INTRO LIB RESEARCH
INTRO LIB RESEARCH INT 1
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PHYSICAL REVIEW A 74 013607 2006 Atomic matter of nonzeromomentum BoseEinstein condensation and orbital current order W Vincent Liu Department of Physics and Astronomy University of Pittsburgh Pittsburgh Pennsylvania I 5260 USA Cong39un W Kavli Institute for Theoretical Physics University of California Santa Barbara California 93106 USA Received 16 March 2006 published 13 July 2006 The paradigm of BoseEinstein condensation has been associated with zero momentum to which a macro scopic fraction of bosons condense Here we propose a new quantum state where bosonic alkalimetal atoms condense at nonzero momenta defying the paradigm This becomes possible when the atoms are con ned in the porbital Bloch band of an optical lattice rather than the usual sorbital band The new condensate simul taneously forms an order of transversely staggered orbital currents reminiscent of orbital antiferromagnetism or d density wave in correlated electronic systems but different in fundamental ways We discuss several approaches of preparing atoms to the porbital band and propose an energy blocking mechanism by Fesh bach resonance to protect them from decayng to the lowest sorbital band Such a model system seems very unique and novel to atomic gases It suggests a new concept of quantum collective phenomena of no prior example from solid state materials DOI 101103PhysRevA74013607 I INTRODUCTION Con ning bosonic atoms in an optical lattice can bring out different and new physics beyond the standard Bose Einstein condensation BEC observed in a single trap 12 The super uidiMottinsulator experiment on an optical lat tice 3 based on an early theoretical idea 45 demon strated one such example with bosons Proposals of explor ing various lattice atomic systems many concerning spin have further extended the scope of interest into different di rections experimentally 6712 and theoretically see Refs 13719 and references therein Atomic optical lattices not only can realize many standard solidstate problems but also may bring new and unique as pects speci c to the atomic gas One possible direction is orbital physics 20 This is a new direction that has not yet received as much attention as spin The pioneering experi menm of Browaeys et al 9 and Kohl et a1 10 which demonstrated the occupation of bosonic and fermionic at oms respectively in the higher orbital bands further justify and motivate theoretical interest in the orbital degrees of freedom of cold atoms beyond the conventional sorbital band such as the next three p orbitals In electronic solids such as manganese oxides and other transitionmetal oxides the orbital physics is believed to be essential for understand ing their metalinsulator transitions superconductivity and colossal magnetoresistance The solids are of periodic arrays of ions The quantummechanical wave function of an elec tron takes various shapes when bound to an atomic nucleus by the Coulomb force For those oxides the relevant orbitals of the electron are the ve d wave orbitals usually split into two groups of eg and tzg due to the crystal eld The orbital degree of freedom having intrinsic anisotropy due to various orbital orientations interplaying with the spin and charge gives rise to an arena of interesting new phenomena in the eld of strongly correlated electrons 2122 We will focus on the orbital degree of freedom of cold atoms below In this paper we point out that the current experimental condition makes it possible to study a whole new class of 1050294720067410136079 0136071 PACS numbers 0375Nt 6740 w 7472 h lattice systemithe dilute porbital Bose gasibeyond the conventional sorbital BoseHubbard model We show that the system reveals in the super uid limit a new state of mat ter in which atoms undergo BoseEinstein condensation at nonzero momenta and form a staggered orbital current order simultaneously These features distinguish the porbital atomic gases from the dorbital electronic oxide compounds The unique signature of the state is predicted for the time of ight experiment II ATOMS IN THE pORBITAL STATE Let us study an optical lattice of bosonic atoms in a single internal hyper ne spin state To gain a qualitative under standing we approximate the lattice potential well by a har monic potential around minimum The characteristic har monic oscillator frequency is rub 4V3E 2 where V3 is the threedimensional 3D lattice potential depth and E the re coil energy for the bosonic atoms The recoil energy is de termined by the laser wavelength and atom mass In the pres ence of periodic lattice potentials the boson state can be expanded in the basis of Wannier functions to be denoted as flx with n the Blochband index The lowest Bloch band is swave symmetric with n000 The next band is a pwave band with threefold degeneracy pf with Mxyz corresponding to n100 010 001 The energy splitting between s and p is hwb Note that the three degenerate pwave energy subbands disperse anisotropically when hop ping process included The fact that the next band starts at the p orbital instead of the s tells an important difference between the optical lattice potential and the Coulomb ionic lattice potential in electronic solids In a dilute weakly interacting atomic boson gas con ned in optical lattices bosons intend to aggregate into the lowest s band in the lowtemperature limit with an exponentially small fraction in the higher Bloch bands suppressed by the 2006 The American Physical Society W VINCENT LIU AND CONGJUN WU factor e wbkBT A singleband approximation is then ad equate which was proven successful both theoretically and experimentally 35 Several approaches are available for transferring cold at oms to the rst excited porbital band In the study of a related but different model Isacsson and Girvin 20 sug gested A to use an appropriate vibrational 7T pulse with frequency on resonance with the sp state transition and B to apply the method demonstrated in the experiment of Browaeys et al 9 by accelerating atoms in a lattice We may also add a third possible approach that is C to sweep atoms adiabatically across a Feshbach resonance Kohl et al 10 pioneered this method experimentally by showing fer mionic atoms transferred to higher bands the phenomenon was subsequently explained in theory 23 Whether bosonic atoms can be transferred this way remains to be seen Now suppose that a metastable porbital Bose gas has been prepared on the optical lattice The remaining challenge is that the system is not in the ground state and thus genu inely has a nite lifetime The interactions between two bosonic atoms although weak can cause atoms in the porbital states to decay An elastic decaying process which conserves total energy is that two atoms initially in the porbital band scatter into the nal state of one atom in the n0 ls band and another in the n2 orbital 2s 1d band where n represents the principle energy level quantum number and the states of the harmonic oscillator are labeled in the LandauLifshitz notation 24 For a related model Isacsson and Girvin 20 have studied the decaying rate and estimated that the lifetime is about 10 100 times longer than the time scale of tunneling in an optical lattice However such a lifetime can be still short to achieve condensation and perform experimental detection In the following we pro pose a new mechanism that should suppress the above de caying process and thus extend the lifetime III ENERGY BLOCKING OF THE pORBITAL DECAY We propose a deep fermion optical lattice on top of a relatively shallow optical lattice for bosons such that the characteristic lattice well frequencies are very different wb ltwf Consider loading a fermion density around I lle that is one fermion per site Now one tunes a Feshbach resonance between the boson and fermion 2526 to set the interspecies interaction strength in energy scale between wb and wf In this case the fermions ll up the s band com pletely so essentially behaving as an band insulator whose dynamical effect on bosons becomes exponentially sup pressed by the band energy gap The lowestsorbital wave function is approximately a Gaussian peaked at the center of the lattice well The role of fermions then can be thought as providing a repulsive central potential barrier in addition to the optical lattice potential for bosons on the same lattice site As a result all energy bands are shifted up signi cantly including s and p of course by the Feshbach interaction with the sorbital fermion Because the overlap integrals are different as shown in Fig l the energy shifts are different in magnitude for different orbital states Fig 2 PHYSICAL REVIEW A 74 013607 2006 FIG 1 Color online The overlap between s and p density clouds wave functions squared is smaller than between two s clouds The lowest orbit for a single particle in a single site must be nodeless it is actually not possible to increase the sstate energy higher than the pstate energy However the gap Asp between n0 s and n l p can be signi cantly reduced to very small by a suf ciently strong interspecies Feshbach resonance as illustrated in Fig 2 In Appendix A 2 we give an estimate of the interaction strength needed On the other hand the splittings between the l p state and states in the n 2 level 2s and 1d should not change much because the wave functions of the latter all are spatially more extended than the ls state In summary the lowest two Bloch bands sp are close in energy while all other Bloch bands n 22 have energy far above the rst excited porbital band in the energy scale of Asp The lowenergy quantum theory of the system effectively reduces to a twoband problem The lattice gas under