## Anyons in an exactly solved model and beyond (2005)

Citations: | 33 - 2 self |

### BibTeX

@MISC{Kitaev05anyonsin,

author = {Alexei Kitaev},

title = {Anyons in an exactly solved model and beyond},

year = {2005}

}

### OpenURL

### Abstract

A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are non-Abelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and non-Abelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.

### Citations

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Citation Context ...ohomology and the second to the group cohomology). The four-level classification of symmetry-breaking phenomena is summarized in Table 6. It has the flavor of topological obstruction theory (see e.g. =-=[98]-=-). One may hypothesize that there is a somehow relevant topological space Y such that π1(Y ) = Γ1, π2(Y ) = Γ2, π3(Y ) = 0, π4(Y ) = Z, and all higher homotopy groups vanish. We now argue that the ana... |

863 |
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Citation Context ...ciativity relations are given by 6j-symbols. This theory admits trivial braiding. 4. Examples 3 and 1 (with M = G) can be combined in what is called representation theory of Drinfeld’s quantum double =-=[85, 80]-=-. Models with anyons described by this theory were proposed in [86] (gauge-symmetric Lagrangian) and in [24] (lattice Hamiltonian not constrained by an external symmetry). This construction can be def... |

645 |
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(Show Context)
Citation Context ...levant to anyons. Different pieces of this structure were discovered in a colossal work of many people, culminating in the paper by Moore and Seiberg [4]. Witten’s work on quantum Chern-Simons theory =-=[5]-=- was also very influential. A more abstract approach (based on local field theory) was developed by Fredenhagen, Rehren, and Schroer [6] and by Frohlich and Gabbiani [7]. The most amazing thing about ... |

303 |
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Citation Context ...oup Γ2 consists of all automorphisms of the identity functor. Note that Γ1 may act on Γ2 in a nontrivial 28 As explained to me by Ezra Getzler, higher groupoids may be nicely defined as Kan complexes =-=[99]-=-; this approach also allows one to endow the morphism sets with topology. In the definition of Γ no topology is assumed. It would be interesting to see what changes (if anything) when a natural topolo... |

295 |
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Citation Context ... from Witten’s paper on quantum Chern-Simons fields [5] and the work of Moore and Siberg on conformal field theory [4]. Important mathematical studies in this area were done by Reshetikhin and Turaev =-=[66]-=- and Walker [67]. For our purposes, it suffices to use a construction called unitary modular category (UMC), which constitutes the algebraic core of TQFT [68]. This construction will be outlined in Ap... |

283 |
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Citation Context ...ces may be divided into three categories: the original field-theoretic papers where the relevant mathematical structure was discovered [4, 5], local field theory theory papers [6, 7] (see Haag’s book =-=[78]-=- for a general reference on this subject), and purely mathematical expositions [68, 79, 80]. Unfortunately, all these texts have different focus and/or difficult to read. A nice elementary introductio... |

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Citation Context ...or example, the arrow on the left (labeled by F xyz q ) actually designates the map � q F xyz q ⊗ id V qw u , where id V qw u is the unit operator on V qw u . According to MacLane’s coherence theorem =-=[82]-=-, any sequence of F-moves between two given trees results in the same isomorphism between the corresponding spaces: the equality of all such isomorphisms follows from the pentagon equation. (Actually,... |

247 |
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Citation Context ... area were done by Reshetikhin and Turaev [66] and Walker [67]. For our purposes, it suffices to use a construction called unitary modular category (UMC), which constitutes the algebraic core of TQFT =-=[68]-=-. This construction will be outlined in Appendix E. Actually, the full theory is not necessary to understand the calculations below. On the contrary, one may use these calculations to motivate some of... |

244 |
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Citation Context ...y topological and algebraic structure in CFT is relevant to anyons. Different pieces of this structure were discovered in a colossal work of many people, culminating in the paper by Moore and Seiberg =-=[4]-=-. Witten’s work on quantum Chern-Simons theory [5] was also very influential. A more abstract approach (based on local field theory) was developed by Fredenhagen, Rehren, and Schroer [6] and by Frohli... |

