors We de scribe power law correlation of a given operator A at a wave vector Q by specifying the scaling dimension AAQ de ned from the realspace decay etQ ltAxA0gt 2 W 24 Q iquot The corresponding static structure factor ie Fourier trans form has momentumspace singularity iq QIZAAQ 1 The Q0 entries in Table I come from simple identi ca tions SZQ0 N 910 19x62039a 25 BQ0 96 26 XQ0 N w 27 In particular the last line provides physical meaning to the p sectorithis spinsinglet sector encodes lowenergy uctuations of the chirality A direct way to observe the propagating p mode would be to measure the spectral func tion of the chirality while in the present DMRG study we detect it by a iqi ie V shaped behavior in the static struc ture factor at small wave vector q III DMRG STUDY OF THE SPIN BOSE METAL IN THE 11JzK ng MODEL ON THE ZIGZAG CHAIN We study the ring model Eq 1 on the twoleg triangu lar strip shown in Fig 1 We use the 1D chain picture and take site labels xl L where L is the length of the sys tem We use exact diagonalization ED and density matrix renormalization group DMRG Refs 36738 methods supplemented with variational Monte Carlo VMC Refs 39 and 40 to determine the nature of the ground state of Hamil tonian l A Measurement details We rst describe numerical measurements All calcula tions use periodic boundary conditions in the direction In the ED we can characterize states by a momentum quantum number k On the other hand our DMRG calculations are done with realvalued wave functions This gives no ambi guity when the ground state carries momentum 0 or 7739 and is unique However if the ground state carries nontrivial mo mentum k 07 then its timereversed partner carries k and the DMRG state is some combination of these While the realspace measurements depend on the speci c combination in the nite system the momentumspace measurements de scribed below do not depend on this and are unique Most of the calculations are done in the sector with 5350 which contains any ground state of the SU2invariant system The DMRG calculations keep more than m3200 states per block36 38 to ensure accurate resulm and the density ma trix truncation error is on the order of 106 Typical relative error for the groundstate energy is on the order of 10 4 or smaller for the systems we have studied Using ED we have con rmed that all DMRG results are numerically exact when the system size is L224 The DMRG convergence depends strongly on the phase being studied the system size the type of the correlations and the distance between operators In the Bethechain and VBS2 phases there is still good conver gence for size Ll92 while in the SBM we are limited to L2967144 systems The entanglement entropy calculations are done with up to m6000 states in each block which is necessary for capturing the longrange entanglement in the SBM states where we nd an effective central charge c 2 3 We have already speci ed the main observables in Sec II C cf Eqs 22 and 23 We measure spin correlations bond energy dimer correlations and chirality correlations de ned as follows Com lt ltx Eva 28 mm ltBrxgt6ltx39gt ltBgt2 29 MM WW0 30 For simplicity we set Dxx 0 if bonds Bx and Bx have common sites and similarly for Xxx Structure fac 2051127 SHENG MOTRUNICH AND FISHER tors Cq Dq and X q are obtained through Fourier trans formation 1 r 0q 22 0xx equot1lt h 31 where 0CDX We have 0q0 q and usually show only OSqu The spin structure factor at q0 gives the total spin in the ground state Cq 0 SEQ 51016101 1 32 In all gures we loosely use 811811 and XqX Z to denote Dq and Xq respectively Tuniing to the VMC calculations the trial wave functions are described in broad terms in Sec I and in more detail in Appendix B The states are labeled by occupation numbers of the two Fermi seas N1N2 Since N1N2L2 there is only one variational parameter There are three distinct re gimes i N1L2 N220 ie a single Fermi sea which is appropriate for the Bethechain phase ii N1 N2 at 0 appro priate for the SBM and iii N1N2L4 ie decoupled legs which is a reasonable starting point for the large Jz limit In Appendix B we describe correlations in the generic SBM wave functions and identify characteristic wave vectors 2de2rrNdL al2 and also 772 see Sec II and Table I One observation is that such wave functions correspond to a special case gl in the SBM theory and thus cannot cap ture general exponenm Despite this shortcoming the wave functions capture the locations of the singular wave vectors observed in the DMRG We also try to improve the wave functions by using a gapless superconductor modi cation described in Appendix B 2 and designed to preserve the sin gular wave vectors while allowing more variational param eters This indeed improves the trial energy and provides better match with the DMRG correlations at short scales even if the longdistance properties are still not captured fully When presenting the DMRG structure factors we also show the corresponding VMC results for wave functions de termined by minimizing the trial energy over the described family of states Using the DMRG we nd four distinct quantum phases in the JzJl ngJl plane as illustrated in Fig 2 In the small ng region we have the conventional Bethechain phase at small JZ and valence bond solid state with period 2 VBS2 at larger Jz The SBM phase emerges in the regime Kng gt 0211 and dominates the intermediate parameter space In side this region we discover a VBS state with period 3 VBS3 To fully understand the VBS3 in particular its relationship to the anking spin Bose metals we will need the stability analysis of the SBM in Sec IV while the DMRG results are discussed afterward in Sec V We explore more nely a cut ng211 through the phase diagram Fig 2 and our presentation points are taken from this cut The Bethechain and VBS2 phases are fairly con ventional for this King the VBS2 is close to the decoupled leg state at large JZ values Nevertheless it is useful to see measuremenm in these phases for comparisons Such ex PHYSICAL REVIEW B 79 205112 2009 amples are given in Appendix C while here we focus on the spin Bosemetal points deep in the phase We will discuss more dif cult parts of the phase diagram Fig 2 once we have the overall picture of the SBM B Representative spin Bosemetal points Proceeding along the ngzllzl cut through Fig 2 we start in the Bethechain phase at large negative 12 a repre sentative point is discussed in Appendix C 1 As we change 12 toward positive value the system undergoes a transition at 22 06 The new phase has characteristic spin correlations that are markedly different from the Bethechain phase Fig ure 4 shows a representative point 120 The DMRG calcu lations are more dif cult to converge and are done for smaller size Ll44 than in the Bethechain phase example see also the entanglement entropy discussion below Comparing with the Bethechain state eg Fig 22 in Appendix C 1 there is no q 7739 dominance in the spin struc ture factor Instead we see three singular wave vectors lo cated at ll gtlt27739 144 772 and 61 gtlt27739 144 Our Gutzwiller wave functions determined from the energetics have N1N26lll and the corresponding 2km km kF17739 2 mod 27739 and 2kF1 match precisely the DMRG singular wave vectors The improved Gutzwiller wave func tion reproduces crude shortdistance features better but it has the same longdistance properties as the bare Gutzwiller As discussed in Appendix B our Gutzwiller wave functions do not capture all power laws predicted in the general analytical eory The wave functions appear to have equal exponents for spin correlations at these three wave vectors while the theory summarized in Table I gives stronger singularities at 2kF1 2km and a weaker singularity at 77 2 Very encourag ingly these theoretical expectations are consistent with what we nd in the DMRG where we can visibly see the differ ence in the behaviors at these wave vectors particularly when we reference against the VMC results Similar discussion applies to the bond energy dimer cor relations shown in the middle panel of Fig 4 The dominant features are at 2kF1 and 77 2 and we also see a peak at a wave vector identi ed as 4km which is indeed expected from the SBM theory cf Table I The theory predicts simi lar singularities at 2kF1 and 2km but for some reason we do not see the 2km in the DMRG data even though it is clearly present in the bare Gutzwiller We suspect that this is caused by the narrowness in energy of the second Fermi sea when its population is low so the amplitude of the 2km bond en ergy feature may be much smaller The 2km can still show up in the bare Gutzwiller since as we discuss in Appendix B this wave function knows only about the Fermi sea sizes and not about the spinon energy scales such as bandwidths or Fermi velocities The improved Gutzwiller clearly tries to remedy this although within its limitations The 4km feature is not associated solely with the second Fermi sea and is less affected by this argument indeed 4kF2 4kF1 and both Fermi seas participate in producing this feature as can be seen from Eq A31 Tuniing to the chirality structure factor in the bottom panel of Fig 4 we see a feature at 77 2 which in the SBM 2051128 SPIN BOSEMETAL PHASE IN A SPIN MODEL KW J1 1J20J30 L144 16 D R6 Gutzw6111 L4 u Wm eu 12 10 08 06 L 04 0 2kF2 Spin structure factor ltSq Slqgt 3kk272 kFZkp 2kg 7 q 020 015 010 005 000 005 010 015 Dimer structure factor ltBq qugt 4sz 2kF2 3kRkF2 7r2 q 025 0 2kg 7 020 D R6 Gulzw6111 w GuIZWImprnved Chirality structure factor ltxq xlqgt 0 3kF1kF2 m2 kFZkF1 n q FIG 4 Color online Spin dimer and Chirality structure fac tors at a representative point in the spin Bosemetal phase K ng Jll 120 close to the Bethechain phase measured in the DMRG for system size L144 We also show the structure factors in the Gutzwiller projection of two Fermi seas N1N26l 11 and in the improved Gutzwiller wave function with parameters f0 1 f1065 and f2 05 see Appendix B the parameters are determined by optimizing the trial energy within the given family of states Vertical lines label important wave vectors expected in the SBM theory for such spinon Fermi sea volumes We discuss the comparison of the DMRG VMC and analytical theory in the text theory is expected to have the same singularity as the spin and dimer at this wave vector We also see features at wave vectors kFQ kpl and 3kF1kFZ in all our observables these features are expected to be i8qi ie V shaped in the Gutzwiller wave functions but have weaker singularity in the PHYSICAL REVIEW B 79 205112 2009 SBM theory which is reasonably consistent with the DMRG measuremenm Very notable in the Chirality structure factor is an in shape at small q This can be viewed as a direct evidence for the gapless p mode in the SBM cf Eq 27 On the other hand a feature at 7739 is weaker than V shaped in contrast with the Gutzwiller wave functions but in agreement with the SBM theory expectations in Table I To summarize the correlations in the SBM phase are dra matically different from the Bethechain phase and we can match all the characteristic wave vectors using the Gutzwiller wave functions projecting two Fermi seas We also understand the failure of the wave functions to repro duce the nature of the singularities and the amplitudes while the bosonization theory of the SBM is consistent with all DMRG observations even when the wave functions fail With further increase in Jz continuing along the Kng 2111 cut through Fig 2 the SBM phase becomes promi nently unstable toward the VBS3 phase occupying the range 15 lt12 lt 25 and described in Sec V A Interestingly as we increase JZ still further the SBM phase reappears with its characteristic correlations shown in Fig 5 for a representa tive point 12232 Much of the SBM physics discussion that we have just done at 120 carries over here with the appro priately shifted locations of the singular wave vectors The singularities at wave vectors q10W2kF2 and qhigh2kF1 are now more equally developed and are detected in the spin as well as dimer structure factors The two wave vectors are closer to 772 and are located symmetrically in accordance with the general sum rule 2kF12kF2773 while the com parable strengths re ect the more similar energy bandwidths of the two Fermi seas The apparent lack of the wave vector 77 2 in the DMRG dimer structure factor is similar to that in the Gutzwiller wave function and is a matrix element effect for the rstneighbor