Elsevier

Annals of Physics

Volume 363, December 2015, Pages 385-439
Annals of Physics

Fermionic projected entangled pair states and local U(1) gauge theories

https://doi.org/10.1016/j.aop.2015.10.009Get rights and content

Abstract

Tensor networks, and in particular Projected Entangled Pair States (PEPS), are a powerful tool for the study of quantum many body physics, thanks to both their built-in ability of classifying and studying symmetries, and the efficient numerical calculations they allow. In this work, we introduce a way to extend the set of symmetric PEPS in order to include local gauge invariance and investigate lattice gauge theories with fermionic matter. To this purpose, we provide as a case study and first example, the construction of a fermionic PEPS, based on Gaussian schemes, invariant under both global and local U(1) gauge transformations. The obtained states correspond to a truncated U(1) lattice gauge theory in 2+1 dimensions, involving both the gauge field and fermionic matter. For the global symmetry (pure fermionic) case, these PEPS can be studied in terms of spinless fermions subject to a p-wave superconducting pairing. For the local symmetry (fermions and gauge fields) case, we find confined and deconfined phases in the pure gauge limit, and we discuss the screening properties of the phases arising in the presence of dynamical matter.

Introduction

Within the framework of the standard model of particle physics, the three fundamental forces are described by gauge bosons, which are the excitations of gauge fields. Gauge fields are vector fields, which manifest a very special local continuous symmetry, called local gauge invariance. This symmetry gives the matter fields gauge charges, and its local nature induces local interactions of the gauge currents with the charged matter. The conservation of local charges, manifested by local constraints which are extensions of the well-known Gauss law from electrodynamics, implies a very rich, complicated structure of the Hilbert space of quantum gauge theories, dictated by superselection rules governed by these local charges. This makes such theories, in general, very challenging and difficult to solve—just as much as they are interesting and important for the description of nature.

Described within the framework of quantum field theory, gauge theories come along with a very important computational tool: perturbation theory and its Feynman diagrams. However, despite the great success and accuracy achieved for Quantum Electrodynamics (QED) with perturbative methods, they apply only partially to QCD. Unlike QED, which is the Abelian gauge theory associated with the group U(1), QCD is a non-Abelian, SU(3) gauge theory, which makes it behave in a completely different manner: due to an important property of such non-Abelian theories, Asymptotic Freedom   [1], the strong coupling constant flows to zero for high energies (or short distances)—allowing, therefore, for perturbative calculations in these scales, such as within the nuclei (e.g., the parton model, or Bjorken’s scaling  [2], [3]). On the other hand, at low energies or large distances, the coupling constant is strong and perturbative physics is impossible; this may be seen as both the cause and the effect of Quark Confinement   [4], [5], the phenomenon responsible for holding quarks bound together into hadrons, and for the absence of free quarks in the spectrum of the theory.

This has significant implications on the study of the theory, and, indeed, over the years many non-perturbative techniques have been developed and applied for the study of QCD, and non-Abelian gauge theories in general. One of them, perhaps the most fruitful, is lattice gauge theory (LGT) [4], [6], [7], [8], in which either spacetime, or space, is discretized, allowing either for a regularization of the theory for analytical purposes, or very efficient and fruitful numerical (Monte Carlo) calculations. While having a great success with many different types of calculations and predictions (e.g., low-energy hadronic spectrum  [9], among others), Monte Carlo calculations are problematic in some cases: first, with fermions with a finite chemical potential (required, for example, for the phases of color superconductivity and quark–gluon plasma  [10], [11]), due to the computationally hard sign problem  [12], and second, as the calculations are carried out in Euclidean spacetime, real-time dynamics in Minkowski spacetime cannot be achieved (see, on the other hand, the recent works  [13], [14]).

A complementary way of overcoming the computational difficulties may be the use of tensor network techniques, or tensor network states (TNs), and in particular Matrix Product States (MPS)  [15] and Projected Entangled Pair States (PEPS)  [16], [17], [18]. One may, for example, use TN variational techniques, in which a TN state with variational parameters is used as an ansatz for the ground state of a given Hamiltonian, as well as calculate dynamics of such states in very efficient ways, exploiting methods like DMRG (Density Matrix Renormalization Group)  [19]. This approach has been recently applied with MPS for 1+1 dimensional lattice gauge theories, either Abelian or non-Abelian, and used for the study of their spectrum, dynamics (including string-breaking) and finite temperature effects  [20], [21], [22], [23], [24], [25], [26], [27], [28], [29]. Furthermore also tensor renormalization group techniques have been recently applied to the study of such models  [30]. The MPS studies have shown many of the static and dynamic properties of some well known theories (such as the Schwinger model  [31], [32], for example), and allowed to reach better precision than analogous Monte Carlo calculations and to perform dynamical simulations, holding the promise for even better accuracy and computational possibilities.