240 |
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Citation Context ...q. (172) on page 66), though it is unclear how to deduce it only considering the bulk. The only known argument is to assume that edge modes are described by a CFT, then one can use modular invariance =-=[62]-=-. In fact, the modular invariance alone would suffice. In Appendix D we try to derive it from general principles, but again, encounter a problem. 8 Non-Abelian anyons We continue the study of phase B ... |

176 |
Quantum groups, Graduate Texts
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Citation Context ... the relevant mathematical structure was discovered [4, 5], local field theory theory papers [6, 7] (see Haag’s book [78] for a general reference on this subject), and purely mathematical expositions =-=[68, 79, 80]-=-. Unfortunately, all these texts have different focus and/or difficult to read. A nice elementary introduction can be found in Preskill’s lecture notes on quantum computation [81], but more detail is ... |

131 | Fault-tolerant quantum computation
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(Show Context)
Citation Context ...med by fusion. This “purely topological” scheme is universal for sufficiently complicated phases such as the k = 3 parafermion state [31], lattice models based on some finite groups (e.g. S5 [24], A5 =-=[32, 33]-=- and S3 [34]), and double ChernSimons models [35, 36, 37]. Unfortunately, the model studied in this paper is not universal in this sense. One can, however, combine a topologically protected quantum me... |

127 |
Lectures on Tensor Categories and Modular Functors
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(Show Context)
Citation Context ... the relevant mathematical structure was discovered [4, 5], local field theory theory papers [6, 7] (see Haag’s book [78] for a general reference on this subject), and purely mathematical expositions =-=[68, 79, 80]-=-. Unfortunately, all these texts have different focus and/or difficult to read. A nice elementary introduction can be found in Preskill’s lecture notes on quantum computation [81], but more detail is ... |

127 | Topological gauge theories and group cohomology
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(Show Context)
Citation Context ...stem may have nontrivial properties such as the integer quantum Hall effect. The latter is described by a Chern-Simons term in the effective action for the electromagnetic field. Dijkgraaf and Witten =-=[97]-=- showed that a general topological action for a gauge field in 2 + 1 dimensions is characterized by an element of the cohomology group H 4 (BG, Z), where BG is the classifying space of G. (The 104sEff... |

120 |
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(Show Context)
Citation Context ...e and Seiberg [4]. Witten’s work on quantum Chern-Simons theory [5] was also very influential. A more abstract approach (based on local field theory) was developed by Fredenhagen, Rehren, and Schroer =-=[6]-=- and by Frohlich and Gabbiani [7]. The most amazing thing about anyons is that they actually exist as excitations in some condensed matter systems. Such systems also have highly nontrivial ground stat... |

114 | Why is there a field algebra with compact gauge group describing the superselection structure in particle physics - Doplicher, Roberts - 1990 |

110 |
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- 1989
(Show Context)
Citation Context ...l conservation laws, which are described by fusion rules between superselection sectors. A profound understanding of the first principle and its underlying assumptions is due to Doplicher and Roberts =-=[25, 26]-=-. They proved that any consistent system of fusion rules for bosons is equivalent to the multiplication rules for irreducible representations of some compact group. Fermions also fit into this framewo... |

106 |
Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations
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(Show Context)
Citation Context ...tems. Such systems also have highly nontrivial ground states that are described as possessing topological order. The best studied example (both theoretically and experimentally) is the Laughlin state =-=[8]-=- in the fractional quantum Hall system at the filling factor ν = 1/3. It carries Abelian anyons with exchange phase ϕ = π/3 and electric charge ±1/3. It is the fractional value of the charge that was ... |

98 | Fault-tolerant quantum computation by anyons
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- 2003
(Show Context)
Citation Context ... observable � j∈l σz j for any enclosing path l, though no local order parameter exists. The state (2) can be represented as the ground state of the following Hamiltonian with 5sfour-body interaction =-=[24]-=-: � H = −Je vertices As − Jm � plaquettes Bp, where As = � star(s) σ x j , Bp = � boundary(p) σ z j. (3) Its elementary excitations are Z2-charges with energy 2Je and vortices with energy 2Jm. Certain... |

