Topological Phase Transitions in the Golden String-Net Model
TTopological Phase Transitions in the Golden String-Net Model
Marc Daniel Schulz,
1, 2
S´ebastien Dusuel, Kai Phillip Schmidt, and Julien Vidal Lehrstuhl f¨ur Theoretische Physik I, Technische Universit¨at Dortmund,Otto-Hahn-Straße 4, 44221 Dortmund, Germany Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee, CNRS UMR 7600,Universit´e Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris Cedex 05, France Lyc´ee Saint-Louis, 44 Boulevard Saint-Michel, 75006 Paris, France
We examine the zero-temperature phase diagram of the two-dimensional Levin-Wen string-netmodel with Fibonacci anyons in the presence of competing interactions. Combining high-orderseries expansions around three exactly solvable points and exact diagonalizations, we find thatthe non-Abelian doubled Fibonacci topological phase is separated from two nontopological phasesby different second-order quantum critical points, the positions of which are computed accurately.These trivial phases are separated by a first-order transition occurring at a fourth exactly solvablepoint where the ground-state manifold is infinitely many degenerate. The evaluation of criticalexponents suggests unusual universality classes.
PACS numbers: 71.10.Pm, 75.10.Jm, 03.65.Vf, 05.30.Pr
Quantum phases of matter are often well described bylocal order parameters and Landau-Ginzburg symmetry-breaking theory is an efficient tool to analyze transi-tions between these phases. However, in the late 1980s,a new class of phases that cannot be understood interms of local symmetries has emerged in the context ofhigh-temperature superconductivity [1–3]. These phases,dubbed topological because of their sensitivity to the sys-tem topology, have stimulated many studies in differentdomains (see Ref. 4 for a recent review). One of the mostintriguing properties of topologically ordered phases isthat they are robust against local (not too strong) per-turbations [5, 6]. This stability makes them especially ap-pealing for quantum computation [7] as well as good can-didates for quantum memories [8]. Several experimentshave been proposed to realize the so-called topologicallyprotected qubits [9]. In this perspective, a theoreticalcharacterization of the robustness of topological phasesunder strong perturbations as well as the nature of thephase transitions signaling their breakdown is undoubt-edly an important issue. Thanks to recently proposedexactly solvable lattice models realizing various topolog-ical phases of matter [5, 10, 11], this program has beenundertaken for several models [12–25].The main purpose of this Letter is to go one stepbeyond, by studying the phase diagram of a paradig-matic 2D non-Abelian model. We consider the Levin-Wen model [11] on the honeycomb lattice with Fibonaccianyons (the golden string-net model) in the presence ofthe same perturbation as the one introduced in Ref. 20.We determine the extension of the doubled
Fibonacci(DFib) topological phase and show that it is separatedfrom two other nontopological phases via second-ordertransitions that are analyzed in detail.
