Dimensional Lattice Gauge Theory Research Articles

Measurement-based quantum simulation of Abelian lattice gauge theories

Numerical simulation of lattice gauge theories is an indispensable tool in high energy physics, and their quantum simulation is expected to become a major application of quantum computers in the future. In this work, for an Abelian lattice gauge theory in dd spacetime dimensions, we define an entangled resource state (generalized cluster state) that reflects the spacetime structure of the gauge theory. We show that sequential single-qubit measurements with the bases adapted according to the former measurement outcomes induce a deterministic Hamiltonian quantum simulation of the gauge theory on the boundary. Our construction includes the (2+1)(2+1)-dimensional Abelian lattice gauge theory simulated on three-dimensional cluster state as an example, and generalizes to the simulation of Wegner’s lattice models M_{(d,n)}M(d,n) that involve higher-form Abelian gauge fields. We demonstrate that the generalized cluster state has a symmetry-protected topological order with respect to generalized global symmetries that are related to the symmetries of the simulated gauge theories on the boundary.Our procedure can be generalized to the simulation of Kitaev’s Majorana chain on a fermionic resource state. We also study the imaginary-time quantum simulation with two-qubit measurements and post-selections, and a classical-quantum correspondence, where the statistical partition function of the model M_{(d,n)}M(d,n) is written as the overlap between the product of two-qubit measurement bases and the wave function of the generalized cluster state.

Large- N phase transition in a finite lattice gauge theory

We consider gauge theories of non-Abelian $finite$ groups, and discuss the 1+1 dimensional lattice gauge theory of the permutation group $S_N$ as an illustrative example. The partition function at finite $N$ can be written explicitly in a compact form using properties of $S_N$ conjugacy classes. A natural large-$N$ limit exists with a new 't Hooft coupling, $\lambda=g^2 \log N$. We identify a Gross-Witten-Wadia-like phase transition at infinite $N$, at $\lambda=2$. It is first order. An analogue of the string tension can be computed from the Wilson loop expectation value, and it jumps from zero to a finite value. We view this as a type of large-$N$ (de-)confinement transition. Our holographic motivations for considering such theories are briefly discussed.

Lattice quantum electrodynamics in (3+1)-dimensions at finite density with tensor networks

Gauge theories are of paramount importance in our understanding of fundamental constituents of matter and their interactions. However, the complete characterization of their phase diagrams and the full understanding of non-perturbative effects are still debated, especially at finite charge density, mostly due to the sign-problem affecting Monte Carlo numerical simulations. Here, we report the Tensor Network simulation of a three dimensional lattice gauge theory in the Hamiltonian formulation including dynamical matter: Using this sign-problem-free method, we simulate the ground states of a compact Quantum Electrodynamics at zero and finite charge densities, and address fundamental questions such as the characterization of collective phases of the model, the presence of a confining phase at large gauge coupling, and the study of charge-screening effects.

Real-time simulation of (2+1)-dimensional lattice gauge theory on qubits

Abstract We study the quantum simulation of $Z_2$ lattice gauge theory in 2+1 dimensions. The dual variable formulation, the so-called Wegner duality, is utilized for reducing redundant gauge degrees of freedom. The problem of artificial charge unconservation is resolved for any charge distribution. As a demonstration, we simulate the real-time evolution of the system with two static electric charges, i.e. with two temporal Wilson lines. Some results obtained by a simulator (with no hardware noise) and a real quantum computing device (with sizable hardware noise) are shown.

Wilson Loop Expectations in Lattice Gauge Theories with Finite Gauge Groups

Wilson loop expectations at weak coupling are computed to first order, for four dimensional lattice gauge theories with finite gauge groups which satisfy some mild additional conditions. This continues recent work of Chatterjee, which considered the case of gauge group $\mathbb{Z}_2$. The main steps are (1) reducing the first order computation to a problem of Poisson approximation, and (2) using Stein's method to carry out the Poisson approximation.

Loop, string, and hadron dynamics in SU(2) Hamiltonian lattice gauge theories

The question of how to efficiently formulate Hamiltonian gauge theories is experiencing renewed interest due to advances in building quantum simulation platforms. We introduce a reformulation of an SU(2) Hamiltonian lattice gauge theory---a loop-string-hadron (LSH) formulation---that describes dynamics directly in terms of its loop, string, and hadron degrees of freedom, while alleviating several disadvantages of quantumly simulating the Kogut-Susskind formulation. This LSH formulation transcends the local loop formulation of $d+1$-dimensional lattice gauge theories by incorporating staggered quarks, furnishing the algebra of gauge-singlet operators, and being used to reconstruct dynamics between states that have Gauss's law built in to them. LSH operators are then factored into products of "normalized" ladder operators and diagonal matrices, priming them for classical or quantum information processing. Self-contained expressions of the Hamiltonian are given up to $d=3$. The LSH formalism makes little use of structures specific to SU(2) and its conceptual clarity makes it an attractive approach to apply to other non-Abelian groups like SU(3).