bond when the sizes of the two Fermi seas approach each other On the other hand the strength of the 4km dimer feature is very notable here it can indeed be dominant in the SBM theory for suf ciently small Luttinger parameter g cf Table I The improved Gutzwiller wave function modi es the structure factors in the right direction but clearl does not succeed reproducing them accuratelyias noted before our wave functions do not con tain the full physics expected in the bosonized theory One technical remark that we want to make here is that the DMRG ground state at this point 12232 and size L 144 appears to have nontrivial momentum quantum num ber k at 0 7739 We deduce this by observing that the measured correlations 0xx depend not just on x x but on both x and x and by seeing characteristic beatings as a function of x and x while the qspace structure factors are well con verged On the other hand the VMC wave function shown in Fig 5 has momentum zero see Appendix B and all mea surements depend on x x only If we assume that the beat ings originate from the DMRG state being a superposition of ik and i k we can extract 2k and nd it to be consistent with the state ik constructed from the VMC by moving one spinon across one of the Fermi seas 2k4km 4kF1 It is plausible that such state happens to have a slightly lower energy in the given nite system eg at the same 12232 we nd trivial k for L272 but nontrivial k for L296 likely re ecting nitesize effects on the lling of the last few 2051129 SHENG MOTRUNICH AND FISHER Kmg J 1 J2 32 J8 o L144 A DMRG q Gulzw 4428 quot1 20 55 39aulzw Improved v V 15 2 o J 9 10 E o E 5 05 C 395 m 00 0 3kFikF2 2km TrZ 2km kFZ39kFi TE q 040 DMRG Gulzw 4428 Gulzw Improved Dimer structure factor ltBq qugt o 3kHkF2 2kF2 wz 2km q 008 DMRG Gulzw 4428 Gulzw Im proved 006 004 002 000 002 Chirality structure factor ltxq xlqgt 004 o 3kF1kF2 wz q kF239kF1 TE FIG 5 Color online Spin dimer and chirality structure fac tors at a representative point in the spin Bosemetal phase K ng Jl1 1232 between the VBS3 and VBS2 phases measured in the DMRG for system size L144 We also show results in the Gutzwiller projection of two Fermi seas N1N24428 and in the improved Gutzwiller wave function with parameters f01 f1 075 and f2 O1 see Appendix B for details Vertical lines label important wave vectors expected in the SBM theory We dis cuss the comparison of the DMRG VMC and analytical theory in the text spinon orbitals We have not attempted to construct a trial spinsinglet state with the right momentum quantum number for the present L Still we expect that the structure factors are not very sensitive to such rearrangements of few spinons in the large system limit Indeed we nd that the structure factors have the same features for different system lengths PHYSICAL REVIEW B 79 205112 2009 ring K J 1 J8 0 varying J2 L96 Spin structure factor ltSq Slqgt 020 015 010 005 000 005 010 015 020 025 Dimer structure factor ltBq qugt FIG 6 Color online Evolution of the structure factors in the spin Bosemetal phase between the Bethechain and VBS3 or par tial FM measured by the DMRG for system size L96 We can track singular wave vectors Bose surfaces as spinons are trans ferred from the rst to the second Fermi sea upon increasing 12 In the spin structure factor the wave vector qhigh that starts near 77 is identi ed as 2kF1 and the wave vector qlow that starts near 0 is identi ed as 2kpZ see Fig 4 and text for details The qhigh and qlow are summarized in Fig 8 L272 96 and 144 It is also worth repeating that our struc ture factor measuremenm using Eq 31 do not depend on which speci c combination of ik and i k is found by the DMRG procedure C Evolution of the singular wave vectors in the SBM We further illustrate the spin Bose metal by showing evo lution of the DMRG structure factors and singular wave vec tors as we move inside the phase The spin and dimer struc ture factors are shown in Fig 6 for L96 and varying parameter 1211 inside the SBM phase adjacent to the Bethe chain phase With increasing Jz the singular wave vector qhigh identi ed as 2kF1 in the VMC is moving away from 7739 toward smaller values while the singular wave vector qlow identi ed as 2km is moving to larger values this corre sponds to spinons being transferred from the rst to the sec ond Fermi sea as found in the VMC energetics The spin and dimer correlations show similar behavior at 2kF1 and both have features also at the wave vector 77 2 The lack of visible dimer feature at 2km was discussed for the point 120 ear 20511210 SPIN BOSEMETAL PHASE IN A SPIN MODEL her and it is likely that something similar is at play here On one hand the second band is narrow when we just enter the SBM since the second Fermi sea starts small On the other hand in the region labeled partial FM close to the VBS3 phase in Fig 2 the DMRG nds nonzero magnetization in the ground state We think that this occurs in the second Fermi sea and indicates an effective narrowness of this band near the VBS3 phase as well so the 2km energy feature may indeed be weak in the whole SBM phase between the Bethechain and VBS3 We think that the same physics also stans causing conver gence dif culties in the DMRG for the L296 systems shown in Fig 6 Speci cally we can use Eq 32 to extract the groundstate spin Smt and nd noninteger values of order 1 eg StotStot103 10 19 21 and 34 for the points 120 02 04 06 and 08 in Fig 6 This can only happen if the DMRG is not converged and is mixing several states that are close in energy but have different spins Correspondingly we observe a difference between SirS39Eq and S f struc ture factors recall that we are working in the sector Sfot0 We do not show these graphs but the difference is localized near 2km where the Ser39fq has much sharper feature while the SI 1 has weaker feature On the other hand there is essentially no difference near 2kF1 This strongly suggests that the origin of the convergence dif culties lies with the second Fermi sea The 5187 structure factor that is shown in Fig 6 does not mix different Slot states but only adds the corresponding structure factors and is less sensitive to these convergence issues The fact that the peaks in the top panel of Fig 6 are located symmetrically with respect to 77 2 sug gests that this region is still spinsinglet SBM but is on the verge of some magnetism in the second band Finally the Smt values for the same parameters but system size L48 are indeed well converged to zero however with increasing number of states needed in the DMRG blocks for larger Jz We thus conclude that the points in Fig 6 are spinsinglet SBM The DMRG convergence dif culties for the larger L are in accord with the presence of many lowenergy excita tions see also the discussion of the entanglement entropy below We will further test our intuition that this region is close to some weak subband ferromagnetism in Secs V B and VI by adding antiferromagnetic or ferromagnetic J3 to suppress or enhance the FM tendencies Consider now Fig 7 that shows evolution of the structure factors in the SBM phase between the VBS3 and VBS2 The qhigh2kF1 and q10W2kF2 continue moving toward each other with increasing Jz and the spin structure factors be come nearly symmetric with respect to 77 2 When the qhigh and qlow peaks merge at 77 2 which in the VMC would correspond to decoupled legs one expects18v19 that a new instability will likely emerge resulting in a VBS2 state we discuss a representative point in Appendix C 2 A very notable feature in the SBM dimer structure factor is the strong 4km peak Foretelling a bit this peak can be traced as evolving out of the dimer Bragg peak at 2773 of the VBS3 phase to be discussed in Sec V A Turning this around and approaching the VBS3 phase by decreasing Jz we can view the VBS3 as an SBM instability when the dimer 4km peak merges with the 2kF1 singularity 4km 2kF1277 3 PHYSICAL REVIEW B 79 205112 2009 ng J11J8 0 varying J2 L96 20 A 2kF2 C a ma 15 V 30 E 10 2 7 E 3 0395 J228 30 5 32 39 l m 00 34 39 o m n q Dimer structure factor ltBq qugt FIG 7 Color online Evolution of the structure factors in the spin Bosemetal phase between VBS3 and VBS2 measured by the DMRG for system size L96 The qhigh2kF1 and qlow2kn bracket the 772 and approach each other with increasing 12 they are summarized in Fig 8 Very notable here is the strong 4km peak in the dimer structure factors that evolves out of the Bragg peak present in the VBS3 phase at 2773 see Sec V A and Fig 15 the 2kF1 moves away from the 2773 in the opposite direction Finally in this SBM region the DMRG converges well to spinsinglet states The remark we made when discussing the point 12232 applies to all points shown in Fig 7 with L 96 they show correlations 0xx beating in both x and x which can be interpreted similarly to the earlier 12232 case by assuming superposition of degenerate ground states with opposite momentum quantum numbers Figure 8 summarizes the singular spin wave vectors ex tracted from plom like Figs 6 and 7 superimposed on the phase diagram of the model along the cut ng27 1 Remark ably the singular wave vectors throughout the entire SBM phase are well captured by the improved Gutzwiller wave functions as we have illustrated in Figs 4 and 5 These singular wave vectors are intimately connected to the sign structure of the groundstate wave function indicating a striking coincidence between the exact DMRG groundstate wave function and the Gutzwillerprojected VMC wave function Besides the SBM regions Fig 8 also shows the Bethechain cf Appendix C 1 VBS2 Appendix C 2 and VBS3 Sec V A phases We now mention more dif cult points in the overall phase diagram The lightly hatched SBM region in Fig 8 indicates 20511211 SHENG MOTRUNICH AND FISHER ng J11J3 0 varying J2 L96 Fl N w Singularwavevectors in ltSq Slqgt a a on m r m m 399 R w o m P g m E lt w I w m u g lt w m m JzJ1 FIG 8 Color online Cut through the phase diagram Fig 2 at King111 showing evolution of the most prominent wave vectors in the spin structure factor In the Bethechain phase we have sin gular antiferromagnetic qAF77 cf Fig 22 In the spin Bose metal we have singular qhigh2kF1 and qlow2kn located symmetrically about 772 cf Figs 6 and 7 In the VBS3 we have singular qAF 773 cf Fig 15 In the VBS2 region for large 12 cf Fig 23 we have dominant correlations at 772 corresponding to the decou pled legs xed point which is likely to be unstable toward opening a spin gap Refs 18 and 19 The dotted lines show results for the improved Gutzwiller wave function the discussed rising DMRG dif culties of not converging to an exact singlet for L296 Such DMRG states are shown as open circles with crosses in Fig 2 As we have already men tioned the estimated Smt values are not converged and are of order 1 for L296 but are converged to zero for L248 while glow and qhigh are located symmetrically around 77 2 all this suggests that the phase is spinsinglet SBM On the other hand at points 12212715 not marked in Fig 8 but shown as star symbols in Fig 2 the estimated Smt values are larger and the apparent dominant wave vectors are no longer located symmetrically Here we suspect a modi cation of the ground state likely toward partial polarization of the second Fermi sea this partial FM region is also indicated by cross hatching in Fig 8 D Entanglement entropy and effective central charge in the SBM We explore properties of the SBM phase that can further distinguish it from the Bethechain and VBS states Earlier we have noted that we need to keep more states per block to achieve similar convergence for the SBM phase in compari son with the Bethechain and VBS phases indicating stron ger entanglement between subsystems in the SB Bosonization analysis nds that the SBM xedpoint theory has three free Boson modes One can associate a central charge 1 with each mode Despite the fact that they have different velocities so the full system is not conformally invariant we expect that the total entanglement entropy should have a universal behavior described by a combined central charge c3 In general for a onedimensional gapless state with con formally invariant correlation functions in spacetime the entanglement entropy for a nite subsystem of length X in PHYSICAL REVIEW B 79 205112 2009 SBM J2 00 x Entanglement entropy SX L Subsystem length X FIG 9 Color online Entanglement entropy at representative points in the Bethechain Jz 1 SBM 120 and 1232 VBS3 122 and VBS2 124 