Hilbert space—
Following the Levin-Wen construction, we considera honeycomb lattice with anyonic degrees of freedom liv-ing on its edges. In the Fibonacci string-net model, these local (microscopic) degrees of freedom can be in two dif-ferent states | (cid:105) or | (cid:105) . The Hilbert space H is restrictedto states that satisfy the so-called branching rules stem-ming from the non-Abelian fusion rules0 × a = a × a for a ∈ { , } , (1)1 × . (2)At each vertex of the honeycomb lattice, the fusion rulesmust not be violated; i.e., if one edge is in state | (cid:105) , thenat least one of the two other edges must be in the samestate. For an arbitrary trivalent graph with N v vertices,the dimension of the Hilbert space is then given bydim H = (1 + ϕ ) N v2 + (1 + ϕ − ) N v2 , (3)where ϕ = √ is the golden ratio (see, e.g., Ref. 26). Model.—
We study the following Hamiltonian [20], H = − J p (cid:88) p δ Φ( p ) , − J e (cid:88) e δ l ( e ) , . (4)The first term is the string-net Hamiltonian introducedby Levin and Wen [11]. It involves the projector δ Φ( p ) , onto states with no flux Φ( p ) through plaquette p [11, 20].The second term is diagonal in the basis introduced abovesince δ l ( e ) , is the projector onto state | (cid:105) on edge e .To help the reader to grasp the physical content ofthis Hamiltonian, let us mention what happens if onereplaces Eq. (2) by the simpler Abelian Z fusion rule1 × J e in the x direction. Indeed, onecan write the Hamiltonian in terms of Pauli matrices,with δ Φ( p ) , = (cid:0) + (cid:81) e ∈ p σ ze (cid:1) / δ l ( e ) , = (cid:0) + σ xe (cid:1) / / a r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r Ising model on the triangular lattice (of plaquettes), withcoupling J e and transverse-field J p . Limiting cases.—
For convenience, let us set J p = cos θ and J e = sin θ . To our knowledge, contrary to the lad-der geometry studied in Ref. 20, the Hamiltonian H isexactly solvable only at the four points for which θ isa multiple of π/
2. In the following, we discuss the low-energy spectrum of H as well as the corresponding phasesaround these special points and give some arguments infavor of transitions between them.( a ) θ = 0 : For J p > J e = 0, the model reduces tothe golden string-net model. Ground states | g (cid:105) are flux-free states satisfying δ Φ( p ) , | g (cid:105) = | g (cid:105) for all p and thushave an energy per plaquette e = E /N p = − N p be-ing the number of plaquettes). Their degeneracy dependson the system topology which is the most salient prop-erty of a topological phase. For the Fibonacci string-netmodel on any trivalent graph, the ground state is uniqueon a sphere [27] whereas it is fourfold degenerate on atorus. Interestingly, one can also compute the degener-acy of the k th excited states (with energy E k = E + k ) D k = (cid:18) N p k (cid:19) (cid:0) p F k − + q F k + r F k F k − (cid:1) , (5)where we introduced the famous Fibonacci sequencedefined for any integer n by F n +1 = F n + F n − ,with F − = 1 and F = 0. The integers ( p, q, r ) de-pend on the surface considered. For instance, one has( p, q, r ) = (1 , ,
0) on a sphere whereas ( p, q, r ) = (4 , , an odd number of excitations can exist on a com-pact surface contrary to the charge-free toric code dis-cussed above where fluxes are always created and annihi-lated by pairs. Note that the binomial coefficient simplyarises from the different ways to choose k plaquettes car-rying the flux excitations among N p .Products of two F k ’s stem from the fact that the “emer-gent” flux excitations are not the microscopic Fibonaccianyons but are achiral combinations of two Fibonaccianyons (details will be given in Ref. 28). The non-Abeliantopological phase in the vicinity of θ = 0 is described by aDFib theory [11, 29–33]. Excitations have a trivial topo-logical spin [11] and can fuse to the vacuum (also calledtrivial particle) [34]. As such, they can also be consideredas bosons [35] and hence condense.( b ) θ = π : For J e = 0 and J p <
0, the low-energy spec-trum is very different. Indeed, in this case, the ground-state manifold is D N p -fold degenerate and spanned by allstates | g (cid:105) satisfying δ Φ( p ) , | g (cid:105) = 0 for all p . As discussedabove, this degeneracy depends on the topology throughits indices ( p, q, r ) so that one might be tempted to con-sider the system as topologically ordered. However, thelocal operator (cid:80) e δ l ( e ) , couples the ground states andsplits the degeneracy for any J e (cid:54) = 0. As a consequence,the system cannot be considered as topologically ordered[36]. − e Non topological Non topological DFib − − π/ π π/ πθ c1 θ c2 θ FIG. 1. (color online). Ground-state energy per plaquette e = E /N p as a function of θ . ED results (black line) for N p = √ × √
13 plaquettes (see inset) are in excellent agree-ment with typical Pad´e approximants (white lines) computedfrom high-order series expansions (see Supplemental Mate-rial).