Entanglement entropy for pure gauge theories in 1+1 dimensions using the lattice regularization

We study the entanglement entropy (EE) for pure gauge theories in 1[Formula: see text]+[Formula: see text]1 dimensions with the lattice regularization. Using the definition of the EE for lattice gauge theories proposed in a previous paper,1 we calculate the EE for arbitrary pure as well as mixed states in terms of eigenstates of the transfer matrix in (1[Formula: see text]+[Formula: see text]1)-dimensional lattice gauge theory. We find that the EE of an arbitrary pure state does not depend on the lattice spacing, thus giving the EE in the continuum limit, and show that the EE for an arbitrary pure state is independent of the real (Minkowski) time evolution. We also explicitly demonstrate the dependence of EE on the gauge fixing at the boundaries between two subspaces, which was pointed out for general cases in the paper. In addition, we calculate the EE at zero as well as finite temperature by the replica method, and show that our result in the continuum limit corresponds to the result obtained before in the continuum theory, with a specific value of the counterterm, which is otherwise arbitrary in the continuum calculation. We confirm the gauge dependence of the EE also for the replica method.

Quantum simulations of gauge theories with ultracold atoms: Local gauge invariance from angular-momentum conservation

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 and Lorentz invariance. In this paper we discuss these special requirements, and present a new method for quantum simulation of lattice gauge theories using ultracold atoms. This method allows 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: compact QED (U(1)), SU(N) and Z_N, which can be used to build quantum simulators in 1+1 dimensions. We also present a new loop method, which uses the elementary interactions as building blocks in the effective construction of quantum simulations for d+1 dimensional lattice gauge theories (d>1), without having to use Gauss's law as a constraint, as in previous proposals. We discuss in detail the quantum simulation of 2+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.

Two dimensional Script N = (2,2) super Yang-Mills theory on the lattice via dimensional reduction

The N=(2,2) extended super Yang-Mills theory in 2 dimensions is formulated on the lattice as a dimensional reduction of a 4 dimensional lattice gauge theory. We use the plaquette action for a bosonic sector and the Wilson- or the overlap-Dirac operator for a fermion sector. The fermion determinant is real and, moreover, when the overlap-Dirac operator is used, semi-positive definite. The flat directions in the target theory become compact and present no subtlety for a numerical integration along these directions. Any exact supersymmetry does not exist in our lattice formulation; nevertheless we argue that one-loop calculable and finite mass counter terms ensure a supersymmetric continuum limit to all orders of perturbation theory.

MATRIX MODELS OF NONCOMMUTATIVE 3D LATTICE GAUGE THEORIES

We investigate through the Morita equivalence the problem of mapping, odd dimensional noncommutative lattice gauge theories onto suitable matrix models. We specialize our analysis to three-dimensional noncommutative QED (NCQED) and scalar QED (NCsQED), for which we explicitly build the corresponding matrix model.

Majorana and Majorana-Weyl fermions in lattice gauge theory

In various dimensional Euclidean lattice gauge theories, we examine a compatibility of the Majorana decomposition and the charge conjugation property of lattice Dirac operators. In $8n$ and $1+8n$ dimensions, we find a difficulty to decompose a classical lattice action of the Dirac fermion into a system of the Majorana fermion and thus to obtain a factorized form of the Dirac determinant. Similarly, in $2+8n$ dimensions, there is a difficulty to decompose a classical lattice action of the Weyl fermion into a system of the Majorana--Weyl fermion and thus to obtain a factrized form of the Weyl determinant. Prescriptions based on the overlap formalism do not remove these difficulties. We argue that these difficulties are reflections of the global gauge anomaly associated to the real Weyl fermion in $8n$ dimensions. For this reason (besides other well-known reasons), a lattice formulation of the N=1 super Yang--Mills theory in these dimensions is expected to be extremely difficult to find.