91 | A modular functor which is universal for quantum computation
- Freedman, Larsen, et al.
(Show Context)
Citation Context ...m memory that is protected from decoherence. Some quantum gates can be implemented by braiding; this implementation is exact and does not require explicit error correction. Freedman, Larsen, and Wang =-=[27]-=- proved that for certain types of non-Abelian anyons braiding enables one to perform universal quantum computation. This scheme is usually referred to as topological quantum computation (TQC). Let us ... |

81 | On fusion categories
- Etingof, Nikshych, et al.
(Show Context)
Citation Context ...ll see on concrete exapmles that these equations have only a finite number of solutions and therefore do not admit small deformations. In general, it is a nontrivial theorem known as Ocneanu rigidity =-=[29, 30]-=-, see Sec. E.6. Thus, we have all essential elements of a quantum computer implemented in a robust fashion: an initial state is made by creating pairs and/or by splitting particles, unitary gates are ... |

71 |
Quasi Hopf algebras, group cohomology and orbifold models, Nuclear Phys
- Dijkgraaf, Pasquier, et al.
- 1990
(Show Context)
Citation Context ...auge-symmetric Lagrangian) and in [24] (lattice Hamiltonian not constrained by an external symmetry). This construction can be deformed by an arbitrary cohomology class � f ∈ H 3 (G, U(1)), see Refs. =-=[87, 88]-=-. 5. A very interesting set of fusion theories is based on a Temperley-Lieb category, see e.g. Chapter XII in book [68]. These theories are also known as “representations of quantum SU(2)”. E.1.4 Calc... |

61 |
Braid group statistics and their superselection rules, in: The algebraic Theory of Superselection Sectors
- Rehren
- 1990
(Show Context)
Citation Context ...obtained expression for c− is not easy to use, nor can we demonstrate that c− is rational. Note that there is a beautiful relation between the chiral central charge and algebraic properties of anyons =-=[7, 61]-=-, which does imply the rationality of c−. We discuss that relation in Appendix E (see Eq. (172) on page 66), though it is unclear how to deduce it only considering the bulk. The only known argument is... |

57 |
The index of a pair of projections
- Avron, Seiler, et al.
- 1994
(Show Context)
Citation Context ...rs. If they were finite, their traces would be integers, and the difference would also be an integer. In the infinite case, one may refer to the general notion of the relative index of two projectors =-=[73, 74]-=-. However, we will proceed in a more pedestrian fashion and simply truncate the matrices. Let Λ = U † ΠU − Π. It is clear that the matrix element Λjk vanishes if |j| or |k| is greater than L, so Λ is ... |

45 |
Nonabelions in the fractional quantum Hall effect
- Moore, Read
- 1991
(Show Context)
Citation Context ...the filling factor ν = 5/2, though it is more fragile and less studied experimentally. There is much evidence suggesting that this system is described by a beautiful theory proposed by Moore and Read =-=[14, 15]-=-. The Moore-Read state admits nonAbelian anyons with charge ±1/4. If 2n such particles are present, the associated Hilbert space has dimension 2 n−1 . (The non-Abelian anyons studied in this paper hav... |

42 |
Nijs, “Quantized hall conductance in a two-dimensional periodic potential,” Phys
- Thouless, Kohmoto, et al.
- 1982
(Show Context)
Citation Context ...d phase B with the magnetic field may be attributed to different topology of fermionic pairing. Topological properties of Fermi-systems were first studied in the theory of integer quantum Hall effect =-=[43, 44]-=-. Let us outline the main result. To begin with, the Hall conductivity of noninteracting electrons in a periodic potential (e.g., in the Hofstadter model with m/n flux quanta per plaquette) is express... |

