We derive an exact representation of the massless Schwinger model on the lattice in terms of dual variables which are configurations of loops, dimers and plaquette occupation numbers. When expressed with the dual variables the partition sum has only real and positive terms also when a chemical potential or a topological term are added -situations where the conventional representation has a complex action problem. The dual representation allows for Monte Carlo simulations without restrictions on the values of the chemical potential or the vacuum angle. arXiv:1502.05479v2 [hep-lat]
The standard perturbative weak-coupling expansions in lattice models are asymptotic. The reason for this is hidden in the incorrect interchange of the summation and integration. However, substituting the Gaussian initial approximation of the perturbative expansions by a certain interacting model or regularizing original lattice integrals, one can construct desired convergent series. In this paper we develop methods, which are based on the joint and separate utilization of the regularization and new initial approximation. We prove, that the convergent series exist and can be expressed as the re-summed standard perturbation theory for any model on the finite lattice with the polynomial interaction of even degree. We discuss properties of such series and make them applicable to practical computations. The workability of the methods is demonstrated on the example of the lattice $\phi^4$-model. We calculate the operator $\langle\phi_n^2\rangle$ using the convergent series, the comparison of the results with the Borel re-summation and Monