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arXiv:1608.03903 · Reviews in Mathematical Physics

Local disorder, topological ground state degeneracy and entanglement entropy, and discrete anyons

Sven Bachmann

Published May 24, 2017• 10 citations
arXivHTMLPDF

Abstract

In this comprehensive study of Kitaev’s abelian models defined on a graph embedded on a closed orientable surface, we provide complete proofs of the topological ground state degeneracy, the absence of local order parameters, compute the entanglement entropy exactly and characterize the elementary anyonic excitations. The homology and cohomology groups of the cell complex play a central role and allow for a rigorous understanding of the relations between the above characterizations of topological order.

DOI
10.1142/s0129055x17500180

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•Jul 9, 2021•0 citations
In this comprehensive study of Kitaev’s abelian models defined on a graph embedded on a closed orientable surface, we provide complete proofs of the topological ground state degeneracy, the absence of local order parameters, compute the entanglement entropy exactly and characterize the elementary anyonic excitations. The homology and cohomology groups of the cell complex play a central role and allow for a rigorous understanding of the relations between the above characterizations of topological order.
Abstract Graphs are topological spaces that include broader objects than discretized manifolds, making them interesting playgrounds for the study of quantum phases not realized by symmetry breaking. In particular they are known to support anyons of an even richer variety than the two-dimensional space. We explore this possibility by building a class of frustration-free and gapped Hamiltonians based on discrete abelian gauge groups. The resulting models have a ground state degeneracy that can be either a topological invariant, an extensive quantity or a mixture of the two. For two basis of the degenerate ground states which are complementary in quantum theory, the entanglement entropy (EE) is exactly computed. The result for one basis has a constant global term, known as the topological EE, implying long-range entanglement. On the other hand, the topological EE vanishes in the result for the other basis. Comparisons are made with similar occurrences in the toric code. We analyze excitations and identify anyon-like excitations that account for the topological EE. An analogy between the ground states of this system and the θ -vacuum for a U (1) gauge theory on a circle is also drawn.
Topologically ordered phases in 2 + 1 dimensions are generally characterized by three mutually related features: fractionalized (anyonic) excitations, topological entanglement entropy, and robust ground state degeneracy that does not require symmetry protection or spontaneous symmetry breaking. Such a degeneracy is known as topological degeneracy and can be usually seen under the periodic boundary condition regardless of the choice of the system sizes L1 and L2 in each direction. In this work, we introduce a family of extensions of the Kitaev toric code to N level spins (N ≥ 2). The model realizes topologically ordered phases or symmetry-protected topological phases depending on the parameters in the model. The most remarkable feature of topologically ordered phases is that the ground state may be unique, depending on L1 and L2, despite that the translation symmetry of the model remains unbroken. Nonetheless, the topological entanglement entropy takes the nontrivial value. We argue that this behavior originates from the nontrivial action of translations permuting anyon species.
To perform topological quantum computation we first need to experimentally realise anyons in a topological system. These systems are characterised by intriguing non-local quantum correlations that give rise to the anyonic statistics. What are the diagnostic tools we have to identify if a given system is indeed topological or not? Different phases of matter are characterised by their symmetries. This information is captured by order parameters. Usually, order parameters are defined in terms of local operators that can be measured in the laboratory. For example, the magnetisation of a spin system is given as the expectation value of a single spin with respect to the ground state. Such local properties can describe fascinating physical phenomena efficiently, such as ferromagnetism, and can pinpoint quantum phase transitions. But what about topological systems? Experimentally, we usually identify the topological character of systems, such as the fractional quantum Hall liquids, by probing the anyonic properties of their excitations (Miller et al. , 2007). However, topological order should be a characteristic of the ground state (Thouless et al. , 1982; Wen, 1995). The natural question arises: is it possible to identify a property of the ground state of a system that implies anyonic excitations? The theoretical background that made it possible to answer this question came from entropic considerations of simple topological models. Hamma et al . (2005) studied the entanglement entropy of the toric code ground state and noticed an unusual behaviour.