42 | On Witten’s 3-manifold invariants
- Walker
- 1991
(Show Context)
Citation Context ...aper on quantum Chern-Simons fields [5] and the work of Moore and Siberg on conformal field theory [4]. Important mathematical studies in this area were done by Reshetikhin and Turaev [66] and Walker =-=[67]-=-. For our purposes, it suffices to use a construction called unitary modular category (UMC), which constitutes the algebraic core of TQFT [68]. This construction will be outlined in Appendix E. Actual... |

38 | Beyond paired quantum Hall states: parafermions and incompressible states in the first excited Laudau level - Read, Rezayi |

35 | The diagonal of the stasheff polytope
- Loday
(Show Context)
Citation Context ...uadrilaterals correspond to trees with two degree-3 vertices, and the pentagons correspond to trees with one degree-4 vertex. Described in Fig. 18b is a concrete geometric realization of Kn (cf. Ref. =-=[84]-=-). It is fairly easy to show that the convex hull of the points (x1, . . .,xn−1) ∈ Rn−1 obtained this way is characterized by one equation and n(n−1) − 1 inequalities: 2 �n−1 j=1 xj = η1n, �l−1 xj � η... |

34 |
Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect,” Phys
- Read, Green
- 2000
(Show Context)
Citation Context ...of this kind, the 3 He-A film, was studied by Volovik [46]. Volovik and Yakovenko [47] showed that the Chern number in this system determines the statistics of solitons. More recently, Read and Green =-=[48]-=- considered BCS pairing of spinless particles with angular momentum l = −1. They identified a “strong pairing phase” with zero Chern number and a “weak pairing phase” with ν = 1. The latter is closely... |

32 | Quantum Field Theory in Condensed Matter Physics (Cambridge - Tsvelik - 1995 |

31 | B.: Charge deficiency, charge transport and comparison of dimensions
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- 1994
(Show Context)
Citation Context ...eorems. 11 A powerful theory, called noncommutative geometry [45] was used by Bellissard at al [1] to prove rigorously that the Hall conductivity is quantized provided the electron are localized (see =-=[72, 73]-=- for 11 Note for experts: we are effectively trying to build a K-theory on a manifold by considering functions that are not continuous, but rather constant on cells that are dual to simplices. This se... |

25 | Correlation functions for symmetrized increasing subsequences
- Rains
- 2000
(Show Context)
Citation Context ... −a34 0 ⎞ ⎟ ⎠ = a12a34 + a14a23 − a13a24. (Pf A) 2 = det A, Pf(WAW T ) = Pf(A) det(W). (137) We will also need a generalization of the Pfaffian to infinite matrices, which is called Fredholm Pfaffian =-=[75]-=-. Let A and B be real skew-symmetric, B invertible, and A−B have a well-defined trace. Then Pf(A, B) def = � det(1 + K) = exp =1 + 1 � Tr K + − 2 1 � � 1 Tr ln(1 + K) 2 4 Tr K2 + 1 � (Tr K)2 + . . ., ... |

24 |
The finite group velocity of quantum spin systems
- Lieb, Robinson
- 1972
(Show Context)
Citation Context ...We set �Hj = � ∞ e −∞ iHt Hje −iHt w(t) dt. (157) It is easy to check that � j � Hj = H and � Hj|Ψ〉 = 0. A locality condition of the form (151) can be established using the bound of Lieb and Robinson =-=[77]-=- on correlation propagation (cf. [16]). For our purposes, we can actually use an arbitrary gapped Hamiltonian X = � j Xj made up of local constraints for the given state |Ψ〉. It turns out that the chi... |

23 |
Statistics of quasiparticles and the hierarchy of fractional quantized
- Halperin
- 1583
(Show Context)
Citation Context ... predicted in original Laughlin’s paper and confirmed by several methods, in particular by a shot noise measurement [9, 10]. The statistical phase is a subtler property which is deduced theoretically =-=[11, 12]-=-; a nontrivial experimental test has been performed recently using quasiparticle tunneling [13]. A different kind of state is observed at the filling factor ν = 5/2, though it is more fragile and less... |