Owing to this huge degeneracy, we have not been ableto analyze the vicinity of this point. However, numericalresults obtained by exact diagonalizations clearly showthat ( i ) the degeneracy is lifted as soon as the coupling J e (cid:54) = 0 and ( ii ) the ground state for θ = π ± is unique andadiabatically connected to the polarized ground statesfound at θ = π/ π/
2, respectively, (see discussionbelow). This result is in stark contrast with the scenariodescribed in Ref. 20 on the ladder where a gapless phaseis observed for θ ∈ [ π, π/ ∂ θ e at θ = π (for all system sizes) indicating that the twogapped phases ( θ = π + and θ = π − ) are separated by afirst-order phase transition.( c ) θ = π/ J p = 0, the Hamiltonian H is diago-nal in the canonical basis of states satisfying the branch-ing rules. For J e >
0, the ground state is unique whateverthe topology and corresponds to the fully polarized statewhere all edges are in the state | (cid:105) (with eigenenergy E = − N e , where N e is the total number of edges). Firstexcited states are obtained by flipping six links aroundone hexagon. They behave as trivial hard-core bosonsthat become dynamical when the coupling J p is switchedon. Thus, near θ = π/
2, the system is gapped but nottopologically ordered, making the occurrence of a phasetransition in the interval [0 , π/
2] compulsory.( d ) θ = 3 π/ J e < J p = 0, the Hamil-tonian is also diagonal and the unique ground state isthe fully polarized state where all edges are in the states | (cid:105) ( e = 0). Note that such a state would be forbiddenby the Abelian Z fusion rules. First excited states areobtained from the ground state by flipping a single link.As previously, these localized excitations are trivial hard-core bosons that become mobile when J p (cid:54) = 0 so that oneexpects a phase transition in the interval [3 π/ , π ]. Phase diagram.—
To determine the zero-temperature / θ c1 . π/ θ c2 π F i r s t e x c i t a t i o n e n e r g i e s DFib Non Non DFibtopological topological θ θ
FIG. 2. (color online). First four excitation energies obtainedfrom the ED results (black lines) for N p = 3 × phase diagram, we combined two different approaches.First, we performed high-order series expansions in thethermodynamical limit by means of several methods [37–39], around the exactly solvable points θ = 0, π/
2, and3 π/ e as well as the quasiparticle dis-persion from which the low-energy gap ∆ can be ex-tracted. Lengthy expressions of these series expansionscan be found in the Supplemental Material. This methodallows one to accurately compute the critical couplingsfor which the gap vanishes. These points are associatedto second-order transitions but might not be relevant iffirst-order transitions are present (see Refs. 17 and 19for details about this issue in a similar context). Second,we perform exact diagonalizations (ED) for lattices withperiodic boundary conditions. As can be seen in Figs. 1and 2, series expansions and ED data are in very goodagreement except in the vicinity of the transition pointswhere finite-order and finite-size effects are important.Combining these two methods (ED and series expan-sions) we found that the DFib topological phase near θ = 0 ranges from θ c2 (cid:39) − .