study is also assumed to be in the tightbinding limit such that the tunneling amplitude is smaller than Asp so much smaller than the bandlevel split ting between the p and all other higher bands ie IHltASP lt wb Any elastic scattering process that scatters atoms out of the porbital band must conserve the total energy Under the proper condition described above the decay rate of the porbital Bose gas must be suppressed by the energy conser vation law any two atoms initially in the nl porbital band cannot scatter out of the band by the twobody scatter ing process Of course the scattering within the three sub bands of p orbits are allowed This idea is one of our main results see Appendix A for technical detail 31 p v 77 s l Asp by l boson fermion7 L b Feshbach S FIG 2 Color online Energy band shift due to Feshbach reso nance between bosons and fermions a The bosonic ls band is moved close to the 1 band while the 2s and 1d bands not shown are both pushed higher b The fermions ll up their own lowest s band as an band insulator The relative shift of the fermionic band energy is less signi cant than that of the bosonic case for wf can be made far larger than both ml and the Feshbach tunable energy shift 013607 2 ATOMIC MATTER OF NONZERO MOMENTUM BOSE aagk W t t FIG 3 Color online Anisotropic hopping matrix elements of p orbital bosons on a cubic lattice The longitudinal t is in general far greater than ti because the overlap integral for the latter is exponentially suppressed The i symbols indicate the sign of two lobes of p orbital wave function IV LATTICE GAS 0F pORBITAL BOSONS The quantum theory of bosonic atoms prepared in the porbital state is effectively described by a pband Bose Hubbard model A standard derivation see Appendix B gives the Hamiltonian H E tH w m 1 5WbLaraebe Hc Luv UE 5 l n where a is lattice constant Here b 19 are annihilation and creation operators for bosons in lattice site 1 and orbital state pf index label uvxyz n and L are the boson density and angular momentum operators nrEMerbM and LM iEVEmbrblr This model is invariant under Ul phase transformation cubic lattice rotations and time reversal transformation The model is determined by the following parameters I and IL are the nearestneighbor hopping matrix elements in longitudinal and transverse directions respectively with re spect to porbital orientation and U is the onsite repulsive interaction due to the intrinsic nonresonant swave scatter ing between two bosons In quantum chemistry tm are named the 077 bond respectively Their precise de nition is given in the Appendixes By de nition of the Hamiltonian both I and IL are positive note that tHgttL for the tunneling overlap is sensitive to orientation Fig 3 The quantum physics of porbital bosons seems to have never been studied before except very recently by Isacsson and Girvin 20 in a different interesting limit in which the transverse tunneling IL is completely suppressed Different from ours in symmetry and ordering their model had in nite and subextensive local gauge symmetries and columnar or derings Another difference is that it is not obvious whether their onsite interaction were SO3 invariant The interaction in the lattice Hamiltonian 1 is ferro orbital Ugt0 suggesting that the bosons at the same site prefer to occupy the same orbitalpolarized state carrying maximal angular momentum This is analogous to the Hund s rule for electrons to ll in a degenerate atomic energy shell which favors a spinpolarized con guration Next we shall show that the ferroorbital interaction together with the pband hopping gives rise to an orbital ordered BEC PHYSICAL REVIEW A 74 013607 2006 a 39 39 39 b1 jib py band minima x E f 0 12 I pX minima 1 o 1 3911 I 396 1 1 X kxjy in units of na kx FIG 4 Color online Illustration of p band dispersions a The energy dispersion of the px orbital band as an example on the kx axis in the rst Brillouin zone in comparison with the s band Inset its dispersion along the line O O the energy zero line is shifted arbitrarily for display Note that the s and p bands have different locations for minimum energy The band gap be tween the two levels is tuned small by the Feshbach resonance b The p band minima in the kX ky plane the pz s minima are not shown but can be obtained by rotating 90 out of the plane Orbital BEC should occur at a subset of the minima spontaneously break ing the lattice translational and orbital rotational symmetry in addition to the Ul V ORBITAL BOSEEINSTEIN CONDENSATION A weakly interacting Bose gas is expected to undergo BEC becoming super uid at low temperatures In the optical lattice such a state has been rmly established both experi mentally and theoretically for bosonic atoms occupying the s band the widely studied BoseHubbard model In our pband model when the interaction is weak and repulsive 0lt U ltIH the noninteracting term of the Hamiltonian dominates The pband bosons have the energy dispersion e k2EVtH5W til 5Wcoska where u V label the three subbands and a is the lattice constant There are two new aspects of the porbital bosons The rst is that BEC takes place at nonzero momenta While the paradigm of BBC should occur at zero momentum there is no real reason that has to be so Only the lowestenergy state matters The pband energy dispersion shows an exceptional and remarkable case in which the lowestenergy state of bo son happens to be at nite momenta Q I de ned as Gamp 70 w m705 amp y m for the respective porbital states Fig 4 Note that Q are identi ed by a reciprocal lattice vector 2Q 0mod 27Ta The condensate at momentum Q is essentially equivalent to that at Q on the lattice Therefore the sum of atoms over all modes does not show any net current ow in any direction This feature is important for a possible experimen tal test of the state that we shall elaborate below The second aspect is that the orbital degeneracy of pband bosons opens possibilities of novel orbital physics Orbital ordering typically involving dorbital fermionic electrons that are argued to be essential to understanding a class of strongly correlated transitionmetal oxides has become a topical subject in condensedmatter physics 21 In our pband model condensing into any of these momenta Q or their linear superpositions equally minimizes the total kinetic 013607 3 W VINCENT LIU AND CONGJUN WU energy It thus provides a rst bosonic version of such kind from atomic physics Being ferroorbital the interaction in the Hamiltonian will be shown later to favor a condensate of angular momentum ordering to have 9E 0 Keeping the above aspects in mind we seek a condensate described by the following order parameter of total six real variables ltbxkQgt cosX ltbykoygt pei iT399 isinX lt3 bzszzgt 0 with TTxTyTz the generators of 80 3 orbital rotation in the following matrix representation an adjoint 3 for the group theory experts de ned by its elements TMV dewv u Vxyz Such a parametrization manifests symmetry go is the overall phase of the Ul symmetry the three angle variables 0 6x 0y Q are the orbital rotation of 803 and X changes sign under time reversal p is the modulus eld xed b the total boson density in the conden sate ng through p Vng where V is the 3D lattice volume in the units of a3 so V dimensionless We now proceed to calculate the mean eld interaction energy While the full Hamiltonian only has cubic lattice symmetry the interaction term being on site enjoys how ever a continuous 803 rotation invariance Therefore the symmetry dictates that the mean value of the interaction term be independent of go and 0 The former is due to the exact U This argument reduces our calculation essentially to a problem for a singlevariable order parameter namely x Then a straightforward evaluation of the mean value of the interaction energy determines 1 1 HUtermgtEvUn821 8l1122X The interaction energy is minimized at X Xi E if The two minima xXi are degenerate and discrete re ecting the timereversal symmetry Of course shifting the value of Xi by 7T gives other minima of the same energy but the 7T shift can be absorbed away by an overall Ul phase adjustment or an orbital rotation The order parameter points to one of the two discrete minima spontaneously breaking the timereversal symmetry The resultant quantum state is an axial super uid having a macroscopic angular momentum ordering The order parameter is then found to be say for XXjf bxszQ v b 1 ltbykQygt v i lt4 ltbkQZgt 0 with a degenerate manifold characterized by phase go and rotational angles 0 This is our porbital BEC p0BEC It breaks the Ul phase lattice translation orbital 803 rota tion and timereversal symmetries Quantum or thermal uc tuations which will be studied in the future are expected to break the 803 rotational symmetry and align the orbital condensate to speci c lattice directions PHYSICAL REVIEW A 74 013607 2006 px1py px Ipy px my 13 my px ipy px ipy px ipy px ipy FIG 5 Color online The real space con guration of the TSOC state which exhibits a staggered and uniform orbital current pattern in the xy plane and along the z axis respectively VI STAGGERED ORBITAL CURRENT The p0BEC contains one novel feature that is absent in the conventional BEC To illustrate this we assume that the lattice has a small anisotropy such that the state is pinned to the xy plane ie the pxiipy order see the phase diagram in Appendix C The novel feature is contained in the struc ture factor of the boson number and angular momentum op erators Taking for example the state of X Xjf we found Lq00n85anxQy The momentum dependence of the angular momentum operator reveals that the p0BEC is also an orbital current wave in analogy with a commensurate spindensitywave order in antiferromagnets In real space the order