phases taken from the cut KringJl1 measured in the DMRG for system size L72 with periodic boundary conditions We use Eq 33 to t data over the range 65X 66 which gives the best estimates as c10 Bethe chain c31 SBM at 120 c32 SBM at 1232 c16 VBS3 and c21 VBS2 side a system of length L with periodic boundary conditions varies as4 L x SXL 10gltsingt A 33 where A is a constant independent of the subsystem length and c is the effective central charge The virtue of the en tanglement entropy is that it does not depend on the mode velocities and in principle measures the number of gapless modes directly from the groundstate wave function Figure 9 shows the entanglement entropy SXL as a function of X for different quantum phases for a nite system length L72 The results are obtained from the DMRG for representative points taken from the same cut ngzllzl discussed earlier The entropy can be well tted by the ansatz Eq 33 with different c values The Bethechain state at Jzz l gives central charge c 210 consistent with one gapless mode On e other hand the entanglement entropy for either of the two SBM ex amples 120 and 12232 is much larger and can be tted by close values c3l and c32 respectively The closeness of the central charges in these two different SBM states cf Figs 4 and 5 indicates the universal behavior of the en tanglement which is independent of the details such as the relative sizes of the spinon Fermi seas Interestingly the VBS2 point at 124 is tted by c2l which is related to the fact that the wave function is close to the decoupledlegs limit see Appendix C 2 and Fig 23 and 20511212 SPIN BOSEMETAL PHASE IN A SPIN MODEL 45 L24 L48 gtlt L60 x 4 L72 El L96 I 35 J2 00 SBM J2 10 Bethe 39 Entropy of subsystem SX L 1 10 20 40 d LnsinnXL FIG 10 Color online Entanglement entropy for various sys tem sizes for the Bethechain Jz l and SBM 120 points cf Fig 9 plotted versus scaling variable d sinWTX We also show ts to Eq 33 done for the larger sizes The Bethechain data collapse well and are tted with cl The SBM data are tted with c3l we show the same t as in Fig 9 for this SBM point The collapse of different L is less good which is likely due to imprecise scaling of the discrete shell lling numbers with L see text for details this L72 system does not know about the eventual spin gap and very small dimerization Finally the tted effective central charge for the VBS3 example is around c16 The oscillatory behavior of SX re ects translational symmetry breaking in the DMRG state Note that the entropy values are larger here compared with the Bethechain or VBS2 cases a fact which is probably due to a mix of degenerate states in the DMRG wave function However the overall X dependence is clearly weaker than in the VBS2 and is approaching the Bethechain behavior for large X To better understand nitesize effects we focus on the SBM and Bethechain phases and discuss the universal de pendence of the entropy on the scaling variable d sin L X Figure 10 shows SXL as a function of d for several system sizes for the SBM point 120 and the Bethechain point JZ At the SBM point the data for the two larger sizes L 60 and 72 collapse onto one curve which can be reason ably tted by SXL313logd088 strongly suggest ing the effective central charge c23 The smaller sizes L 24 and 48 have somewhat shifted entropy values compared with the L260 and 72 collapse but show roughly similar slope for the largest d The differences are likely due to nitesize shell lling effecm Indeed we can measure the structure factors and characterize the presumed SBM states by the spinon occupation numbers of the two Fermi seas we nd N1N2102 204 264 and 315 for L24 48 60 and 72 respectively and these numbers in each Fermi sea do not precisely scale with L PHYSICAL REVIEW B 79 205112 2009 On the other hand the Bethechain case does not have such effects and data for all sizes collapse The results can be well tted by SXLl3logd099 as shown in the same gure for system sizes L24796 We note that while the entropy for the Bethechain phase for L296 is fully con verged by keeping up to m4200 states in the DMRG block the entropy for the SBM for L72 is still increasing slowly with the number of states kept and we estimate that the error in SXL2L is around a few percent when m6000 states are kept per block comparing to an extrapolation to mzw Indeed the SBM data for L72 are bending down slightly from the tted line at the larger d corresponding to X L 2 as can be seen in Fig 10 which is probably because the data are less converged To summarize the entanglement entropy calculations es tablish the SBM as a critical phase with three gapless modes and clearly distinguish it from the Bethechain and VBS2 phases or the decoupledlegs limit We also note that the structure factor measurements and detection of all features as in Figs 6 and 7 did not require as much effort and was done for larger sizes than the entropy however to characterize the longdistance power laws accurately one would probably need to capture all entanglement which we have not attempted IV STABILITY OF THE SPIN BOSEMETAL PHASE NEARBY PHASES OUT OF THE SBM A Residual interactions and stability of the SBM We account for residual interactions between lowener y degrees of freedom in the SBM theory Sec II by consider ing all allowed shortrange interactions of the spinons The fourfermion interactions can be conveniently expressed in terms of chiral currents 1 Q JPab fivmbem JPzb Efivmo39aafivbg 34 We assume that interactions that are chiral say involving only right movers can be neglected apart from velocity renormalizations The most general fourfermion interactions which mix right and left movers can be succinctly written as Di E ngJRzbJleb AElbowum 35 111 017 E ngJRlzb 39 JLab KngRim 39 JLbba 36 111 with w11w220 convention w12w21 from Hermiticity and Alzzkm from Rlt gtL symmetry There are eight iride pendent couplings w fZ T N3quot A33quot and Mg We treat these interactions perturbatively as follows First we bosonize the interactions and obtain terms quadrch in 196 and 19w as well as terms involving products of four exponentials et39 w and emf We next impose the condition that 99 is pinned and compute the scaling dimensions of the exponential operators The w fzquot terms give W E 0lequ WEIRlz39JL12 H0 37 20511213 SHENG MOTRUNICH AND FISHER cos24w 2cos2 f cos2 9 38 w 172cos2ltp07 F cos26U 2F cos260 39 where F77n771l7721772y 40 The wigquot terms have scaling dimension lg6122 and are irrelevant in the bare theory Eq 18 and henceforth dropped The detailed W expression will be used later when we analyze phases neighboring the SBM The remaining exponentials only depend on the elds 6M and pm so that the charge and spin sectors decouple Since the rest of 01 is quadratic D z g 01 takes the same form as L8 in Eq 19 except with g0voagv where Ur AirWNW Rah7739 Hzquot2 2 4 41 8 01 v2 mfg702 f17r xgzqr 2 v gm 02 Wm xgzw mag7r 42 In the spin sector the remaining interactions are given by if E AgarRm JLmz 172JR1139JL22 JLll Jim 11 43 When we write this in the bosonization the JEJE pieces con tribute to the harmonic part of the action Vz ngtwwr 1960 44 3tltaltp w M M 1 45 4772 x10quot 20 gt510quot x203 while the JEJZJ JZ produce nonlinear potential V52 Azcosa ew 46 2Nsz cos2 60COS2ltp0 47 A oneloop renormalization group RG analysis gives the following ow equations m z W 2 N192 48 d6 2771211 d6 77vlvz39 When A A32 and A172 are positive they scale to zero and the quadratic SBM Lagrangian LEBM Eq 18 is stable We also require that the renormalized g is smaller than 1 so that the w f quot terms in Eq 37 remain irrelevant In Sec IV B we consider what happens when g gt 1 or when some of the N171 A372 and A172 change sign and become marginally relevant The above stability considerations are complete for ge neric incommensurate Fermi wave vectors At special com PHYSICAL REVIEW B 79 205112 2009 mensurations new interactions may be allowed and can po tentially destabilize the SBM Such situations need to be analyzed separately and in Secs IV D and IV E we consider cases relevant for the VBS3 and chirality4 phases found by the DMRG in the ring model Secs V A and V B We want to make one remark about the allowed interac tions which will be useful later Let us ignore for a moment the pinning of 6 for example let us think about the elec tron interactions in the approach of Sec II B Three of the eight 6 and p elds namely pm pm and 6 do not appear as arguments of the cosines and the action has continuous symmetries under independent shifts of these The rst two symmetries correspond to microscopic conservation laws for the total charge Q f19x6p and the total spin SZ f19k60 On the other hand the invariance under the shifts of 6 corre sponds to the conservation of XNR1 NL1 NR2NL2 f19kltpw where N P11 denotes the total number of fermions near Fermi point Pk This is not a microscopic symmetry but emerges in the continuum theory for generic kFl km lndeed writing the total momentum PNR1 NL1kF1 NRz NL2kF2XkF1 NR2 NL27r2 we see that any at tempt to change X violates the momentum conservation ex cept for special commensurate kFl We thus conclude that pm an pm can never be pinned by the interactions while 69 cannot be pinned generically except at special commen surate points B Capped paramagnets when ggt1 We now consider phases that can emerge as some insta bilities of the spin Bose metal We use heavily the bosoniza tion expressions for various observables given in Appendix A As we have already mentioned when ggtl the interac tions in Eq 37 are relevant and the SBM phase is unstable We can safely expect that as a result of the runaway ows the variables pk and 67 will be pinned The situation is less clear with the remaining parts of the potential since we can not pin simultaneously 6 7 and p0 Still it is possible that the situation is resolved by pinning one variable or the other For example depending on whether w fz and w l 2 have the same or opposite signs it is advantageous to pin 6 7 or pm If either pinning scenario happens there remain no gapless modes in the system It is readily established in the cases below that all spin correlations are short ranged ie we have fully gapped paramagnets also 87 at 0 in all cases so the translational symmetry is necessarily broken 1 wfzwf2gt0 and pinned p9 939 Period2 VBS Consider the case when the 60 is pinned Using Appen dix A we see that 3 obtains an expectation value while Em and MW are short ranged It is natural to identify this phase as a period2 valence bond solid shown in Fig 11 The pinning values and therefore some details of the state will differ depending on the sign of the coupling w fz but in either case the ground state is twofold degenerate Here and below when we nd pinning values of appropriate gas and 6 s mini mizing a given potential we determine which solutions are physically distinct by checking if they produce distinct phases mod 27739 in the bosonization Eq 5 More practi 20511214 SPIN BOSEMETAL PHASE IN A SPIN MODEL A39A VA VA VA VA XOXOXOXOZOXOXOXOXOXOX FIG 11 Valence bond solid with period 2 where thicker lines indicate stronger bonds To emphasize the symmetries of the state we also show second and thirdneighbor bond energies but details can be different in different regimes For example the VBS2 state in the K ng0 case has dominant rstneighbor dimerization On the other hand in the model with K ng111 1305 the putative VBS2 region between the Bethechain and SBM phases in Fig 16 has signi cant thirdneighbor modulation but only very small rst neighbor one cally following Ref 31 Sec IVEl the chiral fermion elds remain unchanged under page pmwn39 Rm La 91 gt emwme Qm where 6PM can be arbitrary integers This gives redundancy transformations for the pi 039 elds that we use to check if the minimizing solutions are physi cally distinct 2 wi zwi 2lt0 and pinned p9 Period4 structures Consider now the case when the pm is pinned We nd that either 672 in Eq A42 or MW in Eq A43 but not both obtains an expectation value Thus we either have a period4 VBS or a period4 structure in the chiralities Which one is realized depends on details of the pinning As described in Appendix A we work with the 1 eigen state of the operator F our Eqs A42 and A43 already assume this With this choice to minimize the potential in Eq 37 we require cos2ltp COS2 0 i l 49 Depending on the sign of w fz we have a w fz gt 0 COS2ltpF cos2 90 50 b w fz lt 0 cos2ltp