Building Projected Entangled Pair States with a Local Gauge Symmetry

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Abstract

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.

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Most cited references 38

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Goals and opportunities in quantum simulation

J Cirac, Peter Zoller (2012)

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Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations

Matthias Troyer, Uwe-Jens Wiese (2004)

Quantum Monte Carlo simulations, while being efficient for bosons, suffer from the "negative sign problem'' when applied to fermions - causing an exponential increase of the computing time with the number of particles. A polynomial time solution to the sign problem is highly desired since it would provide an unbiased and numerically exact method to simulate correlated quantum systems. Here we show, that such a solution is almost certainly unattainable by proving that the sign problem is NP-hard, implying that a generic solution of the sign problem would also solve all problems in the complexity class NP (nondeterministic polynomial) in polynomial time.

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Classifying quantum phases using Matrix Product States and PEPS

David Perez-Garcia, Norbert Schuch, Ignacio Cirac (2010)

We give a classification of gapped quantum phases of one-dimensional systems in the framework of Matrix Product States (MPS) and their associated parent Hamiltonians, for systems with unique as well as degenerate ground states, and both in the absence and presence of symmetries. We find that without symmetries, all systems are in the same phase, up to accidental ground state degeneracies. If symmetries are imposed, phases without symmetry breaking (i.e., with unique ground states) are classified by the cohomology classes of the symmetry group, this is, the equivalence classes of its projective representations, a result first derived in [X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 83, 035107 (2011); arXiv:1008.3745]. For phases with symmetry breaking (i.e., degenerate ground states), we find that the symmetry consists of two parts, one of which acts by permuting the ground states, while the other acts on individual ground states, and phases are labelled by both the permutation action of the former and the cohomology class of the latter. Using Projected Entangled Pair States (PEPS), we subsequently extend our framework to the classification of two-dimensional phases in the neighborhood of a number of important cases, in particular systems with unique ground states, degenerate ground states with a local order parameter, and topological order. We also show that in two dimensions, imposing symmetries does not constrain the phase diagram in the same way it does in one dimension. As a central tool, we introduce the isometric form, a normal form for MPS and PEPS which is a renormalization fixed point. Transforming a state to its isometric form does not change the phase, and thus, we can focus on to the classification of isometric forms.

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Publication date Created: 2015-11-26

Publication date Updated: 2016-04-08

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DOI: 10.1088/1367-2630/18/4/043008

ArXiV ID: 1511.08426

SO-VID: ec9aa6fb-ee98-4556-9a8b-ceacd9f271d0

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http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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Journal reference New J. Phys. 18, 043008 (2016)

Comments Updated version with a new section, an extended introduction and additional comments and references

Categories quant-ph hep-lat

ScienceOpen disciplines: Quantum physics & Field theory,High energy & Particle physics

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ScienceOpen disciplines: Quantum physics & Field theory, High energy & Particle physics

Building Projected Entangled Pair States with a Local Gauge Symmetry

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Journal of Green Building

Most cited references 38

Goals and opportunities in quantum simulation

Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations

Classifying quantum phases using Matrix Product States and PEPS

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