Phase structure of (2+1)-dimensional compact lattice gauge theories and the transition from Mott insulator to fractionalized insulator

Large-scale Monte Carlo simulations are employed to study phase transitions in the three-dimensional compact Abelian Higgs model in adjoint representations of the matter field, labeled by an integer q, for $q=2,3,4,5.$ We also study various limiting cases of the model, such as the ${Z}_{q}$ lattice gauge theory, dual to the three-dimensional (3D) spin model, and the 3D $\mathrm{XY}$ spin model which is dual to the ${Z}_{q}$ lattice gauge theory in the limit $\stackrel{\ensuremath{\rightarrow}}{q}\ensuremath{\infty}.$ In addition, for benchmark purposes, we study the square lattice eight-vertex model, which is exactly solvable and features nonuniversal critical exponents. We have computed the first, second, and third moments of the action to locate the phase transition of the compact Abelian Higgs model in the parameter space $(\ensuremath{\beta},\ensuremath{\kappa}),$ where $\ensuremath{\beta}$ is the coupling constant of the matter term and $\ensuremath{\kappa}$ is the coupling constant of the gauge term. We have found that for $q=3,$ the three-dimensional compact Abelian Higgs model has a phase-transition line ${\ensuremath{\beta}}_{\mathrm{c}}(\ensuremath{\kappa})$ which is first order for $\ensuremath{\kappa}$ below a finite tricritical value ${\ensuremath{\kappa}}_{\mathrm{tri}}$ and second order above. The $\ensuremath{\beta}=\ensuremath{\infty}$ first order phase transition persists for finite $\ensuremath{\beta}$ and joins the second order phase transition at a tricritical point $({\ensuremath{\beta}}_{\mathrm{tri}},{\ensuremath{\kappa}}_{\mathrm{tri}})=(1.23\ifmmode\pm\else\textpm\fi{}0.03,1.73\ifmmode\pm\else\textpm\fi{}0.03).$ For all other integer $q>~2$ we have considered, the entire phase-transition line ${\ensuremath{\beta}}_{c}(\ensuremath{\kappa})$ is critical. We have used finite-size scaling of the second and third moments of the action to extract critical exponents $\ensuremath{\alpha}$ and $\ensuremath{\nu}$ without invoking hyperscaling, for the $\mathrm{XY}$ model, the ${Z}_{2}$ spin and lattice gauge models, as well as the compact Abelian Higgs model for $q=2$ and $q=3.$ In all cases, we have found that for practical system sizes, the third moment gives scaling of superior quality compared to the second moment. We have also computed the exponent ratio for the $q=2$ compact $U(1)$ Higgs model along the critical line, finding a continuously varying ratio $(1+\ensuremath{\alpha})/\ensuremath{\nu},$ as well as continuously varying $\ensuremath{\alpha}$ and $\ensuremath{\nu}$ as $\ensuremath{\kappa}$ is increased from $0.76$ to $\ensuremath{\infty},$ with the Ising universality class $(1+\ensuremath{\alpha})/\ensuremath{\nu}=1.763$ as a limiting case for $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\beta}}\ensuremath{\infty},\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\kappa}}0.761,$ and the $\mathrm{XY}$ universality class $(1+\ensuremath{\alpha})/\ensuremath{\nu}=1.467$ as a limiting case for $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\beta}}0.454,\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\kappa}}\ensuremath{\infty}.$ However, the critical line exhibits a remarkable resilience of ${Z}_{2}$ criticality as $\ensuremath{\beta}$ is reduced along the critical line. Thus, the three-dimensional compact Abelian Higgs model for $q=2$ appears to represent a fixed-line theory defining a new universality class. We relate these results to a recent microscopic description of zero-temperature quantum phase transitions within insulating phases of strongly correlated systems in two spatial dimensions, proposing the above to be the universality class of the zero-temperature quantum phase transition from a Mott-Hubbard insulator to a charge-fractionalized insulator in two spatial dimensions, which thus is that of the 3D Ising model for a considerable range of parameters.

Spin resolution of glueballs in(2+1)-dimensional lattice gauge theory

Conventional lattice gauge theory assigns the lowest spin compatible with the symmetry channel of a given operator to the state coupling to that operator. Operators on a cubic lattice, however, are only defined on angles of pi/2, hence states with spin equal modulo 4 may overlap significantly. This paper explores a new technique for generating lattice operators that may be placed onto the lattice at angles other than pi/2, thereby resolving this modulo 4 ambiguity. Calculations of the mass of states with spin equal t o 0, 2, and 4 are performed in the positive parity and charge conjugation channe l and compared to the spectrum from previous lattice calculations. These masses compare well for spin 0 and 2, and for spin 4 the mass agrees with a state conv entionally assigned spin 0, raising the possibility of mis-identification of the spin of states coupling to some traditional operators.