cos2 90 51 a In this case 672 at 020 ie we nd period4 valence bond order Note that em can take four indepen dent values eme39 e 4e393 4 where we have as sumed that 99 is xed by Eq 14 To visualize the state we examine the corresponding contributions to the rst and secondneighbor bond energies 88mg coslt x I a 0 0 52 2 4 SBM2 cosltgxg 04gt 53 One can either use symmetry arguments or write out the microscopic hopping energies explicitly to x the phases as above see Eq A8 which generalizes to nth neighbor bond as B e39 Q26Q for Q at 7739 Each line also shows schemati cally the sequence of bonds starting at x0 for 12 74 The four independent values of oz correspond to four trans lations of the same VBS state along x The pattern of bonds is shown in Fig 12 where the more negative energy is as PHYSICAL REVIEW B 79 205112 2009 Amnnnnunv FIG 12 Top valence bond solid with period 4 suggested as one of the instabilities out of the spin Bosemetal phase Sec IV B 2a Thick lines indicate stronger bonds Bottom in the two leg triangular ladder drawing we see roughly independent sponta neous dimerization in each leg sociated with the stronger dimerization When viewed on the twoleg ladder this state can be connected to a state with independent spontaneous dimerization in each leg b Here we have 672gt0 X z at 0 ie period4 struc ture in the chirality x The pattern is xcoslt xo gt 54 There four independent are values of 007326 m4 1374 e e corresponding to four possible ways to reg ister this pattern on the chain The state is illustrated in Fig 13 When drawn on the twoleg ladder chiralities on the upward pointing triangles alternate along the strip and so do chiralities on the downward pointing triangles 3 Gapped phases in the spinon language With an eye toward what might happen in the 2D spin liquid it is instructive to discuss the above phases in terms of the spinons To this end we can rewrite the W term Eq 37 as follows W w fz wig4 P1112 Hc 55 wiz WE4f 21fL2fZlfR2 HCl 56 Here Pzzfg fzu flif2aT creates a Cooper pair in band a The preceding two sections can be then viewed as follows When w fzw fz gt 0 we minimize the rst line by pairing and condensing the spinons once everything is done we get the period2 VBS state On the other hand when w fzw f2 lt 0 we minimize the second line by developing expectation values in the particlehole channel Using 000000000000 FIG 13 Top chirality order with period 4 suggested as one of the instabilities out of the SBM phase Sec IV B 2b In the 1D chain picture the chirality pattern is given by Eq 54 x is associated with the Ia 1 xx1 loop and the arrows on the links show one gauge choice to produce such uxes in the spinon hopping Bottom in the twoleg triangular ladder drawing we see alternating chiralities on the up triangles along the strip and alter nating chiralities on the down triangles 20511215 SHENG MOTRUNICH AND FISHER fizifm fiifRz HC 21522579 Lszz a 57 we get either the period4 dimer or period4 chirality order depending on the sign of w fz C Nearby phases obtained when some of the jRjL interactions Eq 43 become marginally relevant Let us now assume g lt 1 so the singlet p sector is not a priori gapped We consider what happens when some of the couplings N171 N272 and N172 in Eq 43 change sign and be come marginally relevant We analyze this as follows Con sider the potential V Eq 47 again working with the 1 eigenstate of the operator If one or several of the cou plings becomes negative we have runaway ows Eq 48 to more negative values We then consider pinned eld con gurations that minimize the relevant part of the V Lithis is what happens in the spin sector Next we need to include the interactions Eq 37 with the singlet p sector since they can become relevant once some of the 039 elds are pinned We now consider different possibilities 1 ti 1gt0 xgzgt0 and xfzlt0 In this case only the A172 is relevant and ows to large negative values We therefore pin the elds 90 and pm To minimize V i the pinned values need to satisfy Eq 49 The spin sector is gapped and all spin correlations decay expo nentially we also have 87 7k 0 so the translational symme try is broken Next we include the interactions Eq 37 Using Eq 49 the important part is W 4w2 3w 172cos260cos2ltpg 58 The pg is dynamical at this stage but the 90 is pinned and the W now has scaling dimension 1 g The possibilities are the following a g lt 1 2 The Wterm is irrelevant and the singlet sector remains gapless One manifestation of the gaplessness is that Big2 and XI have power law correlations characterized by scaling dimension 1 4g Thus we have a coexistence of the static period2 VBS order and power law VBS and chirality correlations at the wave vectors in 2 b g gt 1 2 The W term is relevant and pins the eld pg leaving no gapless modes in the system Such fully gapped situation has already been discussed in Sec IV E 2 This gives either the period4 VBS or period4 chirality phase 2 Afllt0 xgzlt0 and xf2gt0 In this case the N171 and 272 are relevant and ow to large negative values while 172 is irrelevant Then both 910 and 620 are pinned and satisfy cos22610cos226201 The spin sector is gapped and all spin correlations are short ranged All correlations at 77 2 are also short ranged The translational symmetry is broken since 87 0 Including the interactions with the singlet sector as in Sec IV C 1 we have the followin a If g lt 1 2 the p sector remains gapless and 82km and 32sz have power law correlations with scaling dimension PHYSICAL REVIEW B 79 205112 2009 g 4 These coexist with the static period2 VBS order b If g gt12 we also pin pg and the situation is essen tially the same as in Sec IV E 1 This gives the fully gapped period2 VBS phase 3 Afllt0 xgzgt0 and xf2gt0 In this case only the A171 is relevant and pins 610 Spin correlations at 2kF1 and all correlations at 77 2 are short range We now include the interactions Eq 37 the important part is W 4w2 3w 172 cos2610 cos2620cos2 pg 59 Both the 2039 and p modes are dynamical at this stage and the W has scaling dimension 1 21 g a glt23 The W term is irrelevant and we have two gapless modes in this phase S B2kFZ have the same scaling dimension 1 2 g 4 as in the SBM phase while 82km has scaling dimension g 4 Furthermore Bhave scaling di mension 12 b g gt2 3 The W term is relevant pinning both 920 and pg This is the already encountered fully gapped period2 VBS state The case with N171gt0 A32lt0 and N172gt0 is considered similarly Finally in the case N172lt0 and either N171lt0 or N272lt0 we cannot easily minimize the potential equation Eq 47 since we have noncommuting variables under the relevant cosines We do not know what happens here although one guess would be that one of the relevant terms wins over the others and the situation is reduced to the already considered cases To summarize we have found several phases that can be obtained out of the spin Bose metal 1 fully gapped period2 VBS 2 and 3 fully gapped period4 phases one with bond energy patteni and the other with chirality pattern 4 period2 VBS coexisting with one gapless mode in the singlet p sector and power law correlations in Em MW 5 period2 VBS coexisting with one gapless mode in the singlet sector and power law correlations in 82k 82km and 6 phase with two gapless modes one in the spin sector and one in the singlet sector It is possible that some of the gapless phases will be further unstable to effects not consid ered here The above essentially covers all natural possibilities of gapping out some or all of the lowenergy modes of the generic SBM phase Thus as discussed at the end of Sec IV A we cannot pin pm because of the spin rotation invari ance The SU2 spin invariance also imposes restrictions on the values of the variables that are pinned these conditions are automatically satis ed in the above cases since our start ing interactions are SU2 invariant Furthermore we cannot pin 9 because of the emergent conservation of WWW One exception is when the Fermi wave vectors take special com mensurate values we discuss this next 20511216 SPIN BOSEMETAL PHASE IN A SPIN MODEL a II 11 1 11 b If 11 1 FIG 14 Color online Valence bond solid states with period 3 Thick lines indicate stronger bonds Remaining effective spin12 degrees of freedom are also s own Coexisting with the transla tional symmetry breaking we have 1x power law spin correlations with the antiferromagnetic dynamic pattern as shown The two cases have slightly different microscopics but are qualitatively simi lar on long length scales D Period3 VBS state as a possible instability of the SBM in the commensurate case with kF11Tl3 In the ring model Eq 1 the DMRG observes transla tional symmetry breaking with period 3 in the intermediate parameter range anked by the spin Bose metal on both sides Motivated by this we revisit the spinongauge theory in the special case with kFlzrr3 then kmz Srro Com pared to fermion interactions present for generic incommen surate Fermi wave vectors we nd one allowed term V6 6fi221fi221fimfmfmlfiz1a HC 60 4n cos2610sin3 19 9 61 The pinned 99 value is kept general at this stage The scal ing dimension is AV6l29g4 Let us study what hap pens when g lt2 3 and V6 becomes relevant so M6 ows to large values Then 6 7 and 610 are pinned while the conju gate elds ltpfg and p10 uctuate wildly There remains one gapless mode 620 that is still described by Eq 20 We can use the bond energy and spin operators to charac terize the resulting state First of all 82km develops long range order Since 2kF1 2277 3 we thus have a valence bond solid with period 3 Using Eqs A8 and A14 the micro scopic bond energy is 5130 cos2610sin2kplx kF1 19 199 62 In Eq 61 we write 39 F 99 3I9p g 469 and use the pinning condition on 469 Eq 17 There are two cases a M6 cos46m lt0 In this case V6 is minimized by in equivalent pinning values 2610 7739 w69776 776 2773 7764773 For 6w697r6 the period3 pattern of bonds is l l 5Bx 1 5 63 while the other two inequivalent pinning values give transla tions of this pattern along the chain A lower bond energy is interpreted as a stronger antiferromagnetic correlation on the bond Then the above pattern corresponds to dimerizing every third bond as shown in Fig 14a b M6 cos4 99 gt0 In this case V6 is minimized by ewzo 6w697r67r627r37r647r3 For 6 697r6 the period3 pattern of bonds is PHYSICAL REVIEW B 79 205112 2009 l l 88xl 2 2 64 while the other two inequivalent pinning values give transla tions of this along the chain This pattern corresponds to every third bond being weaker as shown in Fig 14b Continuing with the characterization we note that 52km and all operators at 77 2 have exponentially decaying corre lations On the other hand 5392sz and 32sz have lx power law correlations because of the remaining gapless 620 mode Since 2kmzw 3 we have period6 spin correlations on the original lD chain The physical interpretation is simple Consider rst Fig 14a where every third bond is stronger A caricature of this state is that spins in the strong bonds form singlets and are effectively frozen out The remaining free spins are sepa rated by three lattice spacings and are weakly antiferromag netically coupled forming a new effective lD chain Thus we naturally have Bethechainlike staggered spin and bond en ergy correlations in this subsystem which coexist with the static period3 VBS order in the whole system The situation in Fig 14b where every third bond is weaker is qualita tively similar Here we can associate an effective spin 12 with each threesite cluster formed by strong bonds These effective spins are again separated by three lattice spacings and form a new weakly coupled Bethe chain Note that while the theory analysis has the Fermi wave vectors tuned to the commensuration the resulting state is a stable phase that can occupy a nite region in the parameter space as found by the DMRG in the ring model see Sec V A We can construct trial wave functions using spinons as follows In the mean eld we start with the band parameters t1 and t2 such that kF17T 3 and then add period3 modula tion of the hoppings The kFl Fermi points are connected by the modulation wave vector and are gapped out The km Fermi points remain gapless just as in the Bethechain case the corresponding bosonized eld theory provides an adequate description of the longwavelength physics pre dicting lx decay of staggered spin and bond energy corre lations The above wave function construction and theoretical analysis