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# INTRO STRUCT & PROP MATRL 101

UCSB

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This 3 page Class Notes was uploaded by Miss Alberto Prohaska on Thursday October 22, 2015. The Class Notes belongs to MATRL 101 at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 65 views. For similar materials see /class/226931/matrl-101-university-of-california-santa-barbara in Material Science and Engineering at University of California Santa Barbara.

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Date Created: 10/22/15

PRL 101 050401 2008 PHYSICAL REVIEW LETTERS week ending 1 AUGUST 2008 Collective States of Interacting Fibonacci Anyons Simon Trebst1 Eddy Ardonne2393 Adrian Feiguin1 David A Huse4 Andreas W W Ludwig5 and Matthias Troyer6 1Microsoft Research Station Q University of California Santa Barbara California 93106 USA 2California Institute of Technology Pasadena California 91125 USA 3Nordita Roslagstallsbacken 23 SE106 91 Stockholm Sweden 4Department of Physics Princeton University Princeton New Jersey 08544 USA 5Physics Department University of California Santa Barbara California 93106 USA 6Theoretische Physik Eidgenossische Technische Hochschiile Zi39i39rich 8093 Zi39i39rich Switzerland Received 4 February 2008 published 30 July 2008 We show that chains of interacting Fibonacci anyons can support a wide variety of collective ground states ranging from extended critical gapless phases to gapped phases with groundstate degeneracy and quasiparticle excitations In particular we generalize the MajumdarGhosh Hamiltonian to anyonic degrees of freedom by extending recently studied pairwise anyonic interactions to threeanyon exchanges The energetic competition between two and threeanyon interactions leads to a rich phase diagram that harbors multiple critical and gapped phases For the critical phases and their higher symme end points we numerically establish descriptions in terms of twodimensional conformal eld theories A topological symmetry protects the critical phases and determines the nature of gapped phases DOI 101lOSPhysRevLett101050401 Twodimensional topological quantum liquids such as the fractional quantum Hall FQH states harbor exotic quasiparticle excitations which due to their unusual ex change statistics are referred to as anyons Interchanging two anyons may result in not only a fractional exchange phase but may also give rise to a unitary rotation of the original wave function in a degenerate groundstate mani fold This latter case of nonAbelian statistics is proposed to be exploited in the context of topological quantum computation 12 Intense experimental efforts 375 are currently under way to demonstrate the nonAbelian char acter of quasiparticle excitations in certain FQH states as proposed theoretically 67 Given a set of several nonAbelian anyons we can ask what kind of collective states are formed if these anyons are interacting with each other A rst step in this direction has recently been taken by studying chains of Fibonacci anyonsquot with nearestneighbor interactions 8 Fibonacci anyons represent the nonAbelian part of the quasiparticle statistics in the k 3 Zkparafermion ReadRezayi state 6 an effective theory for FQH liquids at lling fraction V 125 9 A single Fibonacci anyon carries a topologi cal charge 739 Two such anyons may combine fuse so the pair has charge 739 or has no charge which is denoted I This is analogous to two SU2 spinlZ s combining to either spinl or spinzero total spin A twoanyon interac tion assigns different energy to the two possible charges of the pair just as a Heisenberg exchange interaction does for the two possible total values of spin of a pair of SU2 spinlZ s For a chain ofFibonacci anyons with a uniform pairwise nearestneighbor interaction of either sign it has been explicitly shown 8 that the Hamiltonian has a topo logical symmetry which was predicted to stabilize one of the gapless phases In this Letter we give a broader per 0031 9007 08 10150504014 0504011 PACS numbers 0530Pr 0365Vf 7343Lp spective on possible collective phases of interacting Fibonacci anyons and phase transitions between them The speci c model we will focus on has in addition to the twoanyon term an additional threeanyon interaction both ofwhich may arise from tunneling 10 We nd a rich groundstate phase diagram that harbors multiple critical gapless and gapped phases The topological symmetry introduced in Ref 8 that measures the topological ux through a ring of Fibonacci anyons plays an essential role in determining the nature of the observed phases and phase transitions In particular we nd that this topological symmetry protects all the critical phases against spatially uniform local perturbations These extended critical phases can be described in terms of 2D conformal eld theories CFT with central charges 0 710 and c 45 and can be respectively mapped exactly onto the tricritical Ising and 3state Potts critical points of the generalized hard hexagon model 811 At the phase transitions out of the tricritical Ising phase into adjacent gapped phases the system exhibits even higher symmetries which we identify as letracrilical Ising and 3state Potts critical points This demonstrates that these 2D classical models share an iden tical nonlocal symmetry which is the classical analog of the topological symmetry in the 1D quantum chains At the transition into the gapped phases this topological symme try is spontaneously