has the staggering pattern LxrLyr0 Lzr ng xya We shall call it transversely staggered orbital current T80C for the direction of Lz alternates only in the x y directions The reader should bear in mind that at the mean eld level the direction of orbital ordering is arbitrary for a fully 803invariant interaction In real experiments the presence of a symmetrybreaking perturbation such as a weak anisotropy in lattice potentials we assumed or the ef fect of quantum uctuations is expected to pin down the direction We next provide an explanation for the appearance of T80C in real space Fig 5 The interaction favors a maxi mum angular momentum at each site 80 the orbit con gu ration on each site is pxiipy corresponding to angular mo mentum quanta per atom 0n the other hand the longitudinal and transverse hopping amplitudes are of oppo site sign In order to maximally facilitate the intersite hop ping the phases of 1967y orbits should be staggered in the longitudinal direction and uniform in the transverse direc tions As a result L3 or the orbital current exhibits a stag gered uniform pattern in the x y z directions This state bears some similarity to its fermionic counterpart of the or bital antiferromagnetism or d density wave DDW 27 pro posed for the hightemperature cuprates A major difference is that the DDW current ows on bonds around each plaquette with staggered magnetic moments through lattice whereas the current here circulates inside each site 013607 4 ATOMIC MATTER OF NONZERO MOMENTUM BOSE a high 3 I E med E 3 xgt low 2 4 kX units of na b I b a C ii I L I I 4 6 2 kX units of na FIG 6 Color online Prediction of density of atoms for time of ight experiment a Density integrated over the z axis is shown in a quarter of kX ky momentum plane with other quarters obtained by re ection symmetry b Density shown along the kx axis As suming a free expansion the atom density at distance r is ltnbrgtltxI kgt263ltk G QxI yltkgt263ltk G Qy where k mrf m39 T is the time of ight and G runs over all three dimensional reciprocal lattice vectors The absence of peak at k 0 distinguishes the p OBEC from paradigmatic BBC The highest peak is not necessarily the closest to zero momentum origin a unique feature of orbital BEC depending on the size lb of the bosonic Wannier function Zr Parameters are lbaOl the 6 function is replaced by a Lorentzian line for display VII EXPERIMENTAL SIGNATURE In a timeof ight experiment which has been widely used to probe the momentum distribution of cold atoms the pOBEC will distinguish itself from a conventional sBEC with unique structural factor There are two new aspects Fig 6 The rst is that the condensation peaks are not located at zero momentum nor at any other momenta related to zero by a reciprocal lattice vector G The second is that unlike the swave case the pwave Wannier function superposes a non trivial pro le on the height of density peaks As a result the highest peaks are shifted from the origin a standard for the swave peak to the reciprocal lattice vectors whose magni tude is around llb In the following we show the details of the calculations that led to the above predictions We consider the density distribution of the timeof ight experiment assuming ballistic expansion In our porbital case it can be written as m 3 ltnrgtz E p k pkltbLlbVlgt 5 uv where uVxyz kmr t p k is the Fourier tranS form of the porbital band Wannier function pp 1 and l k mod primitive reciprocal lattice vectors aking into account that bosons condense into the pxipy orbital state spontaneously according to Eq 4 we arrive at PHYSICAL REVIEW A 74 013607 2006 W oc 2 I pkgtlz 3ltk Q G Iltpykl2 G X53k Qy G 6 2ml7r M 21 77 a 7 a 7 a That means that the Bragg peaks are at and ZmW Own 2177 T 7 where mnl are integer numbers The Fourier transform of the pwave Wannier orbits ex hibits the following nontrivial form factor pxk 392 l 1ltkx 0ky 0kz 2 7 W106 392 0C klb29klb227 2 0k2 CC I a 7 where boal refer to the ground and rst excited states of the onedimensional harmonic oscillator respectively and lb is the oscillator length of the boson optical lattice potential As a result unlike the conventional sorbital case where the highest weight is located at the origin of the reciprocal lattice with a distribution width about llb the porbital case has the highest weight shifted from the origin to the reciprocal lattice vectors around llb In Fig 6 where we used the parameter llb 101 0 w 3777 the highest intensity thus appears at the second Bragg peak VIII CONCLUSION AND DISCUSSION In conclusion we have proposed a new state of matter in which porbital bosonic atoms condense at separate nonzero momenta according to orbital orientations defying the zero momentum hallmark characteristic of all standard Bose Einstein condensates This new state is a prediction for a porbital Bose gas on lattice We have shown how bosons in Feshbach resonance with fermions can be effectively blocked from occupying the s band by the energy conserva tion Our idea may be generalizable to a mixture of two species of bosons Realizing this new state pOBEC will change the standard way of thinking BBC as normally at tached to zero momentum The topic of ultracold atomic gases has been ourishing at the interface between atomic and condensedmatter physics The latest progress in Feshbach resonance and optical lattice has further extended its scope of interest One current focus in correlated quantumcondensed matter is the spontaneous timereversal symmetrybreaking ground states for ex ample the ddensity wave state 27 proposed as a compet ing order for the pseudogap phase of the highTC supercon ductivity and the incommensurate staggered orbital current phase suggested as the mechanism for the hiddenorder tran sition in the heavyfermion system URuZSiZ 28 Unfortu nately experimental observation of these states so far re mains elusive and controversial The pOBEC state we proposed here is perhaps the rst bosonic example of this kind from ultracold atomic gases The extraordinary control lability of the atomic system widely recognized by far opens up the possibility of observing this kind of novel states for the rst time 013607 5 W VINCENT LIU AND CONGJUN WU Note added Upon the completion of this manuscript there appeared an independent work 29 that also proposes a porbital BEC based on a related but different mechanism ACKNOWLEDGMENT We acknowledge the Aspen Center for Physics where this work was initiated during the Workshop on Ultracold Atomic Gases W V L is supported in part by ORAU Ralph E Powe Junior Faculty Enhancement Award C W is supported by the NSF under Grant No PHY99707949 APPENDIX A THE MICROSCOPIC MODEL FOR A BOSEFERMI MIXTURE Our model system is a gas of two species of atoms one being bosonic and another fermionic con ned in two over lapping sublattices with separate potential heights The Hamiltonian is V2 H J39 d3x E W 2 V410 1 gres lZ b f new man advMm A1 where the indexes azb and f label the boson and fermion species respectively ges is the interspecies interaction tuned by a Feshbach resonance and g is a weak repulsive interac tion between bosons themselves The singlecomponent fermions do not interact between themselves at short range befx are the 3D optical lattice potentials constructed by counterpropagating laser beams We assume 3 VxVoE sin2ka abf A2 W1 with kL the wave vector of the light The recoil energy for each species is Eaz 2 lama assumed different between boson and fermion for different masses In the presence of such periodic potentials the boson and fermion operators ak can be expanded on the basis of the Wannier functions 38 with n the band index Including the lowest and rst excited Bloch bands swave and p Mwave bands we write wax 2 max r 2 22mm 0 A3 w E Amix r E fwggx 10 A4 where zxyz label the three porbital bands In the har monic approximation the s and porbital states are directly given by the harmonic oscillator HO eigenfunctions x gtx d y lt1 zHO with n 000 for sband and n100 010 aird 061 for pk py and pl respectively in Cartesian coordinates The basis functions are kept separate between fermion and boson for they can have different lat tice potential depths and atomic masses PHYSICAL REVIEW A 74 013607 2006 1 Hartree approximation of interspecies interaction We examine possible con gurations of the band occupa tion that may minimize energy There are several both fer mions and bosons in the 3 band ss fermions in the 3 band and bosons in the p band FsBp and fermions in the p band and bosons in the 3 band Fsz The three con gurations have different interspecies interaction energies per boson fermion pair as follows 1 greJ000ooo gres E W A5 I w W12 7 IFsBp grey0001100 lg 2 W A6 b If W12 IFsz greJ100ooo E Wgtlt7 A7 lb If where In Efd3xl clxl2l flxl2 and lbyf are the harmonic oscillator len s for boson and fermion respectively la E mawa 2 Condition for bandgap closing To further achieve a simpler effective model let us exam ine the singleparticle Hamiltonian at a single siteisay at riin tuni for the boson and fermion We shall show quan titatively how one species shifts the energy levels of another through the Feshbach interaction First consider the effects of an 3 band of fermions on bosons At Hartree approxima tion the singleparticle onsite energy is shifted due to the interspecies Feshbach interaction with a term in the Hamil tonian 3 5 Hfingle site 5 angtb2rbsr E 5 WxnfgterbM7 M A8 where nf is the number of fermions per site assumed all in the 3 band Likewise an 3 band of bosons shift up both the s and pband energies of fermion yielding a similar term in Hamiltonian Hzmgle site Hfingle siter Wx H W x A9 and nb is the number of bosons per lattice site all in the s For 17gt rub the energy shift