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arXiv:2208.13200 · Physical review. B./Physical review. B

Complete characterization of non-Abelian topological phase transitions and detection of anyon splitting with projected entangled pair states

Wen-Tao Xu, José Garre-Rubio, Norbert Schuch

Published Nov 23, 2022• 19 citations
arXivHTMLPDF

Abstract

Topological phases - systems with a global ordering in the entanglement - exhibit exotic excitations known as anyons which can display highly nontrivial (non-Abelian) statistics. Through transitions between such phases, these anyons can undergo rather surprising changes. Perhaps most surprisingly, one anyon can split into two or more anyons, some of which might condense or become confined. The authors propose here a complete set of order parameters which allow one to extract the full information about how the properties of anyons change through phase transitions, and to apply them to a range of topological phase transitions.

DOI
10.1103/physrevb.106.205139

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Topological phases - systems with a global ordering in the entanglement - exhibit exotic excitations known as anyons which can display highly nontrivial (non-Abelian) statistics. Through transitions between such phases, these anyons can undergo rather surprising changes. Perhaps most surprisingly, one anyon can split into two or more anyons, some of which might condense or become confined. The authors propose here a complete set of order parameters which allow one to extract the full information about how the properties of anyons change through phase transitions, and to apply them to a range of topological phase transitions.
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.
We use projected entangled pair states (PEPS) to study topological quantum phase transitions. The local description of topological order in the PEPS formalism allows us to set up order parameters which measure condensation and deconfinement of anyons and serve as substitutes for conventional order parameters. We apply these order parameters, together with anyon-anyon correlation functions and some further probes, to characterize topological phases and phase transitions within a family of models based on a Z4{\mathbb{Z}}_{4}Z4​ symmetry, which contains Z4{\mathbb{Z}}_{4}Z4​ quantum double, toric code, double semion, and trivial phases. We find a diverse phase diagram which exhibits a variety of different phase transitions of both first and second order which we comprehensively characterize, including direct transitions between the toric code and the double semion phase.
Topological phases are exotic states of matters which do not order locally yet exhibit a global ordering in their entanglement, making their characterization a challenging task. In this work, the authors construct an explicit holographic duality between topological phases in the bulk and the structure of the entanglement spectrum at its boundary, thereby establishing a one-to-one correspondence between different bulk topological orders and symmetry-breaking and symmetry-protected orderings of the entanglement. This allows classification and detection of topological phases holographically through ``entanglement spectroscopy'' at the boundary, and construction of order parameters that measure the condensation and confinement of anyons, and thus permits characterization of the nature of topological phase transitions.
We propose a unified scheme to identify phase transitions out of the Z2{\mathbb{Z}}_{2}Z2​ Abelian topological order, including the transition to a non-Abelian chiral spin liquid. Using loop gas and and string gas states [H.-Y. Lee, R. Kaneko, T. Okubo, and N. Kawashima, Phys. Rev. Lett. 123, 087203 (2019)] on the star lattice Kitaev model as an example, we compute the overlap of minimally entangled states through transfer matrices. We demonstrate that, similar to the anyon condensation, continuous deformation of a Z2{\mathbb{Z}}_{2}Z2​-injective projected entangled-pair state (PEPS) also allows us to study the transition between Abelian and non-Abelian topological orders. We show that the charge and flux anyons defined in the Abelian phase transmute into the $\ensuremath{\sigma}$ anyon in the non-Abelian topological order. Furthermore, we show that contrary to the claim in [Phys. Rev. B 101, 035140 (2020)], both the LG and SG states have infinite correlation length in the non-Abelian regime, consistent with the no-go theorem that a chiral PEPS has a gapless parent Hamiltonian.
Anyon condensation transitions remain challenging to characterize especially when the condensing anyons are non-abelian. We develop two complementary approaches for extracting information about the transition from the categorical data of the parent topological order and the condensate. The first adapts a Ginzburg-Landau procedure developed for 1+11+11+1D transitions in the SymTFT setting to 2+12+12+1D anyon condensation transitions. The construction applies when the anyons participating in the condensate have integer quantum dimensions. The second gives a criterion for identifying which point and loop excitations become critical at a continuous transition. The key step is to determine which anyon loops remain condensed throughout. All other loops become critical, while a point excitation becomes critical if it braids trivially with every condensed loop. We illustrate both proposals in several abelian and non-abelian examples.