broken which results in a nontrivial groundstate degeneracy in the gapped phases Twoanyon interactionsiFor a uniform chain of Fibonacci anyons the Hamiltonian introduced in Ref 8 energetically favors one or the other of the possible fusion products of two neighboring Tparticles which by the fusion rule 739 X 739 1 739 can be either a l or a 739 The energy of the former is lower for a coupling that is termed antiferromagnetic AFM in analogy to the familiar 2008 The American Physical Society PRL 101 050401 2008 PHYSICAL REVIEW LETTERS week ending 1 AUGUST 2008 S U 2 spinschains while that of the latter is lower with a coupling termed ferromagnetic FM The underlying Hilbert space is spanned by an orthonormal basis of states each state corresponding to one possible labeling of the chain 8 of repeated fusions with 739 Each site along this chain of fusions has either a l or a 739 with a constraint forbidding two adjacent l s By performing a sequence of local basis transformations and projection onto one ofthe two fusion channels for each pair of neighboring anyons the resulting twoanyon inter action Hamiltonian can be written as a sum of local 3site operators H JZZiH which take the explicit form 139 denotes the rst in a triple of adjacent sites HE Tm 7271111 ilT r r r 7 32I717TTTI HQ 1 where Ta projects onto the state Ia eg T1 I171 X 17139 and gt 1 xEVZ is the golden ratio 8 Here we want to explore a larger space of models than that given by this uniform chain with only nearestneighbor twoanyon interactions One way is to let the strength J2 of the interaction alternate along the chain as illustrated in Fig 1a Two chains can be coupled to form a twoleg ladder Another way is to add a spatiallyuniform three anyon interaction as indicated in Fig lb which because of its rich phase diagram we discuss in the following MajumdarGhosh ChaimiThree SU2 spin12 s can combine to a total spin 32 or 12 For a uniform SU2 spin12 chain Majumdar and Ghosh MG showed that an AEM coupling favoring total spin 12 for each set of three neighboring spins gives rise to a gapped phase with the two possible dimer coverings being the exact ground states 12 In the same spirit we have asked what possible phases can be stabilized by a spatially uniform three particle interaction term in our anyonic generalization of the SU2 Heisenberg model ie a term that energetically favors each set of three adjacent Eibonacci anyons to fuse together 10 into either a l or a 739 as illustrated in Fig lb Like the pairwise interaction term such a three particle interaction term respects both the translational and topological symmetries We nd that the energetic com petition between such two and threeanyon interactions gives rise to the rich groundstate phase diagram shown in Fig 2 which we discuss in some detail in the following Similar to the derivation for the pairwise interaction term 1 we can obtain a local form H J3ZiH of the three anyon interaction term by a sequence of basis transforma agt mm b 252395 FIG 1 color online Illustration of a the alternating chain and b the generalized MajumdarGhosh chain with a three anyon interaction term Shaded enclosures indicate the fusion products that are energetically biased by the Hamiltonian tions and projections which then takes the explicit form of a 4site interaction between consecutive labels along the chain of fusions Hi 731111 T1111 T1111 T11 2 72T1111 VIUJAH T1111 27717gtlt7177 HQ 52I71777777I ITTlTgtltTTTT Ho 2 where the site i denotes the rst position in each quad of sites The full Hamiltonian with competing fusion terms then becomes H123 Z J2H J3H where we pa rameterize the couplings by the angle 0 as J2 0030 and J3 sim9 We study periodic chains of L anyons The phase diagram of this model shown in Fig 2 exhibits two critical phases that contain the two exactly solvable points 0 O 7739 These extended critical phases can be described by 2D conformal eld theories and are thereby related to 2D classical critical points to which an exact mapping was established at the two solvable points 8 For AEM pair interaction J2 gt 0 this is the tricritical Ising model c 710 while for FM pair interaction J2 lt 0 it is the critical point of the 3state Potts model c 45 In particular we note that the critical phases found at the exactly solvable points are Stable upon intro ducing a small threeanyon fusion term While the J3term respects both translational and topological symmetries all translational invariant operators with scaling dimension lt2 at the exactly solvable points are found to break the an e I2 mum 3mm Potts 3state Potts c7 with S3rsymmetly c4l5 incommensurate phase 0 solution Z phase g F 7 93n2 etmctitical Ising MajumdatVGhosh 645 FIG 2 color online The phase diagram of our anyonic MajumdarGhosh chain on the circle parameterized by 6 with pairwise fusion term I cos and threeparticle fusion term J3 sin Besides extended critical phases around the exactly solvable points 6 0 7139 that can be mapped to the tricritical Ising model and the 3state Potts model there are two gapped phases gray lled The phase transitions red circles out of the tricritical Ising phase exhibit higher symmetries and are both described by CFTs with central charge 45 In the gapped phases exact ground states are known at the positions marked by the stars In the lower left quadrant a small sliver of an incommen surate phase occurs and a phase which has 2 symmetry These latter two phases also appear to be critical 0504012 PRL 101 