for the fermionic bands is small compared with the band energy spacing wf the 3 band re mains lower than the p The situation is opposite for boson in this Hartree treatment The ordering of energies of the s and p bands if one naively trusts the Hartree approximation can be even reversed Here we provide a Hartree estimate for how strong the interspecies Feshbach resonance is needed to close the spband gap for bosons The condition for the effect can be met by requiring that the sband Hartree energy be higher than that of the p 0136076 ATOMIC MATTER OF NONZEROMOMENTUM BOSE 0 w lt w A10 from Eq A8 This condition is satis ed when the Feshbach resonance scattering length am related to ges by ges 27rammbmfmbmf is suf cient large such that a All res I 1 mi 1 12 5 2 min gt 2 lb E ares 2 nfmb mf lb The derivation implicitly assumed nf S1 to avoid higher band complication due to Pauli exclusion This condition can be achieved without tuning the scattering length ares to very large via Feshbach resonance if the minimally required ares is made suf ciently small That can be done by tuning the depth of optical potentials to make an adequately small lb and keep lfltlb in the same time As we will discuss next our condition for bandgap closing can in principle avoid the threebody loss problem in the experiments by means of operating the system adequately far from the resonance point In the experiment of resonant atomic gases there is al ways the issue of a nite lifetime Near the Feshbach reso nance a weakly bound dimer relaxes into deep bound states after colliding with a third atom The released binding energy transformed into the kinetic energy of atoms in the outgoing scattering channel that will escape from the trap Such a process of threebody collisions determines the lifetime of the trapped gas The relaxation rate is believed to be highest in the bosonboson resonance and lowest in the fermion fermion resonance with the rate for the bosonfermion reso nance in between 30ithough there is no explicit calcula tion for the bosonfermion resonance yet to the best of our knowledge More relevantly two recent experiments have observed the bosonfermion interspecies Feshbach reso nances in the systems of 6Li2339Na atoms 25 and 40K87Rb atoms 26 respectively which seem to be very stable This is part of the reason that we are proposing a BoseFermi mixture as opposed to a generally lessstable singlestatistics Bose gas Another important reason is of course that the fermion component of the mixture can be made be a band insulator with virtually no dynamical effects on bosons other than providing a central potential barrier see the main text On general grounds we expect that the larger the detun ing from the resonance is the smaller the threebody loss rate should be obeying some powerlaw suppression In our model the estimated scattering length between fermions and bosons is of the order of lbmfmbmf see above to enable the effect of spband gap closing for bosons It can be made suf cient small For example this can be as small as 50 bohr radius for a Na Li mixture taking lattice constant a 2400 nm and the 2339Na oscillator length lb 01a That means that the system does not have to be operated close to resonance This makes the relaxation lifetime practically in nitely long so the threebody loss problem is experimen tally avoidable Finally in contact with the discussion of the energy blocking of the porbital decay in the main text we do not require that the spband gap Asp be vanishing but just small compared with that between the p and higher bands There fore the above quantitative evaluation of the resonance in PHYSICAL REVIEW A 74 013607 2006 teraction strength is but an estimate for the scale 3 Is an equilibrium nodal BoseEinstein condensate possible The mean eld Hartree argument would imply that when the bosonfermion interaction is tuned strong enough aes gta the porbital state has lower energy than the 3 see Fig 2 However if all the effect of fermions on boson is replaced by a singleparticle central potential barrier at every lattice site as in the Hartree approximation the lowest orbit of the singleboson state must be nodeless which is swave but likely extended from simple quantum mechanics The singleparticle Hartree argument must be wrong in predicting the p orbital be a ground state For a bosonic manybody system such as 4He liquid a similar conclusion is expected due to Feynman who drew an analogy between a single particle and a manybody system and argued that the 4He groundstate wave function be nodeless Can the porbital BoseEinstein condensate be a true equi librium ground state instead of a metastable state as we described so far We believe the possibility is not completely ruled out by Feynman s argument Let us explain it Feyn man did not specify the requirement of the form of the two body interaction but a careful examination of his argument would show that the whole argument implicitly assumes a shortrange interaction compared with the average interpar ticle distance For a nite or longrange interaction smooth ing out the wave function will de nitely lower the kinetic energy per each particle coordinate but at the same time will necessarily increase the interaction potential of all neighbor particles within the scope of the interaction range In other words the energy cost or saving is about the completion of oneparticle kinetic energy versus manyparticle interaction energy When the interaction range is long enough we con jecture that the manybody effect becomes dominant An other way to think of our argument is that a longrange in teraction has a strong momentum dependence after Fourier transformation We can always think of a potential that is expandable about zero or some characteristic momentum That corresponds to derivative terms of the realspace inter action potential Once the twobody interaction potential ex plicitly involves derivative terms Feynman s argument seems to fail A longrange interaction is not exotic for cold atoms For atoms in a narrow Feshbach resonance it is known that the effective interaction between two atoms can be longer than or comparable with the average interparticle distance 31 Given the rapid advancement in the control of atomic gases it seems not entirely impossible to realize an equilibrium not just metastable nodal Bose condensate in the future APPENDIX B TIGHTBINDING APPROXIMATION FOR THE pBAND BOSEHUBBARD MODEL In this section we use the tightbinding approximation to derive the general pband BoseHubbard Hamiltonian where each lattice site is approximated as a threedimensional an isotropic harmonic potential with frequencies waVL xyz in three directions respectively 0136077 W VINCENT LIU AND CONGJUN WU The free boson Hamiltonian includes the hopping and on site zeropoint energies ie H0 2 twwjmaeybm Hc E w blrbm B 1 m m where the hopping amplitudes are determined by V2 twp f d3X MX 2 mb VbX Mx 16 tll6LV v l39 In this de nition both I and IL are positive and tHgttL in general Fig 3 Here we have neglected the difference among the values of I and ti in the xyz directions because the major anisotropic effect comes from the onsite energy difference among wxayaz In momentum space H0 reads H10 E em hwpbtkb k 133 M with the energy dispersion of boson e k2 lvtH5MV til 5Wcoska a is the lattice constant The momentum representation of H0 is useful in determining how the pband bosons condense The shortrange interaction between bosonic atoms in the original microscopic model gives rise to an onsite interac tion energy between bosons in the p orbitals It can be clas si ed into three terms 1 Hinll U nr nru 17 13M HinzZ 2 V vnr nrw r ui 1 l Hint3 5 2 V vpl pl prvprw I39Mil with U g d3xl pltxgt 4 vg f d3x pltxgtlzl pltxgt2 135 By straightforward calculation we obtain the following rela tions regardless of the anisotropy of the lattice potential U Ux Uy Uz V ny Vyz V u 3g 4977 z z xyz U 3V B6 where lb 0me1 are the harmonic oscillator length As a result the interaction part can still be reorganized as in Eq 1 as PHYSICAL REVIEW A 74 013607 2006 nU Peri 13311 Nd 1 y FIG 7 Proposed phase diagram for a system of anisotropic lattice potentials sketched as a function of the mean interaction energy nU and the anisotropic ratio 3 wxyy wz U 1 Him 2 n3 3L3 B7 2 139 Note that Him is surprisingly the same as that in the isotropic case of Eq 1 even if the lattice potentials are anisotropic After setting wxayazwb the general Hamiltonian in above reduces to Eq 1 for the isotropic case APPENDIX C THE CASE OF AN ANISOTROPIC LATTICE POTENTIAL Let us consider the case of cylindrical symmetry ie wxwy ywz At ylt 1 particles condense into the pxQxiipyQy state to minimize both the kinetic and inter action energy as discussed in the main text The yl point is subtle Although the full Hamiltonian because of the hop ping term does not possess the 803 symmetry the con densation manifold recovers the 803 symmetry at the mean eld level The quantum uctuation effect is expected to break this 803 down to the cubic lattice symmetry At 71gt l a quantum phase transition takes place from a timereversalinvariant polar condensate to a timereversal symmetrybroken T80C state as the boson density increases see Fig 7 Let us parametrize the condensate as bzszzgt cos X n 4 7 bxszx 1 sin X where n is the boson density Then the mean eld energy per unit volume is C1 2 2 Un2 1 2 EVnwzcos xwxs1n x7 l gsin 2X U U 2 mB n2cos22x chos2x C2 where I wzl y2 and Aw I ywz Minimizing the en ergy we nd 013607 8 RAPID COMMUNIC PHYSICAL REVIEW A 78 010305R 2008 Maximally entangling tripartite protocols for Josephson phase qubits Andrei Galiautdinov Department of Physics and Astronomy University of Georgia Athens Georgia 30602 