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arXiv:1805.05347 · Physical review. B./Physical review. B

Eliminating fermionic matter fields in lattice gauge theories

Erez Zohar, J. I. Cirac

Published Aug 9, 2018• 74 citations
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Abstract

We devise a unitary transformation that replaces the fermionic degrees of freedom of lattice gauge theories by (hard-core) bosonic ones. The resulting theory is local and gauge invariant, with the same symmetry group. The method works in any spatial dimension and can be directly applied, among others, to the gauge groups $G=U(N)$ and $SU(2N)$, where $N\ensuremath{\in}\mathbb{N}$. For $SU(2N+1)$, one can also carry out the transformation after introducing an extra idle ${\mathbb{Z}}_{2}$ gauge field, so that the resulting symmetry group trivially contains ${\mathbb{Z}}_{2}$ as a normal subgroup. Those results have implications in the field of quantum simulations of high-energy physics models.

DOI
10.1103/physrevb.98.075119

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We devise a unitary transformation that replaces the fermionic degrees of freedom of lattice gauge theories by (hard-core) bosonic ones. The resulting theory is local and gauge invariant, with the same symmetry group. The method works in any spatial dimension and can be directly applied, among others, to the gauge groups G=U(N)G=U(N)G=U(N) and SU(2N)SU(2N)SU(2N), where N\ensuremath∈NN\ensuremath{\in}\mathbb{N}N\ensuremath∈N. For SU(2N+1)SU(2N+1)SU(2N+1), one can also carry out the transformation after introducing an extra idle Z2{\mathbb{Z}}_{2}Z2​ gauge field, so that the resulting symmetry group trivially contains Z2{\mathbb{Z}}_{2}Z2​ as a normal subgroup. Those results have implications in the field of quantum simulations of high-energy physics models.
Quantum simulation of lattice gauge theories has been proposed and used as a method to overcome theoretical difficulties in dealing with the nonperturbative nature of such models. In this work we focus on two important bottlenecks that make developing such simulators hard: one is the difficulty of simulating fermionic degrees of freedom, and the other is the redundancy of the Hilbert space, which leads to a waste of experimental resources and the need to impose and monitor the local symmetry constraints of gauge theories. This has previously been tackled in one-dimensional settings, using nonlocal methods. Here we show an alternative procedure for dealing with these problems, which removes the matter and the Hilbert space redundancy, and is valid for higher space dimensions. We demonstrate it for a Z2{\mathbb{Z}}_{2}Z2​ lattice gauge theory and implement it experimentally via the IBMQ cloud quantum computing platform.
Quantum simulations of high-energy physics, and especially of gauge theories, is an emerging and exciting direction in quantum simulations. However, simulations of such theories, compared to simulations of condensed matter physics, must satisfy extra restrictions, such as local gauge invariance and relativistic structure. In this paper we discuss these special requirements, and present a method for quantum simulation of lattice gauge theories using ultracold atoms. This method allows us to include local gauge invariance as a fundamental symmetry of the atomic Hamiltonian, arising from natural atomic interactions and conservation laws (and not as a property of a low-energy sector). This allows us to implement elementary gauge invariant interactions for three lattice gauge theories: U(1) (compact QED), ZN{\mathbb{Z}}_{N}ZN​ and SU(N)(N)(N) (Yang-Mills), which can be used to build quantum simulators in 1+11+11+1 dimensions. We also present a loop method, which uses the elementary interactions as building blocks in the effective construction of quantum simulations for d+1d+1d+1 dimensional lattice gauge theories (d>1d>1d>1), but unlike in previous proposals, here gauge invariance and Gauss's law are natural symmetries, which do not have to be imposed as a constraint. We discuss in detail the quantum simulation of 2+12+12+1 dimensional compact QED and provide a numerical proof of principle. The simplicity of the already gauge-invariant elementary interactions of this model suggests it may be useful for future experimental realizations.
We examine the Kogut-Susskind formulation of lattice gauge theories under the light of fermionic and bosonic degrees of freedom that provide a description useful to the development of quantum simulators of gauge-invariant models. We consider both discrete and continuous gauge groups and adopt a realistic multicomponent Fock space for the definition of matter degrees of freedom. In particular, we express the Hamiltonian of the gauge theory and the Gauss law in terms of Fock operators. The gauge fields are described in two different bases based on either group elements or group representations. This formulation allows for a natural scheme to achieve a consistent truncation of the Hilbert space for continuous groups, and provides helpful tools to study the connections of gauge theories with topological quantum double and string-net models for discrete groups. Several examples, including the case of the discrete D3{D}_{3}D3​ gauge group, are presented.
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We describe an n-dimensional (n≥2) analog of the Jordan-Wigner transformation, which maps an arbitrary fermionic system to Pauli matrices while preserving the locality of the Hamiltonian. When the space is simply-connected, this bosonization gives a duality between any fermionic system in arbitrary n spatial dimensions and a new class of (n-1)-form Z₂ gauge theories in n dimensions with a modified Gauss’s law. We describe several examples of 2d bosonization, including free fermions on square and honeycomb lattices and the Hubbard model, and 3d bosonization, including a solvable Z₂ lattice gauge theory with Dirac cones in the spectrum. This bosonization formalism has an explicit dependence on the second Stiefel-Whitney class and a choice of spin structure on the manifold, a key feature for defining fermions. A new formula for Stiefel-Whitney homology classes on lattices is derived. We also derive the Euclidean actions for the corresponding lattice gauge theories from the bosonization. The topological actions contain Chern-Simons terms for (2+1)D or Steenrod Square terms for general dimensions. Finally, we apply the bosonization to construct various bosonic or fermionic symmetry-protectedtopological (SPT) phases. It has been shown that supercohomology fermionic SPT phases are dual to bosonic higher-group SPT phases.
We propose a Hamiltonian framework for constructing chiral gauge theories on the lattice based on symmetry disentanglers: constant-depth circuits of local unitaries that transform not-on-site symmetries into on-site ones. When chiral symmetry can be realized not-on-site and such a disentangler exists, the symmetry can be implemented in a strictly local Hamiltonian and gauged by standard lattice methods. Using lattice rotor models, we realize this idea in 1+1 and 3+1 spacetime dimensions for U(1)U(1)U(1) symmetries with mixed 't Hooft anomalies, and show that symmetry disentanglers can be constructed when anomalies cancel. As an example, we present an exactly solvable Hamiltonian lattice model of the (1+1)-dimensional "3450" chiral gauge theory, and we argue that a related construction applies to the U(1)U(1)U(1) hypercharge symmetry of the Standard Model fermions in 3+1 dimensions. Our results open a new route toward fully local, nonperturbative formulations of chiral gauge theories.