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arXiv:2101.05289 · Physical Review Research

Wilson loops and area laws in lattice gauge theory tensor networks

Erez Zohar

Published Aug 20, 2021• 7 citations
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Abstract

Tensor network states have been a very prominent tool for the study of quantum many-body physics, thanks to their physically relevant entanglement properties and their ability to encode symmetries. In the last few years, the formalism has been extended and applied to theories with local symmetries too---lattice gauge theories. In order to extract physical properties (such as expectation values and correlation functions of physical observables) out of such states, one has to use the so-called transfer operators, the local properties of which dictate the long-range behavior of the state. In this paper we study transfer operators of tensor network states (in particular, projected entangled pair states) of lattice gauge theories, and consider the implications of the local symmetry on their structure and properties. In particular, we study the implications on the computation of the Wilson loop---a nonlocal, gauge-invariant observable which is central to pure gauge theories, the long-range decay behavior of which probes the confinement or deconfinement of static charges. Using the symmetry, we show how to simplify the tensor contraction required for computing Wilson loop expectation values for such states, eliminate nonphysical parts of the tensors, and formulate conditions relating local properties (that is, of the tensors) to their decay fashion.

DOI
10.1103/physrevresearch.3.033179

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Tensor network states have been a very prominent tool for the study of quantum many-body physics, thanks to their physically relevant entanglement properties and their ability to encode symmetries. In the last few years, the formalism has been extended and applied to theories with local symmetries too---lattice gauge theories. In order to extract physical properties (such as expectation values and correlation functions of physical observables) out of such states, one has to use the so-called transfer operators, the local properties of which dictate the long-range behavior of the state. In this paper we study transfer operators of tensor network states (in particular, projected entangled pair states) of lattice gauge theories, and consider the implications of the local symmetry on their structure and properties. In particular, we study the implications on the computation of the Wilson loop---a nonlocal, gauge-invariant observable which is central to pure gauge theories, the long-range decay behavior of which probes the confinement or deconfinement of static charges. Using the symmetry, we show how to simplify the tensor contraction required for computing Wilson loop expectation values for such states, eliminate nonphysical parts of the tensors, and formulate conditions relating local properties (that is, of the tensors) to their decay fashion.
A bstract We develop a transfer operator approach for the calculation of Rényi entanglement entropies in arbitrary (i.e. Abelian and non-Abelian) pure lattice gauge theory projected entangled pair states in 2+1 dimensions. It is explicitly shown how the long-range behavior of these quantities gives rise to an entanglement area law in both the thermodynamic limit and in the continuum. We numerically demonstrate the applicability of our method to the ℤ 2 lattice gauge theory and relate some entanglement properties to the confinement-deconfinement transition therein. We provide evidence that Rényi entanglement entropies in certain cases do not provide a complete probe of (de)confinement properties compared to Wilson loop expectation values as other genuine (nonlocal) observables.
Tensor network methods are powerful and efficient tools for studying the properties and dynamics of statistical and quantum systems, in particular in one and two dimensions. In recent years, these methods have been applied to lattice gauge theories, yet these theories remain a challenge in ( 2 + 1 ) dimensions. In this article, we present a new (decorated) tensor network algorithm, in which the tensors encode the lattice gauge amplitude expressed in the fusion basis. This has several advantages—firstly, the fusion basis does diagonalize operators measuring the magnetic fluxes and electric charges associated to a hierarchical set of regions. The algorithm allows therefore a direct access to these observables. Secondly the fusion basis is, as opposed to the previously employed spin network basis, stable under coarse-graining. Thirdly, due to the hierarchical structure of the fusion basis, the algorithm does implement predefined disentanglers. We apply this new algorithm to lattice gauge theories defined for the quantum group SU ( 2 ) k and identify a weak and a strong coupling phase for various levels k . As we increase the level k , the critical coupling g c decreases linearly, suggesting the absence of a deconfining phase for the continuous group SU ( 2 ) . Moreover, we illustrate the scaling behaviour of the Wilson loops in the two phases.
Tensor network states, and in particular projected entangled pair states (PEPS) have been a strong ansatz for the variational study of complicated quantum many-body systems, thanks to their built-in entanglement entropy area law. In this work, we use a special kind of PEPS---gauged Gaussian fermionic PEPS (GGFPEPS)---to find the ground state of 2+12+12+1 dimensional pure Z2{\mathbb{Z}}_{2}Z2​ lattice gauge theories for a wide range of coupling constants. We do so by combining PEPS methods with Monte-Carlo computations, allowing for efficient contraction of the PEPS and computation of correlation functions. Previously, such numerical computations involved the calculation of the Pfaffian of a matrix scaling with the system size, forming a severe bottleneck; in this work we show how to overcome this problem. This paves the way for applying the method we propose and benchmark here to other gauge groups, higher dimensions, and models with fermionic matter, in an efficient, sign-problem-free way.
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.
Lattice gauge theories form a very successful framework for studying nonperturbative gauge field physics, in particular in quantum chromodynamics. Recently, their quantum simulation on atomic and solid-state platforms has been discussed, aiming at overcoming some of the difficulties still faced by the conventional approaches (such as the sign problem and real time evolution). While the actual implementations of a lattice gauge theory on a quantum simulator may differ in terms of the simulating system and its properties, they are all directed at studying similar physical phenomena, requiring the measurement of nonlocal observables, due to the local symmetry of gauge theories. In this work, general schemes for measuring such nonlocal observables (Wilson loops and mesonic string operators) in general lattice gauge theory quantum simulators that are based merely on local operations are proposed.
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.