USA John M Martinis Department of Physics University of California Santa Barbara California 93 I 06 USA Received 15 April 2008 published 9 July 2008 We introduce a suit of simple entangling protocols for generating tripartite GreenbergerHomeZeilinger and W states in systems with anisotropic exchange interaction gXX YY ZZ An interesting example is pro vided by macroscopic entanglement in Josephson phase qubits with capacitive g 0 and inductive 0 lt gg lt01 couplings DOI 101103PhysRevA78010305 I INTRODUCTION Superconducting circuits with Josephson junctions have attracted considerable attention as promising candidates for scalable solidstate quantum computing architectures 1 The story began in the early 1980s when Tony Leggett made a remarkable prediction that under certain experimental con ditions the macroscopic variables describing such circuits could exhibit a characteristically quantum behavior 2 Sev eral years later such behavior was unambiguously observed in a series of tunneling experiments by Devoret et al 3 Maru39nis et al 4 and Clarke et al 5 It was eventually realized that due to their intrinsic anharmonicity the ease of manipulation and relatively long coherence times 6 the metastable macroscopic quantum states of the junctions could be used as the states of the qubits That idea had re cently been supported by successful experimental demon strations of Rabi oscillations 7 high delity state prepara tion and measurement 87 3 and various logic gate operations 971214 Further progress in developing a work able quantum computer will depend on the architecture s ability to generate various multiqubit entangled states that form the basis for many important informationprocessing algorithms 15 In this paper we develop several singlestep entangling protocols suitable for generating maximally entangled quan tum states in tripartite systems with pairwise coupling gXX YY ZZ We base our approach on the idea that implementing symmetric states may conveniently be done by symmetrical control of all the qubits in the system This bears a resemblance to approaches routinely used in digital electronics while an arbitrary gate for example a threebit gate can be made from a collection of NAND gates it is often convenient to use more complicated designs with three input logic gates to make the needed gate faster andor smaller The protocols developed in this paper may be directly applied to virtually any of the currently known supercon PACS numbers 0367Bg 0367Lx 8525j HRWA l2Ql 5391 12113 g039150392f trio i 2103 1 with either 0 capacitive coupling case or 0 lt gg lt 01 inductive coupling case Recall that in the RWA an offresonance counter rotating term is ignored in the dynamics This is typically a good approximation for experiments with superconducting qubits because the time to do an operation N 10 us is much slower than the inverse time scale of the qubit transition 01 us These time scales give an amplitude error from the counterrotating drive of order 1 100 and a probability error of order 104 Most theories for qubit logic gates make this approximation 11 THE GREENBERGERHORNEZEILINGER GHZ PROTOCOL A Triangular coupling scheme In the rotating frame interaction picture in the absence of coupling the system s Hamiltonian is represented by a zero ma 1x and us all computational basis states 000 001 010 100 011 101 110 111 have the same effective energy Eeff0 no time evolution The pairwise coupling Himl2E1go of10390391 039oj1 par tially lifts the degeneracy which results in the new energy spectrum Eim 3g23 22g 22g 2 g g2 g gZ NE2 g 2 2 and the corresponding H eigenbasis HGHZ ea Hw ea Hest E 000 ea 111 ea W ea W EBH I DEB 1 1 gt 1 2gt 1 239gt ducting qubit architectures described in the rotating wave 3 approximation RWA by the Hamiltonians of the form 16 where W 100 010 001gt agphysastugaedu lmartinis physicsucsbedu W 011 101 llO 10502947200878l0103054 0103051 2008 The American Physical Society ANDREI GALIAUTDINOV AND JOHN M MARTINIS lwlgtltl100gt l01ogtgt6 IW1 gtl011gt l101gt6 W l100gt l010gt 2l001gt6 lw2 gtl0ngtl101gt 2l110gt6 4 Since the coupling does not cause transitions within each of the degenerate subspaces nor does it cause transitions between different such subspacjs it is impossible to gener ate the lGHZ l000gt llllgt 2 state from the ground state l000gt by direct application of Him Instead we must rst bring the l000gt state out of the HGHZ subspace by for ex ample subjecting it to a local rotation R1 in such a way as to produce a state by that has both l000gt and llll components That is only possible if all onequbit amplitudes 011 3 in e resulting product state by R1l000gtozll0gt Blllgtoz2l0gt32l1oz3l0gt33llgt are chosen to be non zero which means that in the computational basis the state M will have eight nonzero components We now notice that in the H basis the threequbit rota tions are blockdiagonal 3 2 1 XsXs Kg X5 63 is3 i scz Ecsz is3 c3 csz i sc2 i sc2 Ecsz Cl 3sz isl 3c2 Ecsz i sc2 isl 3C2 Cl 332 c is c is 69 EB ZS c ZS c lt3 lt2 lt1 YY Y Y c3 s3 Esc2 s3 c3 Ecs2 Esc2 Escz Ecsz Cl 332 sl 3C2 Ecsz Escz 31 3C2 Cl 332 lt Cs gtlt CS 5 where Xexp ieof2 Yexp ieo 2 k123 and CEcos62 and sEsin62 For 92772 the corre sponding 4gtlt4 blocks acting on the HGHZEBHW subspace are Ecs2 1 i i6 6 1 i 1 6 i6 6 i6 6 1 i 6 i6 i 1 4 4 X77lt l l 1 l YW 6 772 6 6 1 1 6 6 1 1 This shows that Y z provides a convenient choice for R1 We may thus start by generating the socalled uniform superpo sition state l gtumrorm E UnglOOOH l001gt l110gtl111gt leooogt 12HGHZgt 62IWgt W E HGHZ EB HW 7 The entanglement is then performed by acting on l gtunif0m with Uimzexp iHimt thus inducing a phase difference be tween the GHZ and WW components this step works only for g see Sec IV UintYw2l000gt EmMIGHD 6quot 3V32lWgt lW39W a3 2t 82g gjt 8 To transform to the desired GHZ state we rst diagonal ize the Kg and W734 operators to get the unimodular spectra AX emu 6474 6474 gag4 KY my4 ez 4ei 4 my4 9 the eigenbases XlX1gtlX2gtlX3gtlX4gt and 3 lY1gt lY2gt lY3gt lY4gt whose vectors are given by the col umns of Yam 4 and X334 correspondingly Using the X basis we notice that both states GHZgt lX1gt 2 lx4gt 6quot l 36 quot Uimezl000gt 7 T l quot3 W 2 6pm 10 belong to the same twodimensional subspace lX1gtlX4gt Therefore by performing an additional X 72 rotation we can transform UimY zl000gt to Xw2Um1Y72l000gt mew4 lGHZgt a 11 provided the entangling time is set to give l lzw or GHZ 772lg g l Any other GHZ state l000gte39 llllgt can be made out of the standard GHZ state by a Z rotation applied to one of the qubits as usual The protocol may be compared to controlledNOT logic gate implementations 16 that used various sequences RZUCNOTR1e39 4CNOT detUCNOTl With entan gling times CNOTT7r2g 1STltl6 Thus for 0 the entangling operation proposed here will be of same du ration as the fastest possible CNOT We conclude this section by noting that in its present form the GHZ protocol cannot be used to generate the W state This can be seen by writing lWgt lX1gtlX2gt 0103052 MAXIMALLY ENTANGLING TRIPARTITE PROTOCOLS FOR X3 X4 which shows that our XUimY sequence does not result in a W since the nal X z rotation cannot eliminate the le and lX3 components Also lWgtI ilY1gtIY2gtlY3gtilY4gtl 12 and 3e716 2 ilY1gt le mgt im IE 13 and thus no choice of 8 will work for the YUimY sequence either Ywzuimezioom eimlt B Linearcoupling scheme In the case of linear coupling say llt gt2 and 2H3 the energy spectrum is given by Em g g 4 4 4 at o 0 at 2g2 22 2 14 with eigenbasis l000gt l111gt WW CH001eg 010i001 W CltgtH0ngt eltgtglroigt 110 WW CHH001gt adglow 001 W W CHH011gt eltgtg101gt 110 I I gtl001gt l100gtl3 IW gtl011gt l110gt5 15 where C are normalizing constants We have wgt A wgt A Wgti l e g l e g 7 A Cltgt eltgt 67 A CH eltgt 67 16 and similarly for lW Our GHZ sequence then leads to the entangled state Uimleooo amm eHz 32e 3AHW WM erlt39AltgtHwgtltgt WM 17 with azg t 61e t Since tgt0 in order for the Km postrotation to give a GHZ we must restrict coupling to g 0 and set the entangling time to 1 6sz 77 Zlgl An altenia tive GHZ implementation for superconducting qubit systems RAPID COMMUNIC PHYSICAL REVIEW A 78 010305R 2008 with capacitive coupling has recently been considered 18 There individual qubits were conditionally operated upon one at a time 111 THE W PROTOCOL We now turn to the W protocol Equation 16 suggests that control sequence YUimY may still give a W provided a proper adjustment of ilY1 Hz and Y3 ilY4 amplitudes is made by a physically acceptable change of the system s Hamiltonian In the context of Josephson phase qubits such modi cation can be achieved by adding local Rabi terms to Him for instance H tQ2aaio Him The energy spectrum then becomes E3 6m Xltgta err Xe 6ltgt 6ltgt err 67 18 with 4 8 2 t 02 X W8 z i 8 0 02 19 The rst two eigenvectors are llt1gt33gt CIT 1 2mg g I XliGHZ mm IW gt3 20 with normalizing constants Ct kl2 After some algebra we Hangloom errlt4SWIMmweirWm W x m Bmequot 4IY3gt ie 4IY4gtl 21 where A8 QXg 29 X lg gm X8 20 5 B g 0 Xg X quot5g Q X8 E 5 22 and 04 6 Xt 8 2t It is straightforward to verify that additional Ym rotation applied to this state produces a W see Eqs 9 and 12 Ywzummlooo r sgnrg 916quot th 23 provided we set thw Jig g Qz g QZ IV ADDENDUM ISOTROPIC HEISENBERG EXCHANGE gXX YY ZZ Maximally entangling protocols introduced in the previ ous sections are singular in the limit gt g which corre sponds to the isotropic Heisenberg exchange interaction Even though this limit is not met in superconducting qubits for completeness we brie y discuss it here It is obvious that when g the uniform state Y ZIOOO is an eigenstate of the interaction Hamiltonian Consequently 0103053 PHYSICAL REVIEW B