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arXiv:1506.03259 · Physical Review B

Mean-field ansatz for topological phases with string tension

Sébastien Dusuel, Julien Vidal

Published Sep 29, 2015• 21 citations
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Abstract

We propose a simple mean-field ansatz to study phase transitions from a topological phase to a trivial phase. We probe the efficiency of this approach by considering the string-net model in the presence of a string tension for any anyon theory. Such a perturbation is known to be responsible for a deconfinement-confinement phase transition which is well described by the present variational setup. We argue that mean-field results become exact in the limit of large total quantum dimension.

DOI
10.1103/physrevb.92.125150

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We propose a simple mean-field ansatz to study phase transitions from a topological phase to a trivial phase. We probe the efficiency of this approach by considering the string-net model in the presence of a string tension for any anyon theory. Such a perturbation is known to be responsible for a deconfinement-confinement phase transition which is well described by the present variational setup. We argue that mean-field results become exact in the limit of large total quantum dimension.
We study the Wegner-Wilson loops in the string-net model of Levin and Wen in the presence of a string tension. The latter is responsible for a phase transition from a topological deconfined phase (weak tension) to a trivial confined phase (strong tension). We analyze the behavior of all Wegner-Wilson loops in both limiting cases for an arbitrary input theory of the string-net model. Using a fluxon picture, we compute perturbatively the first contributions to a perimeter law in the topological phase as a function of the quantum dimensions. In the trivial phase, we find that Wegner-Wilson loops obey a modified area law, in agreement with a recent mean-field approach.
We use a recently proposed class of tensor-network states to study phase transitions in string-net models. These states encode the genuine features of the string-net condensate such as, e.g., a nontrivial perimeter law for Wilson loops expectation values, and a natural order parameter detecting the breakdown of the topological phase. In the presence of a string tension, a quantum phase transition occurs between the topological phase and a trivial phase. We benchmark our approach for Z2{\mathbb{Z}}_{2}Z2​ string nets and capture the second-order phase transition which is well known from the exact mapping onto the transverse-field Ising model. More interestingly, for Fibonacci string nets, we obtain first-order transitions in contrast with previous studies but in qualitative agreement with mean-field results.
We consider the string-net model on the honeycomb lattice for Ising anyons in the presence of a string tension. This competing term induces a nontrivial dynamics of the non-Abelian anyonic quasiparticles and may lead to a breakdown of the topological phase. Using high-order series expansions and exact diagonalizations, we determine the robustness of this doubled Ising phase, which is found to be separated from two gapped phases. An effective quantum dimer model emerges in the large tension limit, giving rise to two different translation symmetry-broken phases. Consequently, we obtain four transition points, two of which are associated with first-order transitions whereas the two others are found to be continuous and provide examples of recently proposed Bose condensation for anyons.
We consider the string-net model on the honeycomb lattice for Ising anyons in the presence of a string tension. This competing term induces a nontrivial dynamics of the non-Abelian anyonic quasiparticles and may lead to a breakdown of the topological phase. Using high-order series expansions and exact diagonalizations, we determine the robustness of this doubled Ising phase which is found to be separated from two gapped phases. An effective quantum dimer model emerges in the large tension limit giving rise to two different translation symmetry-broken phases. Consequently, we obtain four transition points, two of which are associated to first-order transitions whereas the two others are found to be continuous and provide examples of recently proposed Bose condensation for anyons.
We discuss the emergence of bound states in the low-energy spectrum of the string-net Hamiltonian in the presence of a string tension. In the ladder geometry, we show that a single bound state arises either for a finite tension or in the zero-tension limit depending on the theory considered. In the latter case, we perturbatively compute the binding energy as a function of the total quantum dimension. We also address this issue in the honeycomb lattice where the number of bound states in the topological phase depends on the total quantum dimension. Finally, the internal structure of these bound states is analyzed in the zero-tension limit.
We study the phase diagrams of a family of three-dimensional Walker-Wang--type lattice models, which are not topologically ordered but have deconfined anyonic excitations confined to their surfaces. We add a perturbation (analogous to that which drives the confining transition in Zp{\mathbb{Z}}_{p}Zp​ lattice gauge theories) to the Walker-Wang Hamiltonians, driving a transition in which all or some of the variables associated with the loop-gas or string-net ground states of these models become confined. We show that in many cases the location and nature of the phase transitions involved is exactly that of a generalized Zp{\mathbb{Z}}_{p}Zp​ lattice gauge theory, and use this to deduce the basic structure of the phase diagram. We further show that the relationship between the phases on opposite sides of the transition is fundamentally different than in conventional gauge theories: in the Walker-Wang case, the number of species of excitations that are deconfined in the bulk can increase across a transition that confines only some of the species of loops or string nets. The analog of the confining transition in the Walker-Wang models can therefore lead to bulk deconfinement and topological order.