Dual representation for1+1dimensional fermions interacting with3+1dimensional U(1) gauge fields

Abstract

We study a system of nanowires, i.e., the theory of 1+1 dimensional massless fermions interacting with 3+1 dimensional U(1) gauge fields. When allowing for non-zero chemical potentials, this system has a complex action problem in the conventional formulation. We show that the partition sum can be mapped to a dual representation where the fermions correspond to dimers and oriented loops on 2-dimensional planes embedded in 4 dimensions. The dual degrees of freedom for the gauge fields are surfaces that either are closed or bounded by the fermion loops. In terms of the dual variables the complex action problem is overcome and Monte Carlo simulations are possible for arbitrary chemical potentials. I. INTRODUCTORY REMARKSMonte Carlo simulations of lattice field theories belong to the most powerful tools for the nonperturbative analysis of quantum field theories. However, in the conventional formulation of a lattice field theory a Monte Carlo simulation is not always possible because the action S can be complex and the Boltzmann factor e −S does not have a probabilistic interpretation. This so-called complex action problem (or sign problem) is typical for quantum field theories at finite densities. For overcoming the complex action problem a variety of approaches was considered, such as re-weighting, complex Langevin dynamics or various series expansions. In some cases even a complete solution was achieved by exactly rewriting the partition sum in terms of new degrees of freedom, so called dual variables (see, e.g., [1-6] for reviews).The dual variables are loops for matter fields and surfaces for the gauge field degrees of freedom. The surfaces are either closed or bounded by matter flux. For fermions an additional problem appears: The loops pick up signs coming from the Grassmann nature of the variables as well as from the Clifford algebra. This additional source of problems is the reason that there are only very few results of a real and positive dualization of gauge theories with fermions.In the current paper we generalize the successful dualization [7,8] of massless fermions interacting with U(1) gauge fields in two dimensions to the case of QED with 3+1 dimensional U(1) gauge fields and a set of 1+1 dimensional fermions which may be interpreted as nano-wires. The fermions live on two-dimensional coordinate planes embedded in 3+1 dimensions. At non-zero chemical potential the model exhibits a complex action problem in the conventional representation which we solve by mapping the system to a dual representation. Our dual representation for the set of nanowires also may be viewed as a partial dualization of full QED in four dimensions, since the fermion loops we consider here are a subset of the loops appearing in full QED.The model considered here is not only interesting as a further step in the program of finding dual representations for lattice field theories, but is directly related to the physics of graphene [9]. More specifically, our model of QED 4 with fermion sections corresponds to an array of graphene nan...

Approaches to the sign problem in lattice field theory

“…49 This result can be generalized to a real and positive dual model for a system of 1-dimensional nanowires interacting with the 4-dimensional electromagnetic field. 74 For full QED in four dimensions a positive dualization was presented for a subset of the fermion loops, so called quasi-planar loops, in. 75 It is interesting to note that for the massive Schwinger model (except at strong coupling) the signs come back.…”

Applications of lattice QCD techniques for condensed matter systems

“…A simple but quite realistic lattice model of Weyl semimetals can be obtained from the model Hamiltonian (12) by tuning the Dirac mass m 0 to the critical value (e.g. m 0 = 0) separating the topological insulator and the trivial insulator phases and introducing time-reversal and/or parity-breaking perturbations of the form [86][87][88] δh (k) = b i Σ i + µ A γ 5 (13) where Σ i = −iǫ ijk α j α k /2 is the spin operator and γ 5 = −βα 1 α 2 α 3 is the generator of chiral rotations.…”

“…In this paper, we will review the most important aspects, results and limitations of the Hybrid Monte-Carlo simulations of graphene, and consider the possible applications of HMC to other Dirac materials. In Section 2 we start with a brief a see however 12 for some recent attempts introduction into the HMC algorithm which can be used for simulations of condensed matter systems. We continue with a brief review of the HMC simulations of graphene along with the physical problems which motivated them, from the first attempts with staggered fermions to the most recent simulations with screened Coulomb potential.…”

“…a Dirac mass term) can open the gap. The vanishing of the total flux in the compact Brillouin zone85 implies an equal number of left-and right-handed Weyl nodes.A simple but quite realistic lattice model of Weyl semimetals can be obtained from the model Hamiltonian(12) by tuning the Dirac mass m 0 to the critical value (e.g. m 0 = 0) separating the topological insulator and the trivial insulator phases and introducing time-reversal and/or parity-breaking perturbations of the form[86][87][88] δh (k) = b i Σ i + µ A γ 5…”

Quantum error mitigation for parametric circuits

“…The ultimate applications of the GREC method should include such important problems as quantum computations in lattice QCD and Hubbard model at finite chemical potentials, where the non-physical regions with purely imaginary chemical potentials are accessible by the classical Monte Carlo simulations. For the first steps towards these ambitious goals, one may investigate the application of the GREC algorithm to the lattice φ 4 or Schwinger models, which have similar properties regarding the sign problem within the conventional Monte Carlo methods and also have sign problem-free formulations in terms of dual variables and on quantum computers [29][30][31][32][33][34].…”