VOLUIVIE 53 NUIVIBER 18 1 MAY 1996II Weakcoupling phase diagram of the twochain Hubbard model Leon Balents and Matthew P A Fisher Institute for Theoretical Physics University of California Santa Barbara California 931064030 Received 30 August 199539 revised manuscript received 9 November 1995 We present a general method for determining the phase diagram of systems of a nite number of one dimensional Hubbardlike systems coupled by singleparticle hopping with weak interactions The technique is illustrated by detailed calculations for the twochain Hubbard model providing controlled results for arbitrary doping and interchain hopping Of nine possible states which could occur in such a spin12 ladder we nd seven at weak coupling We discuss the conditions under which the model can be regarded as a one dimensional analog of a superconductor I INTRODUCTION Onedimensional 1D electron systems provide an impor tant testing ground for understanding electroncorrelation ef fects Many methods have been applied to the problem of a single Hubbard chain and there is general agreement that the system remains for repulsive interactions in a Luttinger liquid state with gapless spin and charge modes1 The 1D analog of a superconductor a state with one gapless charge mode and dominant pairing rather than chargedensity wave correlations does not arise in that case Twochain systems are interesting as a rst step towards true 2D materials and may be relevant for some experimen tal systems2 Moreover on a ladder statistics are more im portant since particles can exchange without passing through one another However the theoretical situation in such models is much less clear3 8 Recent simulations sug gest that states with dominant pairing correlations can indeed arise9 In this paper we present a systematic weakcoupling analysis of two Hubbard chains coupled by singleparticle hopping tL Our approach is a controlled renormalization group valid for small U but for arbitrary interchain hopping and lling n 10 The general methods described here may be applied to any system composed of a nite number of Hubbardlike chains with weak shortrange fourfermion in teractions The possible phases of such models can be characterized by the number of charge and spin modes which are gapless at zero momentum For an N chain system the number of gap less charge modes can vary from zero to N and likewise for spin Remarkably of the nine possible phases for two chains seven are realized within the simple Hubbard model at weak coupling re ecting the proliferation of marginal operators Denoting a phase with x gapless charge modes and y gapless spin modes as CxSy the small U phase diagram as a func tion of interchain hopping tL and lling n is shown in Fig 1 Particularly noteworthy is the phase C180 present with purely repulsive interactions positive U This phase has a spin gap and a single gapless charge mode and is thus the 1D analog of either a superconductor SC or chargedensity wave CDW As found by other authors the pairing of up and down spins in this phase is dwavelike in the sense that the pair wave function has opposite sign in the bonding 016318299653181213391000 IUi DJ and antibonding bands k LO779 4 A more precise and general de nition of this type of pairing is given below in terms of bosonization Two alternative physical criteria distinguish the two pos sibilities for an array of weakly coupled ladders depending upon the relative strength of the interladder coupling and of quenched impurities If the impurity interactions dominate localization must be avoided within each ladder indepen dently This requires a very slow decay of pairing correlations11 In particular if the equal time pairing corre lation function AxA101lxquot where Acmcu this requires Klt Kc13 If interladder couplings are stron ger than impurity scattering two or three dimensional phase coherence can set in and further stabilize the SC It must compete however with the formation of a CDW which due to pinning is an insulating phase For 12lt Klt2 the situation is best summarized in Fig 2 which shows a schematic phase diagram at xed disorder as a func tion of interladder pairhopping P and interladder Cou lomb interaction V The exponent K determines the curva ture of the SCCDW phase boundary in this plane VP2 1quot2 The traditional requirement12 of domi nant SC correlations gives Kc1 see below and corre sponds to a straight line on this plot Note that for Klt12 and any weak but still larger than the impurity potential pair hopping the system is a SC conversely for Kgt2 and 131 ti C1s1 T c051gt C150 C232 C1s0 C281 C1324 CISO quot 0 l 0 12 1 11 FIG 1 Phase diagram in the UHO limit 12 133 1996 The American Physical Society 12 134 Vt T M FIG 2 Fate of the C180 phase for various value of K provided the interladder pair hopping matrix element is larger than typical impurity pinning energies any weak interchain interactions it is a CDW Interestingly the spingapped C180 phase occurs in two different regimes Fig 1 one for doping 61in away from half lling and the other when the Fermi energy coin cides with a band edge kF10 In the former case pairing correlations develop upon doping the spingapped Mott in sulator at half lling n 1 as in Anderson s original reso nating valence bond picture for superconductivity in the cuprates13 The critical doping 60 at which C180 gives way to a gapless spin state C281 and C282 is large for weak interchain hopping decreasing from 661 for small t L to 600 as tL gt2t Note that the phase C282 is the 1D analog of a Fermi liquid with all spin and charge modes gapless The presence of the spingapped state C180 near km 0 can be attributed to the coincidence of the Fermi energy with the Van Hove singularity at the 1D band edge II MODEL The twochain Hubbard model is described by the Hamil tonian HH0HU with H0 2 itclacxla Clad itlcladxaH C a xa HUE UcTcxTclicxiclt gtd 21 where 0 cl and d all are fermion annihilation creation operators on the rst and second chain respectively and a Ti is a spin index The parameters t and t L are hopping matrix elements along and between the chains and U is an on site Hubbard interaction Equation 21 has the usual U1gtlt 8U2 chargespin symmetry For weak coupling it is natural to proceed by rst diago nalizing the quadratic portion of the Hamiltonian This is achieved by canonically transforming to bonding and anti bonding band operators 1pmcai 1ida with 139 12 In momentum space H 0 becomes 17 1402 I erpwmpwup 22 where 1fL2f cosp and 2tL2f cosp For t Lgt2t the two bands are completely separated At half lling the system is then a band insulator and when doped becomes an ordinary spin12 Luttinger liquid denoted LEON BALENTS AND MATTHEW P A FISHER 2 C181 see Fig 1 For t Llt2t the bands overlap over some range of energies When the Fermi level lies within this re gion interaction effects must be reexamined in detail It is suf cient to consider the behavior of the system only near the two Fermi momenta kFi de ned by 61kFi u The chemical potential u is xed by the requirement kF1kF2n7T where n is the particle number per site The decomposition 1pm pRlaeikFier szlae ikFix gives up to a constant 1102 fdx vi iaiawaiaiwliaiaxwLia where vi2t sin kFi The allowed four Fermi interactions are highly constrained by symmetry In addition to U 2 invariance these terms must be preserved by time reversal parity chain interchange and spatial translation operations At generic llings the two Fermi momenta are incommen surate and the symmetry under translations is effectively doubled into independent transformations in each band To delineate the couplings in a physical way we employ the notation of current algebra 1 JiR WieiaWRia JiR 5 WriiagaBWRiB 1 LR piglalpRZa LR WrTe1a0aBWR23 MiR i Rit Rilp where 0 denotes Pauli matrices Although we have not ex plicitly indicated it here all the currents in Eq 24 are de ned as normalordered quantities see Appendix A Left moving currents are de ned analogously There are eight al lowed interactions connecting left and right movers for ge neric llings with Hamiltonian densities NRaB l Rial RzB 24 ngiritg1p JlR J1Lg2p JZR J2Lgpr1RJ2LJ2RJ1Lgt gla JiR J1Lg20 J2R J2LgxaJ1R J2L J2R J1LgzpLRLLLrTeLi gmLRLLLLD 25 8ix additional interactions are completely chiral TABLE I Hubbard model coupling constants The gology no tation is given for comparison with Ref 14 Coupling gology Hubbard value Elp gilAAAZigjLAA T U4 gsz giBBBBZTgJZBBBB U4 Exp gilBABZingBA T U4 gsz gAABBZTgAABB U4 E 1 039 28111141414 U E20 28113333 U Exa39 28111343 U gNm 2gAABB U 53 WEAKCOUPLING PHASE DIAGRAM OF THE TWOCHAIN 7 AipUiR Jir A2pJ RJ L rpJ1RJ2R J1LJ2L A10J1R39J1R JiL JiL MAJZR JZR J2LJZLXU39J1RJ2RJ1LJZL39 12135 The couplings in Eq 26 renormalize velocities of vari ous charge and spin modes and can be neglected to leading order in U for what follows Additional operators are needed to treat urnklapp processes at special dopings 737 MM1RMILMILM1R MM1RM2LM1LM2RMIRMZLM1RM1LM1RM1LM2RMIn lt tu1Nla1 NLa NRa Nia 512NlmpNLgaNRalega The singleband urnklapp term is nonzero only if km 7r2 At half lling the three interband urnklapp terms gm ng g39m are nonvanishing III RENORMALIZATION GROUP The Hubbard model values for the coupling constants obtained from Eq 21 are shown in Table 1 To analyze the behavior of the weakly interacting system we employ the renormalizationgroup RG approach In the RG short wavelength modes are progressively eliminated in a system atic way leading to differential equations for the renormal ized coupling constants which describe the physics of the model at longer and longer length scales The ow equations for this system in the absence of u were rst obtained in Ref 14 using conventional diagrammatic methods The full set of