are implicitly in the regime where the residual spin correlations are antiferromagnetic In a given physical sys tem forming such a period3 VBS one can also imagine ferromagnetic residual interactions between the nondimer ized spins Indeed the DMRG nds some weak ferromag netic tendencies in the ring model near the transition to this VBS state This is not covered by our spinsinglet SBM theory but could possibly be covered starting with a partially polarized SBM state E Other possible commensurate points Alerted by the period3 VBS case we look for and nd one additional commensurate case with an allowed new in teraction that can destabilize the spin Bose metal When kF13778 we nd a unique quartic term V4 M4Lfi211fi211 a lemfR25 R H L HCi 20511217 SHENG MOTRUNICH AND FISHER K J11J22J80L96 ring 20 A DMRG q IrIaIVBSVS U 06 15 V 30 E 10 2 3 6 E 1 05 C 395 m in 00 0 7113 Trz T q DMRG 40 7 IrIaIVBSVS Dimer structure factor ltBq qugt FIG 15 Color online Spin and dimer structure factors at a representative point in the VBS3 phase K ng111 122 mea sured in the DMRG for system size L96 we do not show the chirality as it is not informative The most notable features are the dimer Bragg peak at 2773 corresponding to the static VBS order and also the spin singularity at 773 corresponding to the effective spin12 chain formed by the nondimerized spins see Fig 14 The trial VBS3 wave function is constructed as described in the text i771T 7m shrew 69 0sinltpu 4 i77u 772i sin2i9 99 60sinltpw pm 65 For the schematic writing here we have ignored the com mutations of the elds when separating the gas and 6 s The scaling dimension is AV4 l2gl4g This is smaller than 2 for g E 0191 1309 and the interaction is relevant in this range Since we have conjugate variables 9 and p both present in the above potential we cannot easily deter mine the ultimate outcome of the runaway ow It seems safe to assume that 90 and p0 will be both pinned which im plies at least some period2 translational symmetry breaking One possibility perhaps aided by the interactions Eq 37 is that the pp is pinned in this case the situation is essen tially the same as in Sec IV B 2 and we get some period4 structure Another possibility is that the 9 is pinned in this case 4kF1gt 7k 0 and since 4kF1 7r2 we get period4 bond pattern To conclude we note that the commensurate cases in this section and in Sec IV D can be understood phenomenologi cally by monitoring the wave vectors of the energy modes PHYSICAL REVIEW B 79 205112 2009 Krlng J1 1J3 05 varying J2 L96 n r m 1 er E c a 3 E ED 5 3 n2 7 gt 0 1 CL CL gt E E a c m 7c3 g E E U 53 5 MJ E i z m E Eethe j SBM 0 VBS3 SBM VBS2 w 0 1 2 3 4 J2J1 FIG 16 Color online Phase diagram of the JI39JZ39Kring model with additional antiferromagnetic thirdneighbor coupling J3 051 introduced to stabilize spinsinglet states The study is along the same cut KringJl as in Fig 8 and the overall features are similar with the following differences The ground state is singlet throughout eliminating the dif cult partial FM region to the left of the VBS3 and the VBS3 phase is somewhat wider There is a sizable spingapped region between the Bethechain and SBM phases see text for more details A new spingapped phase with period4 chirality order appears inside the SBM to the left of the VBS3 BQ The dominant wave vectors are i2kFa irr2 and i4kF2 14km When 4km matches with 72 we get the kF1377 8 commensuration of this section here also 2kF1 matches with kFZ kFl while 2km matches with 3kF1kF2 When 4km matches with 2kF1 we get the kFlzrr 3 com mensuration of Sec IV D Tracking such singular wave vec tors in the DMRG is then very helpful to alert us to possible commensuration instabilities and both cases are realized in the ring model with additional antiferromagnetic 1305J1 discussed in Sec V B cf Fig 16 V DMRG STUDY OF COMMENSURATE INSTABILITIES INSIDE THE SBM VBS3 AND CHIRALITY4 A Valence bond solid with period 3 As already mentioned in Sec III we nd a range of pa rameters where the SBM phase is unstable toward a valence bond solid with a period of 3 lattice spacings VBS3 In the model with Km J1l this occurs for l5ltJzlt25 cf Fig 8 The characteristic correlations are shown in Fig 15 at a point 122 The dimer structure factor shows a Bragg peak at a wave vector 277 3 corresponding to the period3 VBS order The spin structure factor has a singularity at a wave vector 773 corresponding to staggered correlations in the effective spinl2 chain formed by the nondimerized spins see Fig 14 If we zoom in closer the dimer structure factor also has a feature at 77 3 that can be associated with this effective chain To construct a trial VBS3 wave function we start with the spinon hopping problem that would produce km 277 3 so the rst Fermi sea would be twice as large as the second We then multiply every third rstneighbor hopping by 1 8 and Gutzwiller project for the point in Fig 15 we nd optimal 20511218 SPIN BOSEMETAL PHASE IN A SPIN MODEL 81 This gaps out the larger Fermi sea but leaves the smaller Fermi sea gapless Our wave function is crude and shows a stronger dimer Bragg peak than the DMRG and somewhat different spin correlations at short scales but oth erwise captures the qualitative features as can be seen in Fig 15 The origin of the VBS3 phase can be traced to the insta bility of the SBM at special commensuration cf Secs IV D and IV E Indeed in Fig 7 we can follow the evolution of the singular wave vectors in the SBM phase between the VBS2 and VBS3 As we decrease JZ moving toward the VBS3 the 4km and 2kF1 singular wave vectors in the bond energy approach each other and coincide at 277 3 When this happens there is a unique umklapp term that can destabilize the SBM and produce the VBS3 state as analyzed in Sec IV D The instability requires g lt2 3 for the SBM Luttinger parameter In this case according to Table I the 4km singu larity in the dimer is stronger than the 2kF172 which is in agreement with what we see in the neighboring SBM in Fig 7 The reemergence of the 4km and 2kF1 at the other end of the VBS3 phase is obscured here by the weak ferromagnetic tendency but is present in a model where this tendency is suppressed see Fig 16 B Enhancement of the spinsinglet SBM by antiferromagnetic thirdneighbor coupling 1305 and a new phase with chirality order with period 4 As discussed in Sec III C in the original JlJzng model states in the SBM region near the left boundary of the VBS3 tend to develop a small magnetic moment We con jecture that this occurs in the second spinon Fermi sea and suggest that an antiferromagnetic J3 will stabilize the SBM phase with spinsinglet ground state One motivation comes from the picture of the neighboring VBS3 where the non dimerized spins are loosely associated with the second Fermi sea These s ins are three lattice spacings apart so adding antiferromagnetic 13 should lead to stronger antiferromag netic tendencies among them and also in the physics associ ated with the second Fermi sea We have performed a detailed study adding a modest J3 205 to the original model Eq 1 along the same cut Km J11 Our motivating expectations are indeed bonie out Figure 16 shows the phase diagram together with the evolution of the singular wave vectors as a function of 1211 While the overall features are similar to the phase diagram in the 130 case Fig 8 a few points are worth mentioning First the partial spin polarization is absent in the whole SBM phase between the Bethechain and VBS3 phases The DMRG converges con dently to spinsinglet ground state for L296 All properties are similar to those in Figs 4 and 6 providing further support for the singlet SBM phase in the original 130 model The VBS3 phase and the SBM phase between the VBS3 and VBS2 are qualitatively very similar in the two cases 130 and 13205 and are not discussed further here An interesting feature in the model with 13205 is the presence of a sizable phase with spin gap intervening be tween the Bethechain and SBM phases Our best guess is PHYSICAL REVIEW B 79 205112 2009 that this phase has period2 VBS order although we do not see a clear signature in the dimer correlations Our DMRG states in this region show very weak if any dimerization of the rstneighbor bonds which may explain the lack of clear order in these dimer correlations On the other hand we see a sizable period2 dimerization of the thirdneighbor bonds 50 x3 but have not measured the corresponding bondbond correlations to con rm longrange order We have not explored possible theoretical routes to understand the origin of this phase yet and leave our discussion of this re gion as is We now turn to one more new phase found in the 13 205 model In a narrow region inside the SBM phase not far from the left end of the VBS3 we again nd a spingapped phase We identify this as having period4 order in the chiral ity chirality4 phase Figure 17 presenm point 12215 J3 205 Looking at the singular wave vectors in Fig 16 we see that this point occurs roughly where qbw2km passes 77 4 Analysis in Sec IV E suggests an instability gapping out all modes and leading to some period4 structure Indeed the DMRG spin and dimer structure factors show only some remnanm of features near 2kF1 2km while the chirality shows a sharp peak at 77 2 Looking at realspace correla tions our L296 DMRG state breaks translational symmetry The pattern of chirality correlations is consistent with the period4 order shown in Fig 13 The pattern of dimer corre lations is also consistent with this picture and shows modu lation with period 2 which can be seen as a feature at 7739 in the dimer structure factor in Fig 17 To summarize with the help of modest 13205 we have stabilized the spinsinglet SBM states between the Bethe chain and VBS3 phases By suppressing potential weak fer romagnetism we have uncovered the chirality4 phase which can be understood as arising from the instability of the SBM at the special commensuration discussed in Sec IV E VI ATTEMPT TO BRING OUT PARTIALLY MAGNETIZED SPIN BOSEMETAL BY FERROMAGNETIC THIRD NEIGHBOR COUPLING J3 05 In the original model Eq 1 we do not have a clear understanding of the partial FM states to the left of the VBS3 phase in Fig 2 As discussed in Sec III C we suspect that there is a tendency to weak ferromagnetism in the sec ond Fermi sea However the magnetizations that we measure are not large eg they are signi cantly smaller than if we were to fully polarize the second Fermi sea and it is dif cult to analyze such states Here we seek better control over the spin polarization by adding modest ferromagnetic 132 05 in hopes of stabilizing states with a full spontaneous polar ization of the second Fermi sea which is easier to analyze We do not have a detailed phase diagram as for the J3 0 and 13205 cases We expect it to look crudely similar to Figs 8 and 16 with a narrower if any VBS3 region and with a wider partially polarized SBM region We indeed nd stronger ferromagnetic tendencies in the range 031232 However we cannot claim achieving robust full polarization of the second Fermi sea and understanding all behaviors The largest magnetization and properties closest to our expecta 20511219 SHENG MOTRUNICH AND FISHER KW J11J2 15J8 05 L96 DMRG Spin structure factor ltSq Slqgt 0 KM 7112 37d4 n DMRG 39 Dimer structure factor ltBq qugt 010 DMRG L96 DMRG L48 quot9 005 000 Chirality structure factor ltxq xlqgt 0 KM Tr2 37114 n q FIG 17 Color online Spin dimer and Chirality structure fac tors at a point in the tentative Chirality4 phase K ng2ll21 12 215 and 13205 measured in the DMRG for system size L296 Note the absence of sharp features in the spin correlations which suggests a spin gap while the dimer correlations have only a feature at 77 corresponding to period2 modulation On the other hand the Chirality structure factor shows a Bragg peak at 77 2 that grows with increasing system size The pattern of Chirality correlations in real space is consistent with the order shown in Fig 13 tions are found in the middle region near 12 1 With this cautionary note warranting more work we now present re sults for 12 1 to illustrate our thinking about such states The DMRG study proceeds as follows We start as before working in the Sfot20 sector We measure the spin structure factor and calculate the total spin Smt from Eq 32 which can give a rst indication of a nonzero magnetization How PHYSICAL REVIEW B 79 205112 2009 ever for the larger system sizes the DMRG nds it