21 |
Operator contents of the ADE lattice models
- Pasquier
- 1987
(Show Context)
Citation Context ...rm known as Verlinde formula: 23 N c ab = ∑ x saxsbxs¯cx s1x . (228) 23 Eqs. (226), (228) were first obtained by Pasquier for statistical models and conformal theories associated with Dynkin diagrams =-=[100]-=-. A general derivation in the CFT framework is due to Verlinde [62]. 85E.5 Braiding nondegeneracy = modularity Let us recall the definition of a superselection sector: it is a class of states that ca... |

20 |
Quantized Hall Conductivity in Two Dimensions, Phys
- Laughlin
- 1981
(Show Context)
Citation Context ...n schematically in Fig. 13b. In this case a left-moving gapless mode occurs at qx = 0. In conclusion, let us give a simple (but not completely rigorous) proof of Eq. (56). The idea is due to Laughlin =-=[71]-=-: we put the system on a cylinder and run magnetic flux through it. For simplicity, we consider a cylinder of the smallest radius, which is equal to the lattice period divided by 2π. Thus the Hamilton... |

19 |
H.: Flux Phase of the Half-Filled
- Lieb
- 1994
(Show Context)
Citation Context ...tate energy. It turns out that the energy minimum is achieved by the vortex-free field configuration, i.e., wp = 1 for all plaquettes p. This statement follows from a beautiful theorem proved by Lieb =-=[50]-=-. (Not knowing about Lieb’s result, I did some numerical study suggesting the same answer, see Appendix A.) Thus we may assume that ujk = 1 for all links (j, k), where j belongs to the even sublattice... |

19 | Simultaneous quantization of edge and bulk Hall conductivity
- Schulz-Baldes, Kellendonk, et al.
- 2000
(Show Context)
Citation Context ...ses Bν (for some specific boundary conditions) and a simple proof of Eq. (56) are given in Appendix B. More rigorous and general results, which even apply to disordered systems, can be found in Refs. =-=[53, 54]-=-. It is important to note that the analogy to the quantum Hall effect is not exact. In our model (like in two-dimensional superfluid and superconducting systems [55, 48]) edge modes are described by r... |

18 | Schulz-Baldes: Edge current channels and Chern numbers in the integer quantum Hall effect
- Kellendonk, Richter, et al.
- 2002
(Show Context)
Citation Context ...ses Bν (for some specific boundary conditions) and a simple proof of Eq. (56) are given in Appendix B. More rigorous and general results, which even apply to disordered systems, can be found in Refs. =-=[53, 54]-=-. It is important to note that the analogy to the quantum Hall effect is not exact. In our model (like in two-dimensional superfluid and superconducting systems [55, 48]) edge modes are described by r... |

16 |
Lieb-Schultz-Mattis in higher dimensions,” Phys
- Hastings
- 2004
(Show Context)
Citation Context ... quasiparticles). Otherwise excitations are not localizable and braiding may not be defined. Note that if all excitations are gapped, then all equal-time correlators decay exponentially with distance =-=[16]-=-. An example of anyons in a spin-1/2 system originates from the theory of resonating valence bond (RVB). The idea of RVB was put forward by Anderson [17] and used later as a model of the undoped insul... |

15 |
P.Mathieu, D.Senechal, Conformal Field Theory
- Francesco
- 1997
(Show Context)
Citation Context ...ained below; see also Appendix E. Let us first show a quick way of deriving those properties from conformal filed theory (CFT) in the most important case, ν = ±1. (For a general reference on CFT, see =-=[63]-=-.) Then we will give an alternative derivation, which uses only rudimentary CFT and refers to the operational meaning of braiding and fusion. 8.1 Bulk-edge correspondence The properties listed in Tabl... |

