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arXiv:1702.08469 · Physical review. B./Physical review. B

Entanglement phases as holographic duals of anyon condensates

Kasper Duivenvoorden, Mohsin Iqbal, Jutho Haegeman, Frank Verstraete, Norbert Schuch

Published Jun 12, 2017• 60 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.95.235119

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•Jul 9, 2021•0 citations
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.
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.
A topological order is a new quantum phase that is beyond Landau's symmetry-breaking paradigm. Its defining features include robust degenerate ground states, long-range entanglement and anyons. It was known that RRR- and FFF-matrices, which characterize the fusion-braiding properties of anyons, can be used to uniquely identify topological order. In this article, we explore an essential question: how can the RRR- and FFF-matrices be experimentally measured? By using quantum simulations based on a toric code model with boundaries and state-of-the-art technology, we show that the braidings, i.e. the RRR-matrices, can be completely determined by the half braidings of boundary excitations due to the boundary-bulk duality and the anyon condensation. The FFF-matrices can also be measured in a scattering quantum circuit involving the fusion of three anyons in two different orders. Thus we provide an experimental protocol for measuring the unique identifiers of topological order.
ABSTRACT Topological order is a new quantum phase that is beyond Landau’s symmetry-breaking paradigm. Its defining features include robust degenerate ground states, long-range entanglement and anyons. It was known that R and F matrices, which characterize the fusion-braiding properties of anyons, can be used to uniquely identify topological order. In this article, we explore an essential question: how can the R and F matrices be experimentally measured? We show that the braidings, i.e. the R matrices, can be completely determined by the half braidings of boundary excitations due to the boundary-bulk duality and the anyon condensation. The F matrices can also be measured by comparing the quantum states involving the fusion of three anyons in two different orders. Thus we provide a model-independent experimental protocol to uniquely identify topological order. By using quantum simulations based on a toric code model with boundaries encoded in three- and four-qubit systems and state-of-the-art technology, we obtain the first experimental measurement of R and F matrices by means of an NMR quantum computer at room temperature.
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.
We introduce a holographic framework for analyzing the steady states of repeated quantum channels with strong symmetries. Using channel-state duality, we show that the steady state of a ddd-dimensional quantum channel is holographically mapped to the boundary reduced density matrix of a (d+1)(d+1)(d+1)-dimensional wavefunction generated by a sequential unitary circuit. From this perspective, strong-to-weak spontaneous symmetry breaking (SWSSB) in the steady state arises from the anyon condensation on the boundary of a topological order in one higher dimension. The conditional mutual information (CMI) associated with SWSSB is then inherited from the bulk topological entanglement entropy. We make this duality explicit using isometric tensor network states (isoTNS) by identifying the channel's time evolution with the transfer matrix of a higher-dimensional isoTNS. Built on isoTNS, we further construct continuously tunable quantum channels that exhibit distinct mixed-state phases and transitions in the steady states.
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.