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arXiv:1505.04217 · Physical Review B

Long-range order and symmetry breaking in projected entangled-pair state models

Manuel Rispler, Kasper Duivenvoorden, Norbert Schuch

Published Oct 20, 2015• 18 citations
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Abstract

Projected entangled-pair states (PEPS) provide a framework for the construction of models where a single tensor gives rise to both Hamiltonian and ground state wave function on the same footing. A key problem is to characterize the behavior which emerges in the system in terms of the properties of the tensor, and thus of the Hamiltonian. In this paper, we consider PEPS models with ${\mathbb{Z}}_{2}$ on-site symmetry and study the occurrence of long-range order and spontaneous symmetry breaking. We show how long-range order is connected to a degeneracy in the spectrum of the PEPS transfer operator, and how the latter gives rise to spontaneous symmetry breaking under perturbations. We provide a succinct characterization of the symmetry-broken states in terms of the PEPS tensor, and find that using the symmetry-broken states we can derive a local entanglement Hamiltonian, thereby restoring locality of the entanglement Hamiltonian for all gapped phases.

DOI
10.1103/physrevb.92.155133

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Matrix product operators for symmetry-protected topological phases

Dominic J. Williamson, Nick Bultinck, Michael Mariën +3 more

Projected entangled pair states (PEPS) provide a natural description of the ground states of gapped, local Hamiltonians in which global characteristics of a quantum state are encoded in properties of local tensors. We show that on-site symmetries, as occurring in systems exhibiting symmetry-protected topological (SPT) quantum order, can be captured by a virtual symmetry of the tensors expressed as a set of matrix product operators labelled by the different group elements. A classification of SPT phases can hence be obtained by studying the topological obstructions to continuously deforming one set of matrix product operators into another. This leads to the classification of bosonic SPT states in terms of group cohomology, as originally derived by Chen et al. in [1106.4772]. Our formalism accommodates perturbations away from fixed point models, and hence opens up the possibility of studying phase transitions between different SPT phases. We furthermore show how the global symmetries of SPT PEPS can be promoted into a set of local gauge constraints by introducing bosonic degrees of freedom on the links of the PEPS lattice, thereby providing a natural and general mapping between PEPS in SPT phases and topologically ordered phases.