dif cult to converge to integervalued Smt due to a mixing of states with different total spins and this leads to signi cant uncer tainty To check the value of the groundstate spin we run the DMRG in sectors with different Sfot expecting the groundstate energy to be the same for Sfot20 Stot and then to jump to a higher value for SfotgtStot As an example at a point 1221 for system size L248 we nd that the DMRG energy is the same in the sectors Sfot 20 5 and then jumps so the groundstate spin is deter mined as Stot25 The DMRG convergence is good and the SU2invariant structure factor SqSq is the same mea sured in the different sectors SfotSS indicating that these states indeed belong to the same multiplet The situation is less clear for L296 because of reduced convergence At the point 1221 the extensive energies ob tained by the DMRG in the sectors Sfot2010 are non systematic and are within 0111 of each other with the lowest energy found in the Sfot210 sector Also the SU2invariant structure factors differ slightly and the estimates of Smt vary around Stot 8710 The convergence is best in the highest Sfot210 sector where the total spin is found to be accurately 510210 the improved convergence is indeed expected since there are fewer available lowenergy excited states to mix with eg spinwave excitations of the ferromagnet are not present in the highest SZ sector Interestingly we nd a fully converged state in the Sfot2ll sector with 510211 whose energy is only slightly higher which probably adds to the above convergence dif culties More importantly the energy jumps to a signi cantly higher value in the sector Sfot2l2 Our best conclusion is that the total spin of the ground state is 510210 Turning to the VMC study we consider a family of varia tional Gutzwiller states where we allow different spin up and spin down populations of the two Fermi seas centered around k20 and 7739 we do not attempt any further improvements on top of such bare wave functions In the model parameter region discussed here we nd that the optimal such states have a fully polarized second Fermi sea and an unpolarized rst Fermi sea For the L248 example quoted above the optimal VMC polarization indeed matches the DMRG Smt 25 while for the L296 case the optimal VMC state has 510211 and a state with 510210 is very close in energy Appendix B 3 provides more details on the properties of such Gutzwiller states while here we simply compare the VMC and DMRG measuremenm The DMRG structure factors for the 1221 132 05 and L296 systems are shown in Fig 18 together with the VMC results for the Gutzwiller wave function with 510210 A no table difference from the singlet SBM states of Sec III is that the characteristic peaks are no longer located symmetrically about 77 2 For example in the trial state we have prominent wave vectors 2km and 2kF1 that satisfy 2km2 gtlt 2kF1227r also we have a wave vector km kF1 which is now differ ent from 77 2 It is not easy to discern all wave vectors in Fig 18 because the 2km happens to be close with the km kF1 Nevertheless the overall match between the DMRG and VMC suggests that the trial wave function captures rea sonably the nature of the ground state 20511220 SPIN BOSEMETAL PHASE IN A SPIN MODEL ngJ1J21J3 o596 16 2 DMR sector Sm 10 1A GutzwrllerNm38 N11 N2 20 N210 12 10 08 06 04 Spin structure factor ltSq Slqgt 02 k k 00 F2 F1 0 2km 2km TE q M seclor 50510 03 Gutzwrller wasurfeeu aou 10 02 Dimer structure factor ltBq qugt 39kFZ39kF1 o 2kF2 2kF1 r q FIG 18 Color online Spin and dimer structure factors at a tentative point with partial ferromagnetism K ngl 11 121 and J3 05 measured in the DMRG for system size L96 The cal culations are done in the sector Sfm10 where the DMRG is well converged and gives Stol10 which we think is the true ground state spin The VMC state has the second Fermi sea fully polarized with N2T2O Nu0 while the rst Fermi sea is unpolarized with N U N 1 138 Vertical lines label important wave vectors 2kF1 2km and n F1 Working in the sector SfotzStot also allows more detailed comparison between the DMRG and VMC In this case there is a sharp distinction between the 531 and ngl structure factors The former has singular wave vectors kFI 2km and km kF1 while the latter is lacking the wave vec tor 2km since there is no spin ip process across the second Fermi sea Our measurements are shown in Fig 19 The wave vectors 2km and km kF1 are too close to make a more clearcut distinction nevertheless the VMC reproduces all details quite well In analogy with the SBM theory we expect the 2kF1 and 2km singularities to become stronger compared with the bare Gutzwiller and the km kF1 to become weaker this is roughly consistent with what we see in the DMRG structure factors As discussed in Appendix B 3 however we do not have a complete description of such a partially polarized SBM phase that must incorporate ferromagnetic spin waves as well as the lowenergy SBM modes This is left for future work We also mention that the closeness of the km kF1 warns us that the system is near a commensuration point with kF127r5 where it can be further unstable which requires more study PHYSICAL REVIEW B 79 205112 2009 ng J1 1J2 1J305 L96 0 a emul 0 10 Nsz N250 u In Gutzwrller Nm38 NW 0 0 0 0 b 01 gt structure factor 1 z ltSq Slq 0 0 A m 0 o 0 a DMR seclors zo 10 Gutzwrllerwrpee NW 8 N2 2O N250 v 0 uI 0 J 0 N lt83 82gt structure factor 0 A 0 0 2kg Tr q FIG 19 Color online Separate szSiq and S252 structure factors for the same system as in Fig 18 To summarize by adding modest ferromagnetic 132 05 we have realized the SBM state with fully polarized second Fermi sea con rming our intuition about the origin of the weak ferromagnetic tendencies in the original model dis cussed in Sec more thorough exploration of the hase diagram in the model with 132 05 as well as in the original model in the partial FM region is clearly warranted to develop better understanding of such partially ferromag netic states VII CONCLUSIONS AND FUTURE DIRECTIONS We have summarized the main results and presented much discussion particularly in Sec I Perhaps one point we would like to reiterate is the remarkable coincidence between the sign structure present in the DMRG wave functions for the spin model SBM phase and the sign structure in the spin sector of free fermions on the ladder eg metallic elec trons This sign structure is encoded in the singular wave vectors Bose surfaces and indeed the Gutzwiller projected wave function with just one variational parameter is suf cient to reproduce the locations of all the singularities throughout the observed SBM phase We conclude by mentioning some standing questions and future directions First in the ring model we have focused on the spin Bose metal and dealt with other phases only as needed to sketch the rich phase diagram For example we have not studied carefully the VBS2 region which might 20511221 SHENG MOTRUNICH AND FISHER harbor additional phases We have not studied adequately numerically or analytically the spingap region between the Bethechain and SBM phases in the 13205 model Fig 16 One question to ask here is whether there is a generic insta bility when we start populating the second Fermi sea or whether we can go directly from the Bethechain phase to the SBM More generally we have not studied various phase transitions in the system Next while we understand the longwavelength SBM theory with its single Luttinger parameter g we have found that the Gutzwiller wave functions represent only the special case gl and cannot capture the general situation g lt1 It would clearly be desirable to construct spinsinglet wave functions appropriate for the general case Even thinking about the Gutzwiller wave functions it could be interesting to understand the observed g1 analytically and ask if they may be exact ground states of some Hamiltonians in the spirit of the HaldaneShastry model4zv43 On a separate front we have encountered an interesting possibility of the spin Bose metal with partial ferromag netism occurring in one of the subbands but more work is needed to fully understand the numerical observations and develop analytical theory Even without a spontaneous mo ment in the ground state our observations suggest that in the regime between the Bethechain and VBS3 phases the sec ond band is narrow in energy Some ferromagnetic instability or possibility of spinincoherent regime in one of the bands can lead to anomalous transport properties in such a quantum wire a topic of much current interest2144 46 Looking into future it will be interesting to consider elec tron Hubbardtype models on the twoleg triangular strip and look for possible SBM phase The Hubbard model has been studied in a number of works47 50 but the focus has been mainly on the conventional insulating phases such as the Bethechain and VBS2 states This is appropriate in the strong Mott insulator limit t1 t2lt U where the effective spin model is the 1112 model with J12t U and JZ 2t3 U However at intermediate coupling just on the insu lator side one needs to include multiplespin exchanges and the leading new term is the ring exchange with Kng 220 U3 As we have learned this ring term stabilizes the SBM phase so revisiting the Hubbard model with the in sights gained here is promising In Sec II B we approached the SBM by starting with a metallic twoband electron sys tem C282 and gapping out only the overall charge mode p Then it is natural to look for the SBM near an ex tended such C2S2 metallic phase and we may need to con sider electron models with furtherneighbor repulsion to open wider windows of such phases Last but not least we would like to advance the program of ladder studies closer to 2D It is prudent to focus on the spin model with ring exchanges On the exact numerics front four to six legs are probably at the limit of the DMRG capabilities The VMC approach should still be able to cap ture the critical surfaces if they are present since they are dictated by shortdistance physics on the other hand the bare Gutzwiller will likely fail even more in reproducing correct longdistance behavior We do not know how far the present bosonization approach can sensibly hold going to more legs These are challenging but worthwhile endeavors PHYSICAL REVIEW B 79 205112 2009 given the experimental importance of understanding weak Mott insulators ACKNOWLEDGMENTS We would like to thank L Balents HH Lai T Senthil and S Trebst for useful discussions This work was sup ported by DOE under Grant No DEFG0206ER46305 DNS the National Science Foundation through Grants No DMR0605696 DNS and No DMR0529399 MPAF and the A P Sloan Foundation OIM DNS also thanks the KITP for support through NSF under Grant No PHY0551164 APPENDIX A OBSERVABLES IN THE SBM PHASE We have de ned spin x bond energy Bx and chiral ity xx observables in Sec IIC cf Eqs 22 and 23 Here we nd detailed bosonized forms by a systematic con struction of observables in the SBM and will nd more observables on the way In the gauge theory treatment Sec II A we consider gaugeinvariant objects constructed from the spinon elds in the interacting electron picture of Sec IIB these are opera tors that do not change the total charge We begin with fermion bilinears and rst consider the ones composed of a particle and a hole moving in opposite directions Such bilinears are expected to be enhanced by gauge uctuations since parallel gauge currents experience Amperean attraction We organize these bilinears as follows 1 Q 52km E Efimo39aafiewa A1 7 1 52k E EflmfRam A2 1 i l 1 572 E Efizmo39asfm Efizzao39apfmga A3 1 1 572 E Efierasza Efizszer A4 1 Q l 1 572 E Efizmo39aafmg Efizzao39agfriga A5 1 1 Xwz E Efiuasza Efizszer A6 with S7QSl etc The microscopic spin operator expanded in terms of the continuum fermion elds readily gives the listed SQ The bond energy can be approximated as the spinon hop ping energy Bx thLxfax l Hc recall that we work in the gauge with zero spatial vector potential Expansion in terms of the continuum elds gives up to a real factor A7 20511222 SPIN BOSEMETAL PHASE IN A SPIN MODEL BQ N e39QZEQ A8 Such connection between BQ and EQ is understood below for all Qa n39 The objects 6Q are convenient because of their simpler transformation properties under lattice inversion 6Q lt gt fiQ The physical meaning of the operators 372 and XW can be established on symmetry grounds Thus X is the spin chirality de ned in Eq 23 The expression in