RG equations is more directly obtained using current algebra described in more detail in Appendix A Away from half lling they are 21pB 3 gfp Egg 7 ugh 3 g2p7a gfp Egg 7Bg w 89p 7 3 gfpt E82217 B gm 7 agir 513217 ZBgngw 39 2 a 2 gm 73192177 ng 21182198217 1 3 gm 7193177 51312177 Zgngwxzfgwgm EgMgw gmgoltrgzp80p 80a2 28xagwgt gm Zagipgiu glu 2382p82ugt 31 where gErrv1v2gl aEv1v22v1 3301 Uz202gt 80pagip382p 28xpgt and g00ag1lt7 27 Bg2072gw The dots indicate logarithmic derivatives with respect to the length scale ie g 33g 3 where lnL Equations 31 are valid until maxgl701 To analyze them we employ the following approach Starting with the appropriate initial values cf Table 1 we integrate the equa tions numerically If as gt00 all the couplings approach nite values the procedure is controlled since maxgl00 becomes arbitrarily small as Ugt0 If any coupling diverges we determine the asymptotic behavior of all the couplings with Eqs 31 Speci cally we imagine integrating the ow equations up to a scale at which point the largest coupling gmaxmaxgl satis es UIltgmxltl This allows us to ignore the higherorder terms 0g3 in the RG ow equations As UHO the res caling parameter gt00 so we need only analyze the as ymptotic large behavior of Eq 31 To do so we make the ansatz glkg1017k where 1k is the scale at which the couplings diverge Equa tions 31 then reduce to a set of coupled quadratic equa tions for the gm The search for appropriate solutions is considerably aided by the numerical integration of the ow equations After locating a divergence which xes k we plot 1 7kgl versus from which g10 is extracted from the intercept with the line 1k Applying this procedure for generic llings with Hubbard initial values we found three distinct phases in the regime with both bands partially lled for U0 For a248 the ows are stable with xedpoint values gf jrgfagf gf0 When 43SaS48 the system is singly unstable with g2 7 1B and all other g10 0 For more comparable Fermi velocities 1 ltaS43 all the opera tors except gfoo 0 diverge but in such a way that agwo Bgzw lt0 and gtpo 714g100gt0 The behavior for 12ltalt1 is obtained by interchanging band indices in all quantities The physics of these phases is elucidated through the use of Abelian bosonization123910 With the convention ARLia ned as 151a R1a 15an and 61a Isma an They sat39 isfy lt15x 601 7139 sgnx7y2 A further canonical transformation to lt15 61plt15 6 lt15 611 and lt15 6 lt15 6 7 lt15 611 yields the spincharge separated Euclidean action s17 grwxqswwadaziwzavmm 39 32 12 136 where v p039 Using the scheme discussed in the Introduc tion the noninteracting system with both bands occupied Eq 32 is classi ed as C282 The large a phase found above is also of C282type though it contains the additional marginal couplings gip gxp xlp Axpm and A which makes the behavior highly nonuniversal16 In the intermediate state 43302348 g2 becomes large and negative Using bosonization this interaction ne glecting unimportant gradient terms is Swmgajiiaoa 87at as where the coef cient M is cutoff dependent In the scaling limit it is appropriate to expand the cosine and obtain a true mass M for 620 The resulting phase is therefore C281 For aS 43 an analogous cosine appears in the 10 sector and the asymptotic divergence of glp and gm is such that the interband hopping terms sum to Swim COSE 7p00501UCOSE02aa 34 where as pEq 1pi 2p2 It is natural to assign masses to 61 and 620 after which Eq 34 acts to x 151p 52 up to gapped uctuations since glpgt0 Note that m 15 p appears as the phase difference between pair ing operators in the bonding and antibonding bands so that this value of 151p leads to 1T 1i 2T 2i1lt0 A natural de nition of dwave in this context is just eim Plt0 The resulting state has only a single un gapped charge mode with dual elds 61p 151p and will be labeled C180 It may be characterized by a single dimension less stiffness K 1 p and a velocity v 1p such that the effective action for 61p is K SpTpf vpax6p2via7396p2a with a similar dual Kw 1p form for p1 Equation 35 interpolates smoothly between a CDW at large K1p and a superconductor at small K1p The pairing exponent KK1 p2 while the CDWlike correlator n2xn20ccos2kF1kF2xx2KP plus power laws 1x2 at k0 We are unable to determine K1p and v 1p in a controlled way because they depend on the entire cross over from the noninteracting state to the C180 xed line see Appendix B However heuristic calculations suggest that superconducting uctuations K 12 increase with increasing 1217112 Interestingly the usual powerlaw term at ZkF is missing from the densitydensity correlation function in the C180 phase as noted by Nagaosa8 in a similar model due to strong uctuations of the 61p and 1510 elds It remains to discuss the behavior at several special points in the phase diagram When km 772 umklapp pro cesses imply g2u9E0 For tLgtt this occurs with band 1 empty and one gets the usual C081 spindensity wave as LEON BALENTS AND MATTHEW P A FISHER 2 shown in Fig 1 For t Lltt we must consider interband cou pling via Eqs 31 For almost all ratios of the velocities we nd that a charge gap develops in band 2 simultaneously suppressing the other potential instabilities leading to a C182 phase Surprisingly over the narrow range 06 BS 085 gm renormalizes to zero yielding instead the C180 state see Fig 1 The behavior at half lling is more dif cult to obtain because it requires the inclusion of the gxu gm and gm operators in Eq 27 The RG equations in this case are given in Appendix A Their analysis indicates a completely gapped C080 phase as suggested by a large U picture of coupled antiferromagnetic Heisenberg chains15 The nal remaining special point occurs when the Fermi level lies precisely at the bottom of band 1 It is outside the scope of conventional RG s because the dispersion in band 1 is quadratic with the Hamiltonian 1 Am L f a Go where 61 for the quadratic band but must be taken as a small parameter to control the perturbative treatment The allowed couplings are gzp g2 Azp and the four interband terms I 7172MPJR2JL2J1uaJR2JL2 J17 Eli 2a i23 13 1aHC3 37 where J1 1a 1a and J1 1aaaB IB2 The RG equa tions in this case must be derived via conventional diagram matic techniques see eg Ref 14 since current algebra methods rely on having a linear spectrum at the Fermi en ergy They are 39 i 2 39 i 2 2 39 i is 2 g2p uz2 gZUiigaizuz upi ym z 1 2 1 2 2 uaiiyua 117611721 fut Lile22v4yupi3gU4gpu 38 U C150 1 2 C SO c251 2281 C S 23 C181 c252 I use Vc1so I ClSO kFl 0 DC sz 39l 2 n1 n FIG 3 Cut through the phase diagram at constant t i Note that two ofthe lines in Fig 1 associated with special llings have broad ened into fans for nite U The vertical line at n1 half lling does not broaden because it corresponds to the exact point of particlehole symmetry of the original Hubbard Hamiltonian 53 WEAKCOUPLING PHASE DIAGRAM OF THE TWOCHAIN where m 2r7l y11 mquot um 2 v m v 47139 g2upya 2 p 27rvz we have taken a mo mentum cutoff of 1 Because of the relative simplicity of Eqs 38 we have been able to analytically show an insta bility for Hubbard initial conditions15 Analysis of the ow asymptotics indicates that at this special point uctuation induced attractive interactions vlt0 in band 1 populate it with spinless bound pairs Simultaneously g2Ult0 creates a spin gap in band 2 and u Josephson couples the two bands gapping the out of phase charge mode to leave a C1 SO phase This RG analysis holds for Ult elt 1 but we expect the ten dency to pairing to increase with 6 due to the increased density of states for interband scattering Physically the presence of the Cl SO phase at kn 0 can be attributed to the Van Hove singularity at the band edge Note that this Van Hove mechanism also leads to dwave like pairing in the sense that the interband pair hopping interaction at remains repulsive encouraging zxszxuz2Tzxullt0 as in the other C1S0 phase The above results at special llings are valid only at iso lated points for in nitesimal U For nite U the RG sug gests that the regions of attraction of these phases widen into fans of width 6tiNexp7ctU where c is a constant The nonuniformity of Eqs 38 allows for additional structure within the fan In particular because the instability is driven by the g term we expect a narrow intermediate wedge of C2S1 as shown in Fig 3 In the future it will be interesting to generalize these calculations to threechain systems N 3 to help clarify which features are particular to even N IV COMPARISON WITH PREVIOUS WORK Problems of coupled Luttinger liquids have been investi gated previously by many authors Indeed our RG equations for generic llings were calculated by Varma and Zawad owski in Ref 14 These authors noted the existence of an instability in a large range of parameters Using purely RG methods without the bene t of the organization of a current algebra description or the interpretation allowed by bosonization they did not identify the nature of the C1S0 phase Other authors have also noted the enhancement of pair tunneling processes18 Some years later Penc and Solyom19 PS calculated the twoloop cubicorder RG equations which exhibit xed points instead of instabilities Because these xed points oc cur at values of the coupling constants of order one how ever the calculation of properties from these xed point val ues is uncontrolled Indeed experience with bosonization leads us to believe that the instabilities at the oneloop level signal the development of various gaps which cannot be captured within such a description An approach much closer to our own was taken by Fabrizio10 who employed bosonization to rewrite the effec tive Hamiltonian of the putative xed points of PS These xed point forms contain the same cosine terms