63 (= 5 . to θ c1 (cid:39) . . As we shall now argue, we associate these two pointsto second-order transitions. The first piece of evidencepleading in favor of such a scenario follows from ED andis the behavior of ∂ θ e that clearly decreases with thesystem size near these points (see left and right panels inFig. 3). In addition, the position of the low-energy gapminimum as well as the topological degeneracy splittingshown in Fig. 2 lie in the same region as the position ofthe minimum of ∂ θ e . Let us also note that we did notfind any relevant level crossing in the excitation spec-trum that could lead to a first-order transition. The sec- ond argument comes from the high-order perturbationtheory. As can be seen in the Supplemental Material,the series behave very differently for positive and nega-tive J e . So, let us first discuss the most favorable case J e > J p > θ = 0 and θ = π/ ∂ θ e as a function of N − computed from ED re-sults. As can be seen, all data seem to converge to thesame point θ c1 ∈ [0 . , . J e < J p >
0) is more in-volved for three reasons. First, series expansions of e and ∆ in this region have alternate signs so that the pre-vious criteria based on crossing points cannot be used.Second, contrary to the case J e >
0, the momentum min-imizing the dispersion of the low-energy quasiparticles isnot at the Γ point and only belongs to the reciprocallattice of 3 p × q systems, ( p, q ) ∈ N . The only sys-tem with such characteristics considered in this study isthe 3 × − − −
30 210 0 − − − θ c1 . π/ π π/ π/ θ c2 π∂ θ e DFib Non ∂ θ e Non Non ∂ θ e Non DFibtopological topological topological topological θ θ θ
FIG. 3. (color online). ED results for N p = 2 × × √ × √
13 (solid line) plaquettes.Left and right panels : ∂ θ e decreases with the system size in-dicating second-order transitions at θ c1 and θ c2 . Central panel : ∂ θ e displays a clear jump at θ = π indicating a first-ordertransition. Dips indicated by arrows in the DFib topologicalphase are due to (irrelevant and avoided) level crossings be-tween the four lowest-energy levels that become degeneratein the thermodynamical limit on a torus. . . . / / θ θ c1 FIG. 4. (color online). Position of different quantities as afunction of either the inverse order n − of the correspondingseries expansion [crossing points of e (dots), crossing pointsof the gap (triangles)] or N − [minimum of the gap (squares),minimum of ∂ θ e (diamonds)]. Lines are power-law fits ob-tained by choosing θ c1 such that it maximizes the correlationsfor the crossing points of e (solid lines) that are the most ac-curate results in this work. These crossing points have beenobtained from order n series around θ = 0 and order 2 n − θ = π/
2. For the crossing points of the gap, weused order n series around both limits. analysis from ED data near θ c2 . Third, because of thenature of the low-energy states, maximum orders reach-able around θ = 3 π/ θ = π/ θ = 0, it is possible to perform(dlogPad´e) resummations that lead to a position of thecritical point θ c2 ∈ [5 . , . J e > θ c1 ∈ [0 . , . θ c2 lies in the range[5 . , . Critical exponents.—
The obvious question that arisesnext concerns universality classes associated to the tran-sition points. In the absence of a local order parame-ter, the only meaningful critical exponents for topologicalphase transitions are those associated to the spectrum.Let us remind that for a second-order transition, the gapvanishes, at large linear system size L and at the criticalpoint, as ∆ ∼ L − z where z is the dynamical exponent. Inthe thermodynamical limit, one further has ∆ ∼ | θ − θ c | zν and ∂ θ e ∼ | θ − θ c | − α .As already explained, ED are only useful quantitativelyaround θ c1 although restricted to systems of small sizes.Using the gap data, we find z (cid:39) .