050401 2008 PHYSICAL REVIEW LETTERS week ending 1 AUGUST 2008 topological symmetry 8 This shows that the topological symmetry protects the gaplessness in the vicinity of these points somewhat analogously to the muchdiscussed no tion that a topological symmetry protects a groundstate degeneracy in a gapped topological phase in 2D space For large threeparticle interactions these critical phases eventually give way to other phases such as the two distinct gapped phases indicated by the gray shaded arcs in the phase diagram Remarkably the transition to the gapped phase from the tricritical lsing phase when both interaction terms are AEM apparently has an emergent S3 3state Potts symmetry Our numerical analysis shows that this transition occurs at 0 2 01767 and is described by the parafermion CET with central charge 0 45 indicative of an additional S3symmetry at this point Figure 3 shows the rescaled energy spectrum at this critical point whose universal lowenergy part is in spectacular agreement with the CET predictions Note the relevant operator with zero momentum zero ux and scaling di mension 43 which breaks the S3 symmetry It is the leading operator present in the Hamiltonian away from this special point and drives the system into either the gapped or the tricritical lsing phase In the gapped phase the topological symmetry is spontaneously broken and the resulting ground state which has zero total momentum is twofold degenerate in the thermodynamic limit In the tricritical lsing phase the Z sublatticesymmetry breaking order parameter corresponds to a more relevant continuum operator than the topological order parameter while it is the state corresponding to the Z order parameter which acquires a higher energy in the gapped phase where only the topological symmetry is broken At the transition both order parameters are degenerate see Fig 3 and together i i I 39I lxxxgl quot39 x sewsxugi gidmxi M mam3 ma l 5 MBB u u m5 WWW I I 4453 5 943 3MB 3MB gt sm s 159 4823153 504 t g aw 934 69 9463 4 ma 32MB 2 5 08 315 MAS 3Z15 3 15 7 3 3 MS mag WWW mm 3 4 1443 quot L36 Ma Q w s 915 D imm fields 2Z15828158 o 2 mg pl y lMSB 2 g E O descendants H 43 43 1 a Mrs Wis a a 7 1 E 45 L nx E 70m L nx L nx zst 0 l l l l l l l l l l O 2 4 6 8 10 12 14 16 18 momentum K2139EL FIG 3 color online Energy spectrum at the S3symmetric point 6 2 017677 of the MajumdarGhosh chain The energies have been shifted and rescaled so that the two lowest eigenvalues match the CET scaling dimensions The open boxes indicate the positions of the primary elds of the parafermionic subset of the c 45 CET elds with topological ux T are marked The open circles give the positions of multiple descendant elds as indicated they form the order parameter of a critical 3state Potts model with S3 symmetry 1n the case of EM threeparticle interaction J3 lt O the transition at 0 2 7047277 between tricritical lsing and gapped phases is described in terms of CET by the full 0 45 minimal model representing 13 the tetracritical lsing model Again we have unambiguously identi ed the CET description of this critical endpoint by assigning the low energy states in the energy spectrum similar to Ei 3 shown in the auxiliary material 14 In particular the topological symmetry forces the system onto the integrable renormalization group trajectory 15 owing into the gapped phase or the tricritical lsing xed point driven again by the relevant operator with dimension 43 This operator belongs to the series of least relevant lsing Zsymmetric operators of scaling dimension 2k 1k 3 in the family of multicritical Z Ginzburg Landau theories described by the conformal miminal mod els 16 k 3 for tetracritical lsing The limit k gt 00 corresponds to the ordinary SU2 spin12 chain and in this limit this operator becomes the marginal operator driving the transition into the spontaneously dimerized phase 17 In analogy to the ordinary SU2 spin12 chain at our tetracritical lsing transition into the gapped phase translational symmetry is spontaneously broken in our case however spontaneous breaking of the topological symmetry occurs in addition As a consequence we ob serve a fourfold groundstate degeneracy throughout this gapped phase for chains with even length The nature of this gapped phase is best characterized at the point 0 3772 that is the anyonic analog of the MajumdarGhosh point of the spin12 Heisenberg chain At this point the four ground states for even L take the exact form Ill110111 ITXTTXTTXT gt71I717171 i ITTXTTXTTXgt lllflq39lq39 3 Wang llwfmxTgtquot717171 gt i 71m7mngti 171717gt 4 where TX lll lZIT denotes a normalized su perposition of the states I1 and IT on a single site Note these ground states have total momenta K O or K 7739 indicating the twosublattice ordering There are two states at each momentum one with a Tflux and the other with out Of course we can instead make the simpler linear combinations of these ground states that explicitly break both the topological and sublattice symmetries these four states are ITXTTXTTXT ITITI 7391 and the equivalent states under translation by one site Note that the density of 1 s which for these states is 12 gt2 and 12 respec tively is a simple order parameter that re ects the topo logical symmetry breaking The lowenergy excitations in the gapped phase around the MG point are domain walls between the twosublattice ordered ground states with a low density of 1 sisimilar to 0504013

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