14 |
Fractional statistics and the quantum hall effect
- Arovas, Schrieffer, et al.
- 1984
(Show Context)
Citation Context ... predicted in original Laughlin’s paper and confirmed by several methods, in particular by a shot noise measurement [9, 10]. The statistical phase is a subtler property which is deduced theoretically =-=[11, 12]-=-; a nontrivial experimental test has been performed recently using quasiparticle tunneling [13]. A different kind of state is observed at the filling factor ν = 5/2, though it is more fragile and less... |

14 |
Superconductivity and the Quantum HardCore Dimer Gas
- Rokhsar, Kivelson
- 1988
(Show Context)
Citation Context ... local constraints. (A local constraint is a local operator that annihilates the ground state.) Hamiltonians that are explicitly written as sums of local constraints, e.g., the Rokhsar-Kivelson model =-=[76, 19]-=-, have provided a lot of insight into properties of quantum many-body systems. Proposition D.1. Any gapped local Hamiltonian H = � j Hj whose ground state |Ψ〉 has zero energy can be represented as a s... |

12 | Topological Quantum Computation
- Ogburn, Preskill
- 1999
(Show Context)
Citation Context ...med by fusion. This “purely topological” scheme is universal for sufficiently complicated phases such as the k = 3 parafermion state [31], lattice models based on some finite groups (e.g. S5 [24], A5 =-=[32, 33]-=- and S3 [34]), and double ChernSimons models [35, 36, 37]. Unfortunately, the model studied in this paper is not universal in this sense. One can, however, combine a topologically protected quantum me... |

12 |
Y.: Multichannel Landauer formula for the thermoelectric transport with application to thermopower near the mobility edge
- Sivan, Imry
- 1986
(Show Context)
Citation Context ...arguments. There are two standard ways to calculate the coefficient c−. They both rely on certain assumptions but can be applied to our model, yielding this result: c− = ν . (58) 2 The first argument =-=[58]-=- (adapted to real fermions) assumes translational invariance and the absence of interaction. Each edge mode is described by a free fermion with an energy spectrum ε(q) such that ε(−q) = −ε(q) and ε(q)... |

11 |
Resonating valence bonds: A new kind of insulator
- Anderson
- 1973
(Show Context)
Citation Context ... correlators decay exponentially with distance [16]. An example of anyons in a spin-1/2 system originates from the theory of resonating valence bond (RVB). The idea of RVB was put forward by Anderson =-=[17]-=- and used later as a model of the undoped insulating phase in high-Tc cuprate superconductors [18]. Without electrically charged holes, the problem seems to be described adequately by a Heisenberg-lik... |

11 |
Anyon computers with smaller groups, Phys
- Mochon
- 2004
(Show Context)
Citation Context ...his “purely topological” scheme is universal for sufficiently complicated phases such as the k = 3 parafermion state [31], lattice models based on some finite groups (e.g. S5 [24], A5 [32, 33] and S3 =-=[34]-=-), and double ChernSimons models [35, 36, 37]. Unfortunately, the model studied in this paper is not universal in this sense. One can, however, combine a topologically protected quantum memory with a ... |

8 |
Resonating Valence Bond Phase in the Triangular Lattice Quantum Dimer
- Moessner, Sondhi
- 2001
(Show Context)
Citation Context ...very difficult. Several variants of an RVB state have been proposed, both gapless and gapped. Here we discuss a particular gapped RVB phase, namely the one which is realized on the triangular lattice =-=[19]-=-, but which apparently exists on the square lattice as well. This phase admits quasiparticles of four types: trivial excitations (such as spin waves), spinons (spin-1/2 fermions, which are conserved m... |

8 |
Fractional charge, spin and statistics of solitons in superfluid 3 He film
- Volovik, Yakovenko
- 1989
(Show Context)
Citation Context ...bove when the previous definition is applicable. But in general ν is an arbitrary integer. The first physical example of this kind, the 3 He-A film, was studied by Volovik [46]. Volovik and Yakovenko =-=[47]-=- showed that the Chern number in this system determines the statistics of solitons. More recently, Read and Green [48] considered BCS pairing of spinless particles with angular momentum l = −1. They i... |