terms of bi linears can also be found directly by considering the circula tion of the gauge charge current around the x 1xx1 loop m E 2m rumpus A 110 A9 This is familiar in slave particle treatments14 the circulation produces internal gauge ux whose physical meaning is the spin chirality On the other hand 8 is related to the following microscopic operator 50 x x 51 1 5 WSW A10 At Q7r2 this enters on par with S B and x The bosonized expressions at the 2km are 53 ind mie eet39 slum A11 51 i n mew wemw com A12 53 et wet Jr sin29W A13 22km ie39f et39fw cos29W A14 where the upper or lower sign in the exponent corresponds to 4121 or 2 The pinned value 99 which is determined by minimizing Eq 17 is left general at this stage It is not so important for the qualitative behavior at the 2km and 772 but is crucial at a wave vector 7739 later For each a the S B2kFa structure is similar to that in a single Bethe chain except for the 99 exponentials As in the Bethe chain we expect the spin and VBS correlations to be closely relatediin particular they decay with the same power law The corresponding scaling dimension in the xedpoint theory Eq 18 is 1 Am A1821 5 3 A15 The bosonized expressions at the 77 2 are as follows 5372 hf31 177117721515 5111013 430 A16 4771177216 111M 001 A17 Sly2 571501 l771177215715 COSWA 430 A18 l7711772165 0050a 0017 A19 5172 ei ptl i 771 7721645 Sinwk 430 A20 PHYSICAL REVIEW B 79 205112 2009 177117721815 Sinww 07 A21 672 EWI i771177216 11101 427 A22 177117721515 Sinww 07 A23 Xwz mg31 771177215715W COSWA 430 A24 77117721515 00543 001 A25 Expressions for 872 can be obtained from those for SW essentially by interchanging sines and cosines As before Em is given by Eq A8 The above details are needed particularly when we discuss phases arising as instabilities of the SBM Secs IV B7IV E while in the SBM we immedi ately see that all scaling dimensions are equal 1 1 AISwzl AIBwzl AlDwzl ADM21 2 4g A26 This completes the enhanced bilinears We also men tion without giving detailed expressions nonenhanced bi linears at wave vectors Q i km kn Their scaling di mension is A159 A189 A15Q1 Aug i i 3 A27 which is always larger than the spinon mean eld value of 1 Finally we have bilinears carrying zero momentumi essentially J PM J PM from Eq 34 These give conserved densities and currents and have scaling dimension 1 We spe ci cally mention examples leading to Eqs 25727 l SZQ0 N 211 JZLll Ji222 1222 5xlt90a A28 2 6Q0 N 11211 JL11 11222 JLZZ Tlgxepra A29 2 XQ0 N 11211 JL11 11222 JLZZ 7 7 Xltpp39 A30 One way we can make the identi cations in the last two lines is by using physical symmetry arguments ar we have only considered fermion bilinears Since the theory is strongly coupled we should also study contri butions with more fermion elds We now include four fermion terms focusing on the spin bond energy and chiral ity operators that are measured in the DMRG First there appears a new wave vector 4kF1 4km in the bond energy via 125 fAkFlifinfiilfRnfRn N 5125 A31 20511223 SHENG MOTRUNICH AND FISHER fieztfimemez N Magi A32 The two contributions come with independent numerical fac tors and can also be generated as 2kF12 and 62C 2 Once the 99 is pinned there is only one qualitatively distinct con tribution and the scaling dimension is MBA F11 8 Note that for suf ciently small g lt 23 the power law decay is slower than that of the bilinears 82km There is no compa rable 4kF1 contribution to the spin operator ourfermion terms bring out another important wave vector Q2771 We list independent dominant such contribu A33 tions to S7 B7 and X7 5 sin2 60sin2 p sin290sin299 A34 l3 cos2 90 cos290sin299 A35 cos2 90 f cos2ltpUsin299 A36 f cos2ltpwsin299 A37 X f sm2ltpsm2ep A38 These can be generated by combining the previously exhib ited I bilinears as follows 5S kF162kn39S kFZEZkF1Hc B z62kF152kF2Hc z 2 amp72Hc z wz 772Hc and X7 XmemHc The scaling dimensions are AS AtBi1 A39 AIM 1g A40 The above observables are present if sin299 0 eg if the 99 is pinned as in Eq 14 which we argued is natural when the spin model is describing a Mott insulator phase of a repulsive electron model On the other hand the above contributions would vanish if the pinning potential equation Eq 17 had vglt0 Some other physical observables con taining cos2 99 and having different symmetry properties would be present instead We do not write these out since both the DMRG and the trial wave functions have signatures in the spin VBS and chirality at the wave vector 7739 suggest ing that the pinning equation Eq 14 is realized We have identi ed several more unique observables at 7739 containing sin2 p we do not spell these out here since our primary focus is to understand features in the numerics measuring the familiar S B and x We nally mention that fourfermion terms produce still more wave vectors 3kF1kF2 13kmkF1 for ex ample 3kF1kF2 can be obtained by combining 2kF1 and 7739 2 The scaling dimensions are the same as at ikm kF1 Eq Am For completeness we have also checked sixfermion and eightfermion terms The only new wave vectors where the scaling dimension can be smaller than 2 are 6kFa2kFa 4kFa scaling dimension l29g4 and 8de4de4de PHYSICAL REVIEW B 79 205112 2009 scaling dimension 4g However one needs small g for these to become visible and in any case they always have faster power law decay than at 2km and 4km Finally entries listed as subd in Table I can be constructed eg as XQ Xer which has scaling dimensions 1Ae Table I summarizes our results for the correlations in the spin Bosemetal phase In words we expect dominant spin and VBS correlations at the wave vectors i2kF1 2km de ca ing as lx1g2 and at the wave vectors 772 decaying as lx112g The former decay is more slow since stability of the phase requires g lt 1 Note that the wave vectors 2kF1 and 2km are located symmetrically around 77 2 We also expect a 12 power law at the wave vectors 0 and 7739 Next at the wave vectors ikm kF1 and i3kF1km which are also located symmetrically around i 77 2 we expect a still faster power law 1 x11Zgg2 Furthermore the bond energy shows a power law lng at 4kF1 The spin chirality has similar signatures to the above at i 77 2 0 ikF2 kF1 and 3kF1kF2 but decays as lxZg at 7739 These are the sim plest observables that can be used to identify the SBM phase in a given system Figure 21 shows measurements in the Gutzwiller wave function projecting two Fermi seas and nicely illustrates all singular wave vectors while it appears that such wave functions realize a special case with g 1 We also remark that in the general SBM the presence of the marginally irrelevant interactions Eq 43 will lead to loga rithmic corrections in correlations 1 We conclude by describing our treatment of the Klein factors see eg Ref 32 for more details We need this when determining order parameters of various phases ob tained as instabilities of the SBM Secs IV BilV E The operator F from Eq 40 has eigenvalues i1 For concrete ness we work with the eigenstate corresponding to 1 We then nd the following relation lt77n772gtlt771772gtPure imagmmY A41 This is useful when discussing observables at the in 2 wave vectors for example 594 7711772 gtcoswysinwmkin pa A42 572 I i SinltPFCOSlt90COSltPaJl X Z 8quotplt W71 772 WWW COSlt90COSltP07 i SillltPFSiI1ltSiI1ltPail A43 APPENDIX B DETAILS OF THE WAVE FUNCTIONS 1 Gutzwiller projection of two Fermi seas It is convenient to view the spin wave function as that of hardcore bosons where updown spin corresponds to presentabsent boson In the general spinon construction we occupy k1 l NT orbitals with spin up and kj 1 N orbitals with spin down NT NL is the size of the system After the Gutzwiller projection the boson wave function is 20511224 SPIN BOSEMETAL PHASE IN A SPIN MODEL FIG 20 View of the wave function constructed by lling k states ofs 39nons Here momenta k27m n onn a closed circle Each lled dot is occupied by both spin up and spin down producing spin singlet The projected wave function remains unchanged if all momenta are shifted by the same amount Only the relative con guration matters and here we show symmetric con guration of the two Fermi seas separated by L4 unoccupied k states on either side This is our bare Gutzwiller wave function for the spin Bose metal Irmasz39 1 N detrelkJquetreM B1 where the set pj1NT is a complement to kjij 1 Ni in the Brillouin zone BZ The momentum car ried by this wave function is Ejv11kjpj22j11kl2jii1kL EqEBZq In particular we see that the wave function re mains unchanged if we shift all occupied spinon momenta by the same integer multiple of 277 L We now consider the spinsinglet case when NTzNi L2 k7ki Here L is even and all momenta are integer multiples of 277 L so Equzqzrr For convenience we as sume that L is a multiple of 4 Figure 20 illustrates lled k points for the band in Fig 3 We have two Fermi seas of volume N1 and N in the symmetric con guration ie sepa rated by L 4 unoccupied orbitals on each side around the Brillouin zone Relating to the band Fig 3 the larger N1 corresponds to occupied k points centered around 0 while N2 corresponds to points around 7739 As already noted a solid shift of the occupied states leaves the wave function un changed We can then specify such symmetric state as N1N2 which requires only one parameter since N1N2 NT N izL 2 We can readily verify that such eveneven states carry momentum 0 and are even under site inversion operation while oddodd states carry momentum 7739 and are odd under inversion The relative wave vectors connecting the Fermi points are gauge independent and are observed in various quantities see Fig 21 Speci cally using variational Monte Carlo 40 we measure the spin structure factor and see dominant sin gularities at the wave vectors i2kF1 i2km and ikF1 km i 77 2 which connect Fermi points with opposite group velocities By studying sizes up to L512 and per forming scaling analysis at these wave vectors the singulari PHYSICAL REVIEW B 79 205112 2009 Bare Gutzwiller L256 N1N2 10424 06 Spin structure factor ltSq Slqgt 0 2kF23kF1kF2 712 sz39kF12kF1 7E q 050 040 030 020 r 010 Dimer structure factor lt8q qugt O o 020 4kF2 0 2kF2 3kF1kF2 712 q 030 sz39kF1 2km 7E 0 N n 900 N 0010 lt50 00 no Chirality structure factor ltXq xlqgt O O D 0 3kF1kF2 712 sz39kF1 7E q FIG 21 Color online Spin bond energy and chirality struc ture factors in the bare Gutzwiller wave function with two Fermi seas N1N210424 on the 1D chain of length L256 The expected singular wave vectors are marked by vertical lines here kg and kn are de ned as in Fig 3 and all indicated wave vectors are mod 277 The structure factors are symmetric with respect to qH q and we only show 0541577 The character of the singu larities is consistent with the special case g1 in the SBM theory of Sec II ties appear to have the same power law This is consistent only with the special case g1 in the SBM theory cf Eqs A15 and A26 and Table I Our direct estimates of the scaling dimensions are also consistent with the value A 23 4 expected in this case Turning to other less singular wave vectors we clearly see V shaped 18q features at 0 and 7739 corresponding to scaling dimension 1 which is ex 20511225 SHENG MOTRUNICH AND FISHER pected generally We also see iUQQ kpl with scaling di mension 1 which requires g 1 We next consider VBS correlations and see all of the above wave vectors but we cannot quantify the singularities as accurately The VBS correlations also show singularities at 3kF1kF2 and i4km 14km the former is also ex pected in the spin structure factor but is not visible there probably due to amplitude effect while the 4km is ex pected only in the bond energy Finally we measure spin chirality correlations and see dominant singularity at irr 2 consistent with A3 4 We also see singularities at wave vectors 0 7739 km kn and i3kF1kF2 consistent with A21 again as expected in the special case g The above appears to hold for a range of relative popula tions of the Fermi seas away from the limiting situations of a single or two equal Fermi seas We are then led to conjec ture that such spinsinglet wave functions with two Fermi seas have correlations given by the SBM theory of Sec II and Appendix A with g 1 This conjecture is natural since in the theory the parameter g depends on the ratio of the two Fermi velocities cf Eq 21 while the wave function knows only about the occupiedunoccupied states and does not contain the band energy parameters We leave proving this