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arXiv:0908.1674 · New Journal of Physics

Characterizing symmetries in a projected entangled pair state

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

Published Feb 26, 2010• 75 citations
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Abstract

We show that two different tensors defining the same translational invariant injective projected entangled pair state (PEPS) in a square lattice must be the same up to a trivial gauge freedom. This allows us to characterize the existence of any local or spatial symmetry in the state. As an application of these results we prove that a SU(2) invariant PEPS with half-integer spin cannot be injective, which can be seen as a Lieb–Shultz–Mattis theorem in this context. We also give the natural generalization for U(1) symmetry in the spirit of Oshikawa–Yamanaka–Affleck, and show that a PEPS with Wilson loops cannot be injective.

DOI
10.1088/1367-2630/12/2/025010

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We show that two different tensors defining the same translational invariant injective projected entangled pair state (PEPS) in a square lattice must be the same up to a trivial gauge freedom. This allows us to characterize the existence of any local or spatial symmetry in the state. As an application of these results we prove that a SU(2) invariant PEPS with half-integer spin cannot be injective, which can be seen as a Lieb–Shultz–Mattis theorem in this context. We also give the natural generalization for U(1) symmetry in the spirit of Oshikawa–Yamanaka–Affleck, and show that a PEPS with Wilson loops cannot be injective.
We show that two different tensors defining the same translational invariant injective Projected Entangled Pair State (PEPS) in a square lattice must be the same up to a trivial gauge freedom. This allows us to characterize the existence of any local or spatial symmetry in the state. As an application of these results we prove that a SU(2) invariant PEPS with half-integer spin cannot be injective, which can be seen as a Lieb-Shultz-Mattis theorem in this context. We also give the natural generalization for U(1) symmetry in the spirit of Oshikawa-Yamanaka-Affleck, and show that a PEPS with Wilson loops cannot be injective.
Projected entangled pair states (PEPS) provide exact representations for many nonchiral topologically ordered states whereas their range of applicability to interacting chiral topological phases remains largely unsettled. In this context, the symmetries of the local PEPS tensors are crucial for determining the characteristic topological features of the state. In this paper we examine the constraints that arise when different symmetries are imposed simultaneously on the local tensor such as internal SU(2)\mathrm{SU}(2)SU(2), point group, and translation symmetry. We show how the interplay of these symmetries manifests in the entanglement spectrum which is the main diagnostic tool for chiral topological order. We apply our results to a spin liquid PEPS introduced previously as a chiral generalization of the resonating valence bond state. Our findings explain the discrepancies observed between the entanglement spectrum of this state and the expected edge spectrum described by a chiral conformal field theory. Finally, in a certain parameter region where this PEPS possesses an additional U(1)U(1)U(1) symmetry we are able to resolve these discrepancies and obtain an entanglement spectrum with the expected state countings and conformal weight.
Abstract Projected entangled pair states (PEPS) are very useful in the description of strongly correlated systems, partly because they allow encoding symmetries, either global or local (gauge), naturally. In recent years, PEPS with local symmetries have increasingly been used in the study of non-perturbative regimes of lattice gauge theories, most prominently as a way to construct variational ansatz states depending only on a small number of parameters and yet capturing the relevant physical properties. For the case of one-dimensional PEPS (matrix product states—MPS) a bidirectional connection was established between the internal structure of the tensor network, i.e. the mathematical properties of the constituent tensors, and the symmetry. In higher dimensions this has only been done for global symmetries, where in the local (gauge) case it is known only how to construct gauge-invariant states, but not what the symmetry implies on the internal structure of the PEPS. In the present work we complete this missing piece and study the internal structure of PEPS with a gauge symmetry. The PEPS we consider consist of matter and gauge field tensors placed on the vertices and edges, respectively, of arbitrary graphs.
Tensor network states, and in particular projected entangled pair states (PEPS), suggest an innovative approach for the study of lattice gauge theories, both from a pure theoretic point of view, and as a tool for the analysis of the recent proposals for quantum simulations of lattice gauge theories.In this paper we present a framework for describing locally gauge invariant states on lattices using PEPS.The PEPS constructed hereby shall include both bosonic and fermionic states, suitable for all combinations of matter and gauge fields in lattice gauge theories defined by either finite or compact Lie groups.
We introduce a family of tensor network states that we term semi-injective Projected Entangled-Pair States (PEPSs). They extend the class of injective PEPSs and include other states, like the ground states of the AKLT and the CZX models in square lattices. We construct parent Hamiltonians for which semi-injective PEPSs are unique ground states. We also determine the necessary and sufficient conditions for two tensors to generate the same family of such states in two spatial dimensions. Using this result, we show that the third cohomology labeling of symmetry protected topological phases extends to semi-injective PEPSs.
Projected entangled pair states aim at describing lattice systems in two spatial dimensions that obey an area law. They are specified by associating a tensor with each site, and they are generated by patching these tensors. We consider the problem of determining whether the state resulting from this patching is null, and prove it to be NP-hard; the PEPS used to prove this claim have a boundary and are homogeneous in their bulk. A variation of this problem is next shown to be undecidable. These results have various implications: they question the possibility of a 'fundamental theorem' for PEPS; there are PEPS for which the presence of a symmetry is undecidable; there exist parent hamiltonians of PEPS for which the existence of a gap above the ground state is undecidable. En passant, we identify a family of classical Hamiltonians, with nearest neighbour interactions, and translationally invariant in their bulk, for which the commuting 2-local Hamiltonian problem is NP-complete.