found above Eqs 33734 which allowed Fabrizio to postulate an appropriate set of gaps for the phases he found Three of the phases in his phase diagram Fig 9 of Ref 10 correspond to ours LLl which is our C1S139 LL2 our C2S2 state39 and I our C1S0 phase 12137 We note however that the assignment of a particular phase to a particlar set of Hubbard parameters based on the PS xed points is incorrect Indeed a detailed comparison with our results shows a disagreement in the location of the phase boundaries and more seriously the complete absence of the C2S1 phase In addition using the full PS equations at large U Fabrizio nds two extra phases denoted II and III As we have emphasized however these xed points par ticularly strongcoupling ones have no particular physical signi cance and the existence of additional strongcoupling phases is questionable Our approach does not make use of these unphysical xed points and relies only on the smallcoupling results of the RG Instead we have matched the asymptotic form of the weakcoupling instabilities to the possible inherently strong coupling phases which are characterized by various gaps The nature of the RG instabilities are entirely contained in the oneloop RG equations and because the twoloop correc tions obtained by PS are small in this regime they need not be included in the calculation see the discussion in Sec III ACI IOWLEDGMENTS We are grateful to A Ludwig and especially D Scalapino for innumerable fruitful discussions This work has been sup ported by the National Science Foundation under Grants No PHY8904035 and No DMR 9400142 Note added in proof Recent work with Lin20 has discov ered a very weak instability of the asymptotics leading to the C1S0 phase This implies that in the extreme weak cou pling limit Utlt10 5 the C2S1 phase expands to ll the majority of the C1S0 region in Fig 1 In this limit the C1S0 phase persists only in a narrow sliver near half lling For small but nonin nitestim al coupling 10 5 lt Utlt 1 how ever the asymptotic analysis leading to the phase diagram in Fig 1 is correct and the main conclusions of the text are unchanged APPENDIX A CURRENT ALGEBRA Current algebra methods allow among other things an algebraic calculation of the oneloop RG equations It is dis cussed in some depth in Ref 17 Here we give a very terse description of the method stressing the new points of our calculation All the currents are de ned in terms of the ferm ion elds 11R M a a 12 which obey the operator products an a waxxxwtbimpwmm T 00 Nawaaho 1 A1 l LaaxgtT Lbe gt 27 gt where zu va 7quot ix The operators products should be under stood to hold when two points x r and 00 are brought close together as replacements within correlation functions Equations 11 allow a simple calculation of the operator products of the currents de ned in Eq 24 As an example consider the product J RIJRl Performing all possible con tractions gives 12138 1 J1R21J1R00NAmongWanMammal1140 10130 5132 LEON BALENTS AND MATTHEW P A FISHER 53 A2 516 1 l 1 Ni 27m 5ae5py mi 1Ra 1R i mille lIiRyi iRa 1R lljiRy 1Rei 1171awa 12 8 1 N27rzl2 1 k k mlza J1RO1 Note that the structure arises as a result of extending the normalordering symbol to the full product To perform the nal simpli cation we used the relation 17 17 6 139 E 0k Computation of the full set of operator products is equally simple though tedious The ones needed away from 12 lling are 2 JaRJbRN szuy 6gb gt A4 1 f 12 w k JaRJbRN 2712 28 271 JaR 5gb A5 a a J J 1N7 ii LRLR 2 n3922122239n39z2 27121 A6 k k LILTN 128 16 JIR JZR RR 27122122 2 27122 27121 1 JIR JZR 7 J 7 4 27122 27121 A7 J m J m 1 TN LRLR 27122 27121 A8 LRJaRN 1027TZaLRgt A9 L1 N71R1 1 LT A10 HR 2717a R I a 1 I LRJaRN 1 EL All 1 1 N 7 n1 11 LRJHR 1 ZMHLR A12 1 a 1 I LRJHRNPI TWLR A13 L RTJHRNv I Maui A14 L RJ N i kLk71 16 L A15 R 2m 2 R 4 R A3 DU N i kLkli71 16 Ll A16 R RR 2mg 2 R 4 R J 1 N 7R MaRMbR away 27 5217 A17 25 MambRNmMaR A18 251 MlRJbRN erb A19 2 where the coordinates of the two operators on each lefthand side are consecutively xr and 00 The operator prod ucts MHRLR and M 2 RL are also nonvanishing but do not enter in the oneloop RG away from half lling but see below Similar forms hold for the leftmoving currents but with 20 Hza The renormalizationgroup equations are obtained very simply from Eqs A47A19 We use the functional inte gral formulation in which the terms of Eq 25 appear as interactions in an Euclidean action S E fdxd 7 and 39 d Mzde 31 A20 To perform the RG the exponential is expanded to quadratic order in 5 A typical term takes the form 2 81a 2ZWlt11Rltzgt111ltzgtJ1RltwgtJ1Lltwgtgt A21 where fsz denotes a fourdimensional integral over the two complex planes 2 and w As in any RG we wish to integrate out the shortscale degrees of freedom to derive the effective theory at long wavelengths and low energies Here this is accomplished by considering the contributions to Eq A21 when the two points z and w are close together near the cutoff scale It is then appropriate to employ the operator product expansion which gives 5 1 1 1039 i39 ljk39ljlf Jk 1 I 2 1 1 ZW27rzliw1 27rz 7w IRIIL A22 At this stage it is necessary to more carefully specify the cutoff prescription We will choose a shortdistance cutoff a in space but none in imaginary time For a rescaling factor b we must then perform the integral WEAKCOUPLING PHASE DIAGRAM OF THE TWOCHAIN 12 139 I I d I d l lnb NT T l T x 7 2 7 1 Km 7w T2w2vi x2 2m WLR 2m Mm A23 over the relative coordinates x r Using Eq A23 in Eq A22 gives g1 27ml lIlbelR39JlL A24 2 which when reexponentiated renormalizes g7 and for bed gives the rst term in the ow equation for g in Eq 31 Similar calculations for the remaining terms result in the denominators lzliw lzziw2 2 and zliwlz 7w which yield the factors l27rvl l239n39v2 and l7rvlvz respectively and thereby lead to the a and B factors in Eqs 31 All other forms appearing in the integrals over the relative coordinates give zero contribution re ecting causality of the ballistic fermion propagators For the special case of half lling additional operator products are required To simplify the analysis we note however that at half lling the two velocities are equal v1 szv It is therefore not necessary in this case to dif ferentiate between 21 zzEz Making this assumption the new relations are 6 6 6 6 B B NRa NlihEN 7 2 6J1RJ2R A25 1 5 7 a 27 yJZR 0394 53611 03ml 1 NRQHHRNRNRab Nim aR77 ZTTZNimb 1 tray NRaBJIRNRTNRyb 03 y l NRaEJZRNRTNRLMM a l o 7 N BJIRN7727TZ T 3w 039 if T 2717 2 MW NimgIZRN7 l NRa LRN 7 RULBMZR l NlmpLRN RULngw l NRa Li7 7 moiJ41 1 000 NRQBLRN 7 m fMZR 1 Hum NQBLR772M 2 ML 1 tray 1 NRangyN fa Im l ayeal mwuNgggjgaaw yaa MIRLRN 27 NRaE a BNT MiRLi7 27 Ra Ty B MZRLN NRa a NT MiRLRN 2 Ra Ho 1 a NRa gt 1 MIRLRN27rz 2 T 1 Ha MZRLR77 2717 2 tray 1 B MiRLRNR 2 a Nlmb l N 2712 Ty fuuiamw MIRN as 2 MIRNWW 7 dummy 02 2 LR7LR Ty039a MzRNlmaN 7 Ty g LtLwya T l MZRNRaEN 7 7 With these operator products a full set of RG equations can be derived at half lling using the previously described pro cedure After some lengthy algebra one n s 12140 2 3 2 2 2 811382pJr gw gmi 78212182122 82122 gt 2 3 2 2 2 2 gxp gzp Egza gmi igtulgtuligtuligxugt 1 2 2 2 gla2g1pg1r 5g 4gm 4gmgm gm 1 2 2 2 gm Zgngur 582177482218222 4gzu2igxw 3 gtpg0pgtp nggw gx gzmigzuk gtvg00gtp 1 gorggor2gwgtgw 4gmgm1 8222gt 3 gm Zgzp 58wgtgml 3 2g1p g1agtgm 4gxpgm 3221 Zgzp gwka gzugm 1 3 gm 2g ggm2g1p 5amp1 3222 igmlngr 2g2pgur2gxu 3 1 gm 2gxp ggw2g1r 5g A26 LEON BALENTS AND MATTHEW P A FISHER 53 where we have assumed as dictated by chaininterchange symmetry that glpg2p and glag and that as is the case for generic Ii glug2u0 APPENDIX B STIFFNESS OF THE C150 PHASE In this appendix we attempt to compute the stiffness K p in the Cl SO phase by following the RG ows and usual standard matching procedures We nd however that for v1 122 the value of KM depends on the details of this crossover and hence is not accessible by the present method Our strategy is to integrate Eqs 31 until the diverging couplings are order I in the renormalized action It is then justi ed to assign gaps to the appropriate elds and simply integrate out the massive modes In the Cl SO phase we need only consider the charge sector both spin modes are gapped and may be integrated out without affecting the long wavelength properties of the charge modes From the asymp totic analysis of Sec III we know that the remaining 01 interactions take the form nt 7 1 BJ1RJ1L LIZRJZL iJlRJZL iJZRJlL igcos1W ipgt1 B1 where FgtO and ggtO are 01 constants It is crucial to note that F is not precisely known It depends upon the point at which one stops integrating the perturbative ow equa tions This re ects the fact that what one really wishes to do is calculate crossover properties from the unstable xed point at UO to the nal Cl SO xed line Some such cross over properties may require a knowledge of the RG ows along the entire trajectory between these xed spaces We will see that this is the case for KW From Eq 21 we may easily derive the bosonized rep resentation of the charge sector of the action Following the notation introduced in the text one nds s55 dx er Fra372vaap2 ra372va p2 A72I ia3x6p xaip A21 Bia3x p3x p2i370p3x p2i3707p3x p47rl g cos47r p where vivlv22 Av17v2 and FF27r In the renormalized theory with Fg of order I it is legitimate to expand the cosine around its inimum value qLP GPr 0 Integrating out the a eld effectively re moves all terms containing lt15 3 in Eq 22 only irrelevant Laplaciansquaredtype couplings are generated The 61 eld is isolated by integrating out the 63 and 151 elds to yield B2 1 i 11A72rwiagt12 Mil dx dT Hz a m im l X3x0pzm576pz B3 Comparison with Eq 35 gives
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