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arXiv:2108.13425 · Physical review. B./Physical review. B

Partition function of the Levin-Wen model

Julien Vidal

Published Jan 20, 2022• 52 citations
arXivHTMLPDF

Abstract

Using a description of the Levin-Wen model excitations in terms of Wilson lines, we compute the degeneracy of the energy levels for any input anyon theory and for any trivalent graph embedded on any (orientable) compact surface. This result allows one to obtain the finite-size and finite-temperature partition function and to show that there are no thermal phase transitions.

DOI
10.1103/physrevb.105.l041110

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Using a description of the Levin-Wen model excitations in terms of Wilson lines, we compute the degeneracy of the energy levels for any input anyon theory and for any trivalent graph embedded on any (orientable) compact surface. This result allows one to obtain the finite-size and finite-temperature partition function and to show that there are no thermal phase transitions.
We compute the degeneracy of energy levels in the Kitaev quantum double model for any discrete group GGG on any planar graph forming the skeleton of a closed orientable surface of arbitrary genus. The derivation is based on the fusion rules of the properly identified vertex and plaquette excitations, which are selected among the anyons, i.e., the simple objects of the Drinfeld center Z(VecG)\mathcal{Z}(\mathrm{Vec}_G)Z(VecG​). These degeneracies are given in terms of the corresponding SSS-matrix elements and allow one to obtain the exact finite-temperature partition function of the model, valid for any finite-size system.
The string-net model describes a vast family of topological orders in two spatial dimensions. Here, we consider the effect of thermal fluctuations on these topological phases. In the original string-net model, the description of charge (vertex) excitations can be problematic. Therefore, in order to describe all anyon excitations, we study an extended model [Y. Hu et al., Phys. Rev. B 97, 195154 (2018)]. Building on recent methods, we compute the spectral degeneracies of excited states and obtain the exact partition function. In the thermodynamic limit, the latter is dominated by the trivial (vacuum) anyon, so that topological order is destroyed at any nonzero temperature. In contrast, in a finite-size system, order survives up to a finite temperature, revealing a nontrivial scaling between temperature and size similar to that of the one-dimensional classical Ising model. We confirm this scaling by computing the thermal average of several observables such as Wegner-Wilson loops and topological mutual information.
Abstract Levin-Wen string-net models provide a construction of (2+1)D topologically ordered phases of matter with anyonic localized excitations described by the Drinfeld center of a unitary fusion category. Anyon condensation is a mechanism for phase transitions between (2+1)D topologically ordered phases. We construct an extension of Levin-Wen models in which tuning a parameter implements anyon condensation. We also describe the classification of anyons in Levin-Wen models via representation theory of the tube algebra, and use a variant of the tube algebra to classify low-energy localized excitations in the condensed phase.
We consider a refined version of the string-net model which assigns a different energy cost to each plaquette excitation. Using recent exact calculations of the energy-level degeneracies we compute the partition function of this model and investigate several thermodynamical quantities. In the thermodynamic limit, we show that the partition function is dominated by the contribution of special particles, dubbed pure fluxons, which trivially braid with all other (product of) fluxons. We also analyze the behavior of Wegner-Wilson loops associated to excitations and show that they obey an area law, indicating confinement, for any finite temperature except for pure fluxons that always remain deconfined. Finally, using a recently proposed conjecture, we compute the topological mutual information at finite temperature, which features a nontrivial scaling between system size and temperature, similar to the one-dimensional classical Ising model.
The problem of N interacting spins on a lattice is equivalent to one of N clusters linked in a specific manner. The energy of any configuration of spins can be expressed in terms of the energy levels of this cluster. A new expression is obtained for the probability of occurrence of any configuration. A closed form expression is obtained for the partition function per spin in terms of the energy levels of this cluster with the degeneracies being a function of temperature. On physical grounds it is suggested to be the form of the exact partition function per spin. The partition functions of Ising-like systems all have a common form. For the 3D Ising model seven functions need to be determined to describe the partition function completely. The key to understanding second order phase transitions and critical phenomena lies in the temperature dependence of various degeneracies. It is necessary to develop new techniques to determine the partition function that account for this temperature dependence, as it would represent the underlying physics correctly.
We compute the continuum limit of the spectra for the XX model with arbitrary complex boundary fields. In the case of Hermitian boundary terms one obtains the partition functions of the free compactified boson field on a cylinder with Neumann-Neumann, Dirichlet-Neumann or Dirichlet-Dirichlet boundary conditions. This also applies for certain non-Hermitian boundaries. For special cases we also compute the free surface energy. For certain non-Hermitian boundary terms the results are more complex. Here one obtains logarithmic corrections to the free surface energy. The asymmetric version of the XX model with boundaries (this includes the Dzyaloshinsky-Moriya interaction) is also discussed.