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arXiv:2410.18947 · Journal of Physics A Mathematical and Theoretical

Internal structure of gauge-invariant projected entangled pair states

David Blanik, José Garre-Rubio, András Molnár, Erez Zohar

Published Jan 27, 2025• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/adae83

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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.
Gauge theories form the basis of our understanding of modern physics—ranging from the description of quarks and gluons to effective models in condensed matter physics. In the nonperturbative regime, gauge theories are conventionally treated discretely as lattice gauge theories. The resulting systems are evaluated with path-integral based Monte Carlo methods. These methods, however, can suffer from the sign problem and do not allow for a direct evaluation of real-time dynamics. In this work, we present a unified and comprehensive framework for gauged Gaussian projected entangled pair states (PEPS), a variational ansatz based on tensor networks. We review the construction of Hamiltonian lattice gauge theories, explain their similarities to PEPS, and detail the construction of the ansatz state. The estimation of ground states is based on a variational Monte Carlo procedure with the PEPS as an ansatz state. This sign-problem-free ansatz can be efficiently evaluated in any dimension with arbitrary gauge groups, and can include dynamical fermionic matter, suggesting new options for the simulation of nonperturbative regimes of gauge theories, including quantum chromodynamics. Published by the American Physical Society 2024
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.
This thesis contributes to the understanding of symmetry-enriched topological phases focusing on their descriptions in terms of tensor network states. The Projected Entangled Pair State (PEPS) formalism allows us to locally encode the main properties of the models (like topological order, symmetries and their interplay) in the tensors. We have used that encoding to classify, construct and detect some classes of symmetry-enriched topological phases in 2D PEPS. For that purpose, we have studied what the allowed freedom in the tensors generating the same tensor network state is. These results are the so-called 'fundamental theorems' and they give the necessary knowledge to properly study symmetries (actions that leave the states invariant). We focus our study on the family of PEPS describing quantum double models of GGG, the so-called GGG-injective PEPS, and on global on-site symmetries coming from a finite group QQQ: as a result of the classification we have a finite number of phases closely related to the theory of group extensions. We also provide the maps that appear in group extension theory with physical meaning and we characterize their actions on the ground subspace and on the anyons. We construct a representative of each class of our classification. Moreover, we connect our construction with an interesting physical phenomenon, the so-called anyon condensation, which describes topological phase transitions. We have also proposed a family of gauge invariant quantities and their corresponding order parameters (in the form of expectation values of a local operator in 2D) to detect the corresponding quantum phases, in particular their symmetry fractionalization patterns. We conclude this thesis by touching on some mathematical open problems in PEPS.
This thesis is devoted to the study of quantum many-body systems. This investigation is performed in the framework of Matrix Product States (MPS) and their generalization to higher dimensions, Projected Entangled Pair States (PEPS). In the first part of the work, we discuss the mathematical properties of such tensor network states in depth. In the first chapters we deal with one-dimensional systems, for which we use the connections between MPS and completely positive maps to prove several new results, such as the quantum version of the Wielandt's inequality or the construction of locally invariant states under a symmetry group. Chapters 5-7 are dedicated to higher-dimensional systems, for which we provide the conditions to construct invariant PEPS. The second part of the thesis is dedicated to applications in condensed matter. In chapter 8 we provide several methods of constructing quasi-solvable Hamiltonians with two-body interactions, while in chapter 9 we show that these tensor network states can be used as a laboratory for theoretical condensed matter in, for instance, the characterization of the string-order, the generalization of the Lieb-Schultz-Mattis theorem, the demonstration of new theorems relating entanglement to magnetisation or to long-range interactions, etc.
Matrix Product States (MPS) are a particular type of one dimensional tensor network states, that have been applied to the study of numerous quantum many body problems. One of their key features is the possibility to describe and encode symmetries on the level of a single building block (tensor), and hence they provide a natural playground for the study of symmetric systems. In particular, recent works have proposed to use MPS (and higher dimensional tensor networks) for the study of systems with local symmetry that appear in the context of gauge theories. In this work we classify MPS which exhibit local invariance under arbitrary gauge groups. We study the respective tensors and their structure, revealing known constructions that follow known gauging procedures, as well as different, other types of possible gauge invariant states.
This thesis is divided into two mainly independent parts: In the first part, we derive a criterion to determine when a translationally invariant Matrix Product State (MPS) has long range localizable entanglement, which indicates that the corresponding state has some kind of non-local hidden order. We give examples fulfilling this criterion and eventually use it to obtain all such MPS with bond dimension 2 and 3. In the second part, we show that Projected Entangled Pair States (PEPS) in two spatial dimensions can describe chiral topological states by explicitly constructing a family of such states with a non-trivial Chern number. We demonstrate that such free fermionic PEPS must necessarily be non-injective and have gapless parent Hamiltonians. Moreover, we provide numerical evidence that they can nevertheless approximate well the physical properties of Chern insulators with local Hamiltonians at arbitrary temperatures. We also construct long range, topological Hamiltonians with a flat energy spectrum for which those PEPS are unique ground states. As for non-chiral topological PEPS, the non-trivial, topological properties can be traced down to the existence of a symmetry on the virtual level of the PEPS tensor that is used to build the state. We use the special properties of PEPS to build the boundary theory and show how the symmetry results in the appearance of chiral modes, a ground state degeneracy of the parent Hamiltonian on the torus and a universal correction to the area law for the zero Rényi entropy. Finally, we show that PEPS can also describe chiral topologically ordered phases. For that, we construct a simple PEPS for spin-1/2 particles in a two-dimensional lattice. We reveal a symmetry of the PEPS tensor that gives rise to the global topological character. We also extract characteristic quantities of the edge Conformal Field Theory using the bulk-boundary correspondence.