Exact local fermionic zero modes

We introduce a simple method to find localized exact fermionic zero modes for any local fermionic action. The zero modes are attached to specific local gauge configurations. Examples are provided for staggered and Wilson fermion actions in 2–6 dimensions, at finite and infinite lattice volumes, and for abelian and non-abelian gauge groups. One of our concrete results is that a finite density of almost zero modes must occur in quenched four dimensional lattice gauge theory simulations that use traditional methods. This density is exponentially suppressed in the gauge coupling constant.

Some Analytical Results in (2+1) dimensional L.G.T

We show that, within the framework of suitably chosen approximate effective actions it is possible to evaluate analytically the string tension, the spacelike string tension and the deconfinement temperature of (2+1) dimensional lattice gauge theories. In the case of SU(2) gauge theory our results agree with those obtained through Montecarlo simulations.

Two dimensional lattice gauge theory based on a quantum group

In this article we analyse a two dimensional lattice gauge theory based on a quantum group. The algebra generated by gauge fields is the lattice algebra introduced recently by A.Yu. Alekseev, H. Grosse and V. Schomerus in [1]. We define and study Wilson loops. This theory is quasi-topological as in the classical case, which allows us to compute the correlation functions of this theory on an arbitrary surface.

Critical-index calculation from Monte Carlo data with use of finite-size scaling and optimization.

A method of critical index calculation is proposed. The method is based on matching the temperature values of two different size lattices by finding the optimized solution of the equation t'=${\mathit{b}}_{\mathit{T}}^{\mathit{Y}}$t, where t and t' are the reduced temperatures of the lattices, b is the ratio of lattice sizes, and ${\mathit{Y}}_{\mathit{T}}$ is the thermal critical index. The values of t and t' are obtained from the fourth cumulant of the order parameter, which satisfies the relation ${\mathit{Q}}_{\mathit{r}}$(t,L)=${\mathit{Q}}_{\mathit{r}}$(t'L') around the critical point. By using this method, the critical indices of the two-dimensional Ising model and the (2+1)-dimensional lattice gauge theory are calculated. The advantage of the method is that one can obtain the critical point and the critical indices with high accuracy by performing a relatively small number of Monte Carlo iterations.

Vacuum structure and chiral-symmetry breaking in (2+1)-dimensional lattice gauge theories with fermions.

We study further the vacuum structure and chiral-symmetry breaking in Hamiltonian lattice gauge theory with fermions. Attention is paid to (2+1)-dimensional systems. For the fermion sector, a unitary transformation and the variational method are employed. The antiferromagnetic nature of the unitarily transformed Hamiltonian and mesons as spin waves are presented at strong coupling, and the relevance for high-${T}_{c}$ superconductivity is discussed. The vacuum state of the full theory is assumed to be the combination of the variational fermion vacuum state and the exact ground state of the modified Hamiltonian for pure gauge theory of Guo and co-workers. The existing problems in the modified theory are pointed out as well. The chiral condensates in ${\mathrm{QED}}_{3}$ and ${\mathrm{QCD}}_{3}$ are calculated and the scaling behavior is observed in the crossover regime.

Chiral-Symmetry Breaking in (1+1)-Dimensional Lattice Gauge Theories11The project supported in part by the U.S. National Science Foundation under Grant No. PHY89-04035

The (1+1)-dimensional Hamiltonian lattice gauge theories and the vacuum structure with fermions are studied analytically by using a unitary transformation and the variational method. The matching of the operators in the continuum and on the lattice is discussed. The chiral symmetry is shown to be broken spontaneously. The effect of species doubling is also analyzed. Satisfactory scaling behavior for the chiral order parameter and coincidence with the continuum predictions are obtained.

Variational study of vacuum structure with fermions ind+1 dimensional Hamiltonian lattice gauge theory

The physical vacuum state and general expression for the Hamiltonian ofd+1 dimensional lattice gauge theory are given by incorporating the exact ground state of pure gauge theory and the variational fermion vacuum state. The applications toSU(2) andSU(3) gauge theories in 2+1 and 3+1 dimensions are demonstrated and the fermion condensates\(\left\langle {\bar \psi \psi } \right\rangle \) as functions of 1/g2 are calculated.