conjecture analytically as an open problem Given the preceding discussion it appears that such bare Gutzwiller wave functions cannot capture fully the properties of the generic spin Bose metal as described by the theory of Sec II with general g lt1 It is possible that they are appro priate wave functions for some critical end point of the SBM phase where the parameter g 1 eg for the transitions out of the SBM discussed in Sec IV E While the energetics study with the bare Gutzwiller wave functions gives us rst indications for the SBM phase it is desirable to have more accurate trial states This is what we tuni to next although only with limited success 2 SU2invariant improvement of the Gutzwiller wave functions To allow more variational freedom we consider mean eld with both spinon hopping k Eq 3 and spinon pairing in the singlet channe with real gap function Ak this way the wave function remains spin rotation and time reversal invariant Generic Ak would open up gaps while we want the wave function to be critical One way to main tain gaplessness is to require Ak to vanish at the Fermi points This can be achieved for example by taking Mk fk k with a smooth fk We have tried several simple functions fk eg expanding in harmonics fk 2f cosmic B B3 with few f treated as variational parameters Upon writing out the corresponding Gutzwiller wave function one can see that the dispersion k enters only through its sign lt0 or gt0 so in the case with two Fermi seas like in Fig 20 we PHYSICAL REVIEW B 79 205112 2009 can use the same label N1N2 and expect similar singular wave vectors encoded in the relative positions of the Fermi poinm We refer to such a state as improved Gutzwiller and can view it as a gapless superconductor although with caution because of the nonintuitive effects of the projection For example the SU2 gauge structure of the projective construction implies that fnA8n70 gives the same state as the bare Gutzwiller independent of A In practice we often x f0 and vary f1 f2 For the zigzag ring model in the spin Bosemetal regime such approach improves the trial energy by about 50760 compared to the difference between the bare Gutzwiller en ergy and the exact DMRG groundstate energy The expo nents of the power law correlations in the improved wave functions appear to remain unchan e from the bare case although the numerical amplitudes are redistributed to re semble the DMRG correlations better as can be seen in the examples in Sec III B Figs 4 and 5 Thus this approach is only partially successful since we cannot produce the long distance behavior expected in the generic SBM and tenta tively seen in the DMRG Still the fact that we can signi cantly improve the trial energy while retaining the underlying gapless character gives us more con dence in the variational identi cation of the SBM phase We also mention that in the Bethechain regime where the bare Gutzwiller projects one Fermi sea the gapless super conductor improvement with parameters fo f2 works even better bringing the trial energy much closer to the exact DMRG value and better reproducing shortscale features in the spin correlations see Fig 22 Such good trial states for the competing Bethechain phase give our VMC more accu racy in determining where the SBM phase wins energetically and more con dence interpreting the DMRG results 3 States with fully polarized second Fermi sea Motivated by the possibility of partial ferromagnetism in some regimes discovered in the DMRG study of the ring model see in particular Sec VI we have also considered Gutzwiller projection of states with unpolarized large Fermi sea and fully polarized small Fermi sea The spin correlations here can be understood using a naive bosonization treatment starting with such spinon mean eld state and following the same procedure as for the unpolarized spin Bose metal in Sec II The naive longwavelength theory now has two free Boson modes The dominant correlations are expected to be at wave vectors that connect Fermi points with opposite group velocities Taking the polarization axis to be 2 the spin structure factor SifS39fq has dominant singularities at 2kF1 2km and km kF1 while the SgSfIZ is missing the 2km since there is no spin ip process across the second polar ized Fermi sea We indeed observe such correlations in the wave functions and the dominant power law envelope is consistent with 43 which is what such naive theory would give if we assume equal velocities near all Fermi points and ignore all interactions other than gapping out the overall charge mode Eq 13 We note however that to properly describe such a partially polarized phase in the system with shortrange interactions we would need to also account for 20511226 SPIN BOSEMETAL PHASE IN A SPIN MODEL KnngJ11J21J30L192 Gulzw 960 Gutzw Improved Spin structure factor ltSq DMRG Gulzw 960 39 Gulzw Improved i 005 010 I Dimer structure factor ltBq qugt 015 0 010 008 006 004 002 000 002 004 Chirality structure factor ltxq xlqgt 006 FIG 22 Color online Spin dimer and Chirality structure fac tors at a representative point in the Bethechain phase K ngll 1 Jz l measured in the DMRG for system size Ll92 We also show structure factors in the bare Gutzwillerprojected single Fermi sea state N 1N2960 and in the improved Gutzwiller wave function with parameters f0lf10 and f2 l4 see Ap pendix B for wave function details the ferromagnetic spin wave which is not present in our wave functions42 and not treated in the more general but still naive bosonization theory outlined above We do not pursue this further here APPENDIX C DMRG RESULTS IN CONVENTIONAL PHASES ON THE ZIGZAG CHAIN For ease of comparisons here we show our DMRG mea surements in the conventional Bethechain and VBS2 PHYSICAL REVIEW B 79 205112 2009 ng J11J2 4 JS 0 L192 DMRG Gulzw 4848 quot quot 39 Spin structure factor ltSq Slqgt 030 DMRG 025 Gulzw 4848 u 020 015 010 005 000 005 Dimer structure factor ltBq qugt 010 015 0 FIG 23 Color online Spin and dimer structure factors at a point in the VBS2 phase K nglll 124 measured in the DMRG for system size Ll92 The exhibited trial wave function is the Gutzwiller projection of two equal Fermi seas N1N2 4848 This wave function gives decoupled legs expected in the large 12 limit and does not have any VBS2 order however it reproduces the DMRG data quite well so at this KringJl cut the system is close to the decoupledlegs limit even just outside the phases identi ed on the zigzag chain in earlier works 20 We take representative points from the same cut ngJl 1 Stud ied in detail in Sec III since this allows us to better relate to the SBM phase at such signi cant Kng values 1 Bethechain phase Figure 22 shows spin dimer and Chirality structure fac tors at Jzz l measured in the DMRG for system size L 2192 The DMRG can still obtain reliable results with m 23200 states kept in each block and this is related to the smaller central charge than in the SBM phase as discussed in Sec III D Figure 22 also shows the structure factors in the bare Gutzwillerprojected single Fer i sea state N1N2 96 0 and in the improved Gutzwiller wave func tion see Appendix B the latter achieves signi cantly better trial energy and overall match with the DMRG results At long distances we expect both the spin and dimer correlations to decay with the same power law Sx S0gt l x BxB0gt l x up to logarithmic correc tions We indeed see roughly such power law in the real 20511227 SHENG MOTRUNICH AND FISHER space correlations Some quantitative aspecm are different from the pure Heisenberg chain for which the bare Gutzwiller state is a good approximation Thus the spin structure factor in Fig 22 has a larger amplitude of the q 77 singularity and also develops a hump at wave vectors below 77 2 Both these features are captured by the improved Gutzwiller wave function On the other hand the dimer structure factor has a signi cantly smaller amplitude of the q 7739 singularity than the pure Heisenberg chain and the bare Gutzwiller the improved Gutzwiller wave function moves in the right direction compared to the bare one but still does not capture well the amplitude at 7739 For the chirality correlations we expect xx0 l x3lx4 and we indeed see some fast decay in the realspace data comparable with the power law behavior The corresponding momentumspace singularities at q7r and q 0 are very weak In agreement with this we do not see any features in the chirality structure factor in Fig 22 This Bethechain phase example allows contrasting with the SBM phase in Sec III B where we see different singular wave vectors and prominent features in all these observables including the chirality The experience of being able to im prove signi cantly the shortscale features in the trial wave functions carries over to the SBM although in the Bethe chain phase we have an advantage that our wave functions also capture the longdistance power laws correctly 2 Valence bond solid with period 2 Consider now the large 12 case In the J2gt00 limit we have decoupled legs and each behaves as a Heisenberg spin PHYSICAL REVIEW B 79 205112 2009 chain Finite JlJz and ngJz will couple the two legs and will likely open a spin gaplgv19 producing a VBS state with period 2 Fig 11 Figure 23 shows our measurements at a representative point 124 from the K ngzll 1 cut The spin correlations show a dominant peak at a wave vector q 277 2 and bond correlations have a peak at q7r We com pare with the Gutzwiller projection of two equal Fermi seas in the 1D zigzag chain language or equivalently decoupled legs in the twoleg ladder picture This wave function is thus strictly appropriate only in e JZHOO limit but it clearly reproduces the DMRG data quite well Looking at Fig 23 there is not much direct evidence for the VBS2 order in the DMRG data It is safe to say only that upon exiting the SBM phase along this cut we are close to the xed point of decoupled legs One argument for the VBS2 here could be the continuity to the strong VBS2 phase in the broader phase diagram Fig 2 As is known18 the region Jz0472 along the ng0 axis has strong VBS2 order However this does not preclude possibility of more phases in the model with ring exchanges Thus along the way at points like King03 12215 and ng02 12 12 we also see a dimer feature at q7r2 in addition to a likely Bragg peak at q7r Since our primary interest is the SBM phase we do not explore the states at large JZ further loosely referring to all of them as VBS2 in Fig 2 1Y Shimizu K Miyagawa K Kanoda M Maesato and G Saito Phys Rev Lett 91 107001 2003 2Y Kurosaki Y Shimizu K Miyagawa K Kanoda and G Saito Phys Rev Lett 95 177001 2005 3S Yamashita Y Nakazawa M Oguni Y Oshima H Nojiri Y Shimizu K Miyagawa and K Kanoda Nat Phys 4 459 2008 4M Yamashita N Nakata Y Kasahara T Sasaki N Yoneyama N Kobayashi S Fujimoto T Shibauchi and Y Matsuda Nat Phys 5 44 2009 50 I Motrunich Phys Rev B 72 045105 2005 6SS Lee and P A Lee Phys Rev Lett 95 036403 2005 7T Senthil Phys Rev B 78 045109 2008 23R H McKenzie Comments Condens Matter Phys 18 309 1998 9H Morita S Watanabe and M Imada J Phys Soc Jpn 71 2109 2002 10W LiMing G Misguich P Sindzingre and C Lhuillier Phys Rev B 62 6372 2000 11XG Wen Phys Rev B 65 165113 2002 12W Rantner and XG Wen Phys Rev B 66 144501 2002 13M Hennele T Senthil M P A Fisher P A Lee N Nagaosa and XG Wen Phys Rev B 70 214437 2004 14P A Lee N Nagaosa and XG Wen Rev Mod Phys 78 17 2006 150 I Motrunich and M P A Fisher Phys Rev B 75 235116 2007 16D N Sheng O I Motrunich S Trebst E Gull and M P A Fisher Phys Rev B 78 054520 2008 17M P A Fisher 0 I Motrunich arXiv08122955 unpublished 133 R White and 1 Af eck Phys Rev B 54 9862 1996 19A A Nersesyan A O Gogolin and F H L Essler Phys Rev Lett 81910 1998 20A D Klironomos J S Meyer T Hikihara and K A Matveev Phys Rev B 76 075302 2007 21J S Meyer and K A Matveev J Phys Condens Matter 21 023203 2009 22L B Ioffe and A I Larkin Phys Rev B 39 8988 1989 23T Holstein R E Norton and P Pincus Phys Rev B 8 2649 1973 24M Y Reizer Phys Rev B 40 11571 1989 25R A Lee Phys Rev Lett 63 680 1989 26P A Lee and N Nagaosa Phys Rev B 46 5621 1992 27J Polchinski Nucl Phys B 422 617 1994 28B L Altshuler L B Ioffe and A J Millis Phys Rev B 50 14048 1994 29Y B Kim A Furusaki X G Wen and P A Lee Phys Rev B 50 17917 1994 30R Shankar Acta Phys Pol B 26 1835 1995 31HH Lin L Balents and M P A Fisher Phys Rev B 58 1794 1998 and D N Sheng 20511228

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