2. A finite-size analysisof e yields a surprisingly good data collapse for the 3 × √ × √
13 systems, with θ c1 (cid:39) . z (cid:39) ν (cid:39) . α (cid:39) . α might be re-sponsible for the quality of the data collapse. We empha-size that these values are compatible with the previousestimate of θ c1 as well as with the hyperscaling relation 2 − α = ν (2 + z ). The above exponents are furthermorein agreement with dlogPad´e resummations of the seriesexpansion around θ = 0 which yield zν ∈ [0 . , . θ = π/ θ = 0.Note that as usual, extracting α from series of e doesnot give any conclusive result. Concerning the critical be-havior at θ c2 , we only use dlogPad´e resummation around θ = 0 and we obtain a gap exponent zν ∈ [0 . , . Outlook.—
It is difficult to provide some error bars con-cerning these values. To estimate these errors, we per-formed similar series expansion analysis for the Fibonacciladder (for which exponents are known exactly [20]) andfor the 2D Z string-net model (having either Ising or XY transitions depending on the sign of J e ). The results weobtained [28] lead us to conclude that critical exponentsare to be considered with a precision of about 10%. Asa conclusion, we found two different second-order tran-sitions with universality classes that, to the best of ourknowledge, are as yet unknown in the context of topo-logical phase transitions. Let us mention that a criticalDFib wave function has been proposed [40] but its rel-evance for the present problem requires further studies[28].To gain more understanding about these transitions,different approaches could be used, e.g., variational meth-ods or Monte Carlo simulations, although a naive im-plementation of the latter should suffer from the signproblem. It would also be worth studying similar modelswith a DFib phase [40, 41] as well as different topolog-ical phases. Finally, another important issue concernstransitions between two distinct topological phases [35].Given the ubiquity of Fibonacci anyons in many differentphysics domain such as topological quantum computa-tion, condensed matter, or atomic physics [42], we hopethat the present work will stimulate such investigations.We thank K. Coester, B. Dou¸cot, M. Kamfor, andJ.-B. Zuber for fruitful discussions. K. P. Schmidt ac-knowledges ESF and EuroHorcs for funding through hisEURYI. [1] X.-G. Wen, F. Wilczek, and A. Zee, Phys. Rev. B ,11413 (1989).[2] X.-G. Wen, Phys. Rev. B , 7387 (1989).[3] X.-G. Wen, Int. J. Mod. Phys. B , 239 (1990).[4] X.-G. Wen, arXiv:1210.1281.[5] A. Y. Kitaev, Ann. Phys. , 2 (2003).[6] S. Bravyi, M. B. Hastings, and S. Michalakis, J. Math.Phys. Math. Phys. , 4452 (2002).[9] B. Dou¸cot and L. B. Ioffe, Rep. Prog. Phys. , 072001(2012).[10] X.-G. Wen, Phys. Rev. Lett. , 016803 (2003).[11] M. A. Levin and X.-G. Wen, Phys. Rev. B , 045110(2005).[12] S. Trebst, P. Werner, M. Troyer, K. Shtengel, andC. Nayak, Phys. Rev. Lett. , 070602 (2007).[13] A. Hamma and D. A. Lidar, Phys. Rev. Lett. , 030502(2008).[14] J. Yu, S.-P. Kou, and X.-G. Wen, Europhys. Lett. ,17004 (2008).[15] J. Vidal, S. Dusuel, and K. P. Schmidt, Phys. Rev. B , 033109 (2009).[16] J. Vidal, R. Thomale, K. P. Schmidt, and S. Dusuel,Phys. Rev. B , 081104 (2009).[17] S. Dusuel, M. Kamfor, R. Or´us, K. P. Schmidt, andJ. Vidal, Phys. Rev. Lett. , 107203 (2011).[18] F. Wu, Y. Deng, and N. Prokof’ev, Phys. Rev. B ,195104 (2012).[19] M. D. Schulz, S. Dusuel, R. Or´us, J. Vidal, and K. P.Schmidt, New J. Phys. , 025005 (2012).[20] C. Gils, S. Trebst, A. Kitaev, A. W. W. Ludwig,M. Troyer, and Z. Wang, Nat. Phys. , 834 (2009).