However, the great computational power is not the only reason for TNs to be candidates for the study of gauge theories. In a somewhat change of paradigm, one may describe a physical system from the point of view of its most representative states, instead of starting from its Lagrangian or Hamiltonian formulation. Tensor networks are, indeed, well suited to define families of states, as functions of a set of variational parameters, which fulfill precise symmetry constraints. Therefore they provide a natural way to encode all the symmetries of a system  [33] and to describe its possible thermodynamical phases in terms of representative states, allowing to investigate the main physical properties within the universality class of the problem under scrutiny  [34]. Starting from the tensor network construction, it is also possible to show that such states constitute the ground states of local parent Hamiltonians. These Hamiltonians may be explicitly derived in the simplest cases, and offer, as a function of the variational parameters, suitable examples to study the properties of the thermodynamical phases in a certain universality class. Previous works in this direction include two-dimensional PEPS schemes for lattice pure-gauge theories (without dynamical matter)  [35], as well as a general framework useful to the study of lattice gauge theories with bosonic matter  [36].

The next reasonable step is to consider lattice gauge theories with fermionic dynamical matter, as in the case of high energy physics theories, for example. This paper addresses precisely this problem; more specifically, we analyze how one could utilize PEPS for the study of lattice gauge theories with dynamical matter. Could one classify locally gauge invariant states using PEPS, in a way that allows, eventually, to study the theories described by the Hilbert spaces to which they belong?

For that purpose, we systematically construct, in this paper, fermionic PEPS (fPEPS)  [37] in 2+1 dimensions, which have both local gauge symmetry and fundamental physical symmetries—rotation, translation and charge conjugation. To demonstrate the strength of fPEPS for studying such theories, we consider, as a case study, a truncated compact U(1) gauge theory. Although simple, such states encode all the crucial ingredients for our demonstration: fermionic matter with bosonic gauge fields; nontrivial manifestation of the spatial symmetries, due to the fermions; and a rich, interesting phase diagram.

We shall hereby show that using PEPS, one may capture the symmetry properties of a gauge theory, which are essentially the ones which define it, as can be deduced from its name; that both global and local symmetries may be manifested by PEPS—i.e., that one may use this method to treat both matter and gauge fields; and that PEPS allow to study the phase diagram of a gauge theory, and in some cases, using standard techniques, also to derive parent Hamiltonians. We shall emphasize, on the other hand, that the goal is not to study a compact U(1) lattice gauge theory in 2+1 dimensions, but rather to show that PEPS may be used for the study of gauge theories, once the formalism we introduce is combined with efficient numerical methods. And, eventually, since PEPS allow us to find local parent Hamiltonians, most likely within the universality class of the model in question  [34], one may deduce that even if the parent Hamiltonian of a state in question, using the methods presented below, is not the one of the desired lattice gauge theory, it is highly probable that this parent Hamiltonian will be in the same universality class, which means, that the two Hamiltonians would share many features. As the states in study are exact, i.e. exact ground state of parent Hamiltonians, and one can perform numerical calculations, this could be used as a “lab” for other theoretical methods used in high energy physics.

Another possible avenue of exploring lattice gauge theories, which suggests a way of overcoming these difficulties, is quantum simulation. In recent years, many proposals have been made, for the mapping of lattice gauge theories, Abelian and non-Abelian, to atomic and optical systems (such as ultracold atoms in optical lattices, for example)  [38], [39]. These systems, called quantum simulators, may be built in the laboratory and serve as quantum computers especially tailored for the purpose of lattice gauge theory calculations. Due to experimental requirements, one mostly has to approximate the simulated model by another one, with truncated local Hilbert spaces for the gauge degrees of freedom  [35], [38], [39], [40], [41], [42], [43]. A very important issue is the evaluation of the truncated approximation, which may also be done by the use of tensor network states  [23], [28].

Section snippets

Outline

The work presented in this paper is organized as follows. First, in Section  3, we introduce a class of states for staggered fermions  [44] on a two dimensional spatial lattice, constructed as fermionic Gaussian PEPS  [37]. These states will depend on a set of three parameters we shall introduce, in a way that guarantees both the spatial symmetries of translation and rotation invariance, and a global U(1) symmetry. As PEPS, these states are the ground states of local parent Hamiltonians. By

PEPS construction of globally invariant Gaussian states

Hereby we shall describe the construction of globally invariant Gaussian states which shall fulfill several symmetries, yet to be classified. We assume no acquaintance of the reader with PEPS, and will thus describe in detail the process of constructing the state.

Local gauge symmetry

So far, we have introduced purely fermionic states |ψ(T) which, on top of the spacetime symmetries corresponding to translation and rotation invariance, are invariant under global U(1) transformations. Now we wish to lift the global symmetry to be local, i.e. instead of the global transformation rule (17), with a global transformation parameter (phase) ϕ, we would like to have a local transformation rule, with vertex-dependent parameters (phases) ϕx, ψxeisxϕxψx where sx=(1)x1+x2 accounts

Summary

In this work, we have presented a method to extend the class of symmetric PEPS to local gauge symmetries. In particular, we have focused, as a first demonstration, on a 2+1 dimensional truncated compact QED with fermionic matter, but the methods presented throughout this paper are generalizable to other gauge groups  [75], and, theoretically speaking, also to higher dimensions.

We have shown how to construct, using the Gaussian formalism, globally invariant states for staggered fermions on a

Acknowledgments

The authors would like to thank Marcello Dalmonte, Jutho Haegeman, Tao Shi, Luca Tagliacozzo and Frank Verstraete for helpful discussions. The authors acknowledge support from the EU Integrated Project SIQS. EZ acknowledges the support of the Alexander-von-Humboldt foundation, through its fellowship for postdoctoral researchers.

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