Characterizing symmetries in a projected entangled pair state

Pérez-Garcı́a, David; Sanz, Mikel; González-Guillén, Carlos E.; Wolf, Michael M.; Cirac, J. I.

doi:10.1088/1367-2630/12/2/025010

Cited by 67 publications

(61 citation statements)

References 24 publications

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“…Since the transfer operator has itself a Matrix Product structure, any symmetry of the transfer operator must be encoded locally, i.e., it will show up as a symmetry of the singlesite ket+bra object shown in Fig. 5a [30]. There can be two distinct types of such symmetries: Those which act identically on ket and bra layer, shown in Fig.…”

Section: B the Assumption: Matrix Product Fixed Pointsmentioning

confidence: 99%

Entanglement phases as holographic duals of anyon condensates

et al. 2017

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Anyon condensation forms a mechanism which allows to relate different topological phases. We study anyon condensation in the framework of Projected Entangled Pair States (PEPS) where topological order is characterized through local symmetries of the entanglement. We show that anyon condensation is in one-to-one correspondence to the behavior of the virtual entanglement state at the boundary (i.e., the entanglement spectrum) under those symmetries, which encompasses both symmetry breaking and symmetry protected (SPT) order, and we use this to characterize all anyon condensations for abelian double models through the structure of their entanglement spectrum. We illustrate our findings with the Z4 double model, which can give rise to both Toric Code and Doubled Semion order through condensation, distinguished by the SPT structure of their entanglement. Using the ability of our framework to directly measure order parameters for condensation and deconfinement, we numerically study the phase diagram of the model, including direct phase transitions between the Doubled Semion and the Toric Code phase which are not described by anyon condensation.

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Section: B the Assumption: Matrix Product Fixed Pointsmentioning

confidence: 99%

Entanglement phases as holographic duals of anyon condensates

et al. 2017

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“…For the special case of injective PEPS [6] the MPO P is simply the identity P = 1 1 (i.e. a MPO with bond dimension 1), the symmetry MPOs V (g) can always be factorized into a tensor product of local gauge transformations [41] and the ground state is unique.…”

Section: Global Symmetry In Pepsmentioning

confidence: 99%

Matrix product operators for symmetry-protected topological phases: Gauging and edge theories

et al. 2016

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Projected entangled pair states (PEPS) provide a natural ansatz for the ground states of gapped, local Hamiltonians in which global characteristics of a quantum state are encoded in properties of local tensors. We develop a framework to describe on-site symmetries, as occurring in systems exhibiting symmetry-protected topological (SPT) quantum order, in terms of virtual symmetries of the local tensors expressed as a set of matrix product operators (MPOs) labeled by distinct group elements. These MPOs describe the possibly anomalous symmetry of the edge theory, whose local degrees of freedom are concretely identified in a PEPS. A classification of SPT phases is obtained by studying the obstructions to continuously deforming one set of MPOs into another, recovering the results derived for fixed-point models [X. Chen et al., Phys. Rev. B 87, 155114 (2013)] [1]. Our formalism accommodates perturbations away from fixed point models, opening the possibility of studying phase transitions between different SPT phases. We also demonstrate that applying the recently developed quantum state gauging procedure to a SPT PEPS yields a PEPS with topological order determined by the initial symmetry MPOs. The MPO framework thus unifies the different approaches to classifying SPT phases, via fixed-points models, boundary anomalies, or gauging the symmetry, into the single problem of classifying inequivalent sets of matrix product operator symmetries that are defined purely in terms of a PEPS.

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“…In the case of translational invariant systems, the state is described by a single local tensor, and understanding any property of the system can be mapped to studying a corresponding property of this tensor. In this way, PEPS have been very successful in understanding otherwise intractable questions, such as the characterization of topological order from local symmetries, 4 the way in which global symmetries emerge locally, 5 or the characterization of quantum phases without and with symmetries in one dimension [6][7][8] and beyond, 9 just to name a few. To assess how general these results are, it is therefore important to identify the conditions under which PEPS are robust to perturbations.…”

Section: Introductionmentioning

confidence: 99%

Robustness in projected entangled pair states

et al. 2013

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We analyze a criterion which guarantees that the ground states of certain many-body systems are stable under perturbations. Specifically, we consider PEPS, which are believed to provide an efficient description, based on local tensors, for the low energy physics arising from local interactions. In order to assess stability in the framework of PEPS, one thus needs to understand how physically allowed perturbations of the local tensor affect the properties of the global state. In this paper, we show that a restricted version of the local topological quantum order (LTQO) condition [Michalakis and Pytel, Commun. Math. Phys. 322, 277 (2013)] provides a checkable criterion which allows us to assess the stability of local properties of PEPS under physical perturbations. We moreover show that LTQO itself is stable under perturbations which preserve the spectral gap, leading to nontrivial examples of PEPS which possess LTQO and are thus stable under arbitrary perturbations.

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Characterizing symmetries in a projected entangled pair state

Cited by 67 publications

References 24 publications

Entanglement phases as holographic duals of anyon condensates

Entanglement phases as holographic duals of anyon condensates

Matrix product operators for symmetry-protected topological phases: Gauging and edge theories

Robustness in projected entangled pair states

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