•Dec 17, 2014•6 citations
Projected entangled-pair states (PEPS) provide a framework for the construction of models where a single tensor gives rise to both Hamiltonian and ground state wave function on the same footing. A key problem is to characterize the behavior which emerges in the system in terms of the properties of the tensor, and thus of the Hamiltonian. In this paper, we consider PEPS models with Z2{\mathbb{Z}}_{2}Z2​ on-site symmetry and study the occurrence of long-range order and spontaneous symmetry breaking. We show how long-range order is connected to a degeneracy in the spectrum of the PEPS transfer operator, and how the latter gives rise to spontaneous symmetry breaking under perturbations. We provide a succinct characterization of the symmetry-broken states in terms of the PEPS tensor, and find that using the symmetry-broken states we can derive a local entanglement Hamiltonian, thereby restoring locality of the entanglement Hamiltonian for all gapped phases.
ZN{\mathbb Z}_NZN​ symmetry breaking in projected entangled pair state models
We consider Projected Entangled Pair State (PEPS) models with a global ZN\mathbb Z_NZN​ symmetry, which are constructed from ZN\mathbb Z_NZN​-symmetric tensors and are thus ZN\mathbb Z_NZN​-invariant wavefunctions, and study the occurence of long-range order and symmetry breaking in these systems. First, we show that long-range order in those models is accompanied by a degeneracy in the so-called transfer operator of the system. We subsequently use this degeneracy to determine the nature of the symmetry broken states, i.e., those stable under arbitrary perturbations, and provide a succinct characterization in terms of the fixed points of the transfer operator (i.e.\ the different boundary conditions) in the individual symmetry sectors. We verify our findings numerically through the study of a Z3\mathbb Z_3Z3​-symmetric model, and show that the entanglement Hamiltonian derived from the symmetry broken states is quasi-local (unlike the one derived from the symmetric state), reinforcing the locality of the entanglement Hamiltonian for gapped phases.
Abstract We consider projected entangled pair state (PEPS) models with a global Z N symmetry, which are constructed from Z N -symmetric tensors and are thus Z N -invariant wavefunctions, and study the occurence of long-range order and symmetry breaking in these systems. First, we show that long-range order in those models is accompanied by a degeneracy in the so-called transfer operator of the system. We subsequently use this degeneracy to determine the nature of the symmetry broken states, i.e. those stable under arbitrary perturbations, and provide a succinct characterization in terms of the fixed points of the transfer operator (i.e. the different boundary conditions) in the individual symmetry sectors. We verify our findings numerically through the study of a Z 3 -symmetric model, and show that the entanglement Hamiltonian derived from the symmetry broken states is quasi-local (unlike the one derived from the symmetric state), reinforcing the locality of the entanglement Hamiltonian for gapped phases.
In this thesis, we consider projected entangled pair state (PEPS) models as a framework for two-dimensional strongly-correlated quantum many body systems, where the global properties of the system are concisely encoded in one local tensor. While it was well known how to construct PEPS models with desired symmetries by manifestly encoding the symmetry in the local tensor, our goal in this work is to investigate whether this manifestly encoded symmetry can be spontaneously broken. By defining long-range order as a good criterion for symmetry breaking in a finite volume PEPS, we answer this question in the affirmative. A particularly attractive feature of PEPS models is that they have an exact holographic mapping, which is to say that the large system behavior is described by the ``PEPS boundary'', given by the fixed point(s) of the so called transfer operator. We investigate the implications of long-range order and prove that it leads to a particular symmetry breaking pattern in the fixed point space. We find this pattern by first proving that long-range order implies an asymptotic degeneracy in the transfer operator and then identifying the symmetry breaking mechanism as the one which, from all possible boundary (-matrix) configurations, selects the configurations that correspond to positive-semi-definite matrices, i.e. one-dimensional virtual quantum states. We prove that these correspond to boundary configurations that do not mix under arbitrary perturbations and are thus stable. We study the entanglement properties of these virtual quantum states and show that they can be described by local entanglement Hamiltonians, establishing that also in phases with symmetry breaking, gapped quantum phases have an associated local entanglement Hamiltonian.
We study projected entangled pair states (PEPS) with continuous virtual symmetries, i.e., symmetries in the virtual degrees of freedom, through an elementary class of models with SU(2)\mathrm{SU}(2)SU(2) symmetry. Discrete symmetries of that kind have previously allowed for a comprehensive explanation of topological order in the PEPS formalism. We construct local parent Hamiltonians whose ground space with open boundaries is exactly parametrized by the PEPS wave function, and show how the ground state can be made unique by a suitable choice of boundary conditions. We also find that these models exhibit a logarithmic correction to the entanglement entropy and an extensive ground space degeneracy on systems with periodic boundaries, which suggests that they do not describe conventional gapped topological phases, but either critical models or some other exotic phase.
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 onsite 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 [Chen et al., Phys. Rev. B 87, 155114 (2013)]. 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-point 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.
Projected entangled pair states (PEPS) provide a natural description of the ground states of gapped, local Hamiltonians in which global characteristics of a quantum state are encoded in properties of local tensors. We show that on-site symmetries, as occurring in systems exhibiting symmetry-protected topological (SPT) quantum order, can be captured by a virtual symmetry of the tensors expressed as a set of matrix product operators labelled by the different group elements. A classification of SPT phases can hence be obtained by studying the topological obstructions to continuously deforming one set of matrix product operators into another. This leads to the classification of bosonic SPT states in terms of group cohomology, as originally derived by Chen et al. in [1106.4772]. Our formalism accommodates perturbations away from fixed point models, and hence opens up the possibility of studying phase transitions between different SPT phases. We furthermore show how the global symmetries of SPT PEPS can be promoted into a set of local gauge constraints by introducing bosonic degrees of freedom on the links of the PEPS lattice, thereby providing a natural and general mapping between PEPS in SPT phases and topologically ordered phases.