Topological order is a new paradigm for quantum phases of matter developed to explain phase transitions which do not fit the symmetry breaking scheme for classifying phases of matter. They are characterized by patterns of entanglement that lead to topologically depended ground state degeneracy and anyonic excitations. One common approach for studying such phases in two-dimensional systems is through exactly solvable lattice Hamiltonian models such as quantum double models and String-Net models. The former can be understood as the Hamiltonian formulation of lattice gauge theories and, as such, it is defined by a finite gauge group. However, not much is known about topological phases in tridimensional systems. Motivated by this we develop a new class of three-dimensional exactly solvable models which go beyond quantum double models by using finite crossed modules instead of gauge groups. This approach relies on a lattice implementation of 2-gauge theory to obtain models with a richer topological structure. We construct the Hamiltonian model explicitly and provide a rigorous proof that the ground state degeneracy is a topological invariant and that the ground states can only be characterized with nonlocal order parameters.
Topological phases exhibit unconventional order that cannot be detected by any local order parameter. In the framework of projected entangled pair states, topological order is characterized by an entanglement symmetry of the local tensor that describes the model. This symmetry can take the form of a tensor product of group representations [for quantum double models D(G)D(G)D(G) of a group G]G]G], or in the more general case a correlated symmetry action in the form of a matrix product operator (MPO), which encompasses all string-net models, including those that are not quantum double models. Among other things, these entanglement symmetries allow for the succinct description of ground states and topological excitations (anyons). Recently, the idea has been put forward to use those symmetries and the anyonic objects they describe as order parameters for probing topological phase transitions, and the applicability of this idea has been demonstrated for Abelian groups. In this paper, we extend this construction to the domain of non-Abelian models with MPO symmetries, and we use it to study the breakdown of topological order in the double Fibonacci (DFib) string-net model and its Galois conjugate, namely the non-Hermitian double Yang-Lee (DYL) string-net model. We start by showing how to construct topological order parameters for condensation and deconfinement of anyons using the MPO symmetries. Subsequently, we set up interpolations from the DFib and the DYL model to the trivial phase, and we show that these can be mapped to certain restricted solid on solid (RSOS) models, which are equivalent to the $[(5\ifmmode\pm\else\textpm\fi{}\sqrt{5})/2]$-state Potts model, respectively. Moreover, the order parameter for condensation maps to the RSOS order parameter. The known exact solutions of the statistical models subsequently allow us to locate the critical points of the models, and to predict the critical exponents for the order parameters from conformal field theory. We complement this by numerical study of the phase transitions, which fully confirms our theoretical predictions; remarkably, we find that both models exhibit a duality between the behavior of order parameters for condensation and deconfinement.
Topological phases exhibit unconventional order that cannot be detected by any local order parameter. In the framework of Projected Entangled Pair States(PEPS), topological order is characterized by an entanglement symmetry of the local tensor which describes the model. This symmetry can take the form of a tensor product of group representations, or in the more general case a correlated symmetry action in the form of a Matrix Product Operator(MPO), which encompasses all string-net models. Among other things, these entanglement symmetries allow for the description of ground states and anyon excitations. Recently, the idea has been put forward to use those symmetries and the anyonic objects they describe as order parameters for probing topological phase transitions, and the applicability of this idea has been demonstrated for Abelian groups. In this paper, we extend this construction to the domain of non-Abelian models with MPO symmetries, and use it to study the breakdown of topological order in the double Fibonacci (DFib) string-net and its Galois conjugate, the non-hermitian double Yang-Lee (DYL) string-net. We start by showing how to construct topological order parameters for condensation and deconfinement of anyons using the MPO symmetries. Subsequently, we set up interpolations from the DFib and the DYL model to the trivial phase, and show that these can be mapped to certain restricted solid on solid(RSOS) models, which are equivalent to the ((5±5)/2)((5\pm\sqrt{5})/2)((5±5​)/2)-state Potts model, respectively. The known exact solutions of the statistical models allow us to locate the critical points, and to predict the critical exponents for the order parameters. We complement this by numerical study of the phase transitions, which fully confirms our theoretical predictions; remarkably, we find that both models exhibit a duality between the order parameters for condensation and deconfinement.