[21] C. Gils, J. Stat. Mech. P07019 (2009).[22] A. W. W. Ludwig, D. Poilblanc, S. Trebst, andM. Troyer, New J. Phys. , 045014 (2011).[23] D. Poilblanc, A. W. W. Ludwig, S. Trebst, andM. Troyer, Phys. Rev. B , 134439 (2011).[24] M. H. Freedman, J. Gukelberger, M. B. Hastings,S. Trebst, M. Troyer, and Z. Wang, Phys. Rev. B ,045414 (2012). [25] F. J. Burnell, S. H. Simon, and J. K. Slingerland, NewJ. Phys. , 015004 (2012).[26] S. H. Simon and P. Fendley, J. Phys. A , 105002 (2013).[27] Y. Hu, S. D. Stirling, and Y.-S. Wu, Phys. Rev. B ,075107 (2012).[28] M. D. Schulz et al. , (to be published).[29] L. Fidkowski, M. Freedman, C. Nayak, K. Walker, andZ. Wang, Commun. Math. Phys. , 805 (2009).[30] Z. Wang, Topological Quantum Computation , CBMS Re-gional Conference Series in Mathematics, Number 112(2010).[31] F. J. Burnell and S. H. Simon, Ann. Phys. , 2550(2010).[32] F. J. Burnell and S. H. Simon, New J. Phys. , 065001(2011).[33] F. J. Burnell, S. H. Simon, and J. K. Slingerland, Phys.Rev. B , 125434 (2011).[34] R. Koenig, G. Kuperberg, and B. W. Reichardt, Ann.Phys. , 2707 (2010).[35] F. A. Bais and J. K. Slingerland, Phys. Rev. B , 045316(2009).[36] Z. Nussinov and G. Ortiz, Ann. Phys. , 977 (2009).[37] P.-O. L¨owdin, J. Math. Phys. , 969 (1962).[38] M. Takahashi, J. Phys. C , 1289 (1977).[39] C. Knetter and G. S. Uhrig, Eur. Phys. J. B , 209(2000).[40] P. Fendley, Ann. Phys. , 3113 (2008).[41] P. Fendley, S. V. Isakov, and M. Troyer, arXiv:1210.5527.[42] I. Lesanovsky and H. Katsura, Phys. Rev. A , 041601(2012). SUPPLEMENTAL MATERIAL
In the following, we give the series expansions in the different phases for the ground-state energy per plaquette e and the quasiparticle gaps ∆ ± , for positive and negative signs of the dimensionless parameter t = J e /J p = tan θ respectively. For the sake of clarity, we give below the numerical values of the coefficients with 16 digits. Expansions in the vicinity θ = 0 The ground-state energy per plaquette e near θ = 0 ( J p = 1, J e = 0) has been obtained up to order 11 usingoperator perturbation theory [38], whereas quasiparticle gaps ∆ ± were obtained up to order 9 using perturbativecontinuous unitary transformations. [39] e /J p = − − . t − . t − . t − . t − . t − . t − . t − . t − . t − . t − . t , (6)∆ + /J p = 1 − . t − . t − . t − . t − . t − . t − . t − . t − . t , (7)∆ − /J p = 1 + 0 . t + 0 . t + 0 . t + 0 . t + 0 . t + 1 . t + 4 . t + 11 . t + 31 . t . (8) Expansions in the vicinity θ = π/ The ground-state energy per plaquette e near θ = π/ J e = 1, J p = 0) has been obtained up to order 19 using apartitioning technique provided by L¨owdin [37], whereas quasiparticle gaps ∆ ± were obtained up to order 11 usingoperator perturbation theory [38] on appropriate periodic clusters. e /J e = − − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − , (9)∆ ± /J e = 6 − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − − . · − t − . (10) Expansions in the vicinity θ = 3 π/ The ground-state energy per plaquette e near θ = 3 π/ J e = − J p = 0) has been obtained up to order 9 usingpartitioning techniques [37], whereas quasiparticle gaps ∆ ± were obtained up to order 6 using perturbative continuousunitary transformations. [39] e / ( − J e ) = 3 . · − t − − . · − t − + 2 . · − t − − . · − t − − . · − t − + 4 . · − t − − . · − t − − . · − t − + 3 . · − t − , (11)∆ + / ( − J e ) = 1 − . · − t − + 2 . · − t − − . · − t − + 1 . · − t − − . · − t − + 1 . · − t − , (12)∆ − / ( − J e ) = 1 + 2 . · − t − − . · − t − − . · − t − + 3 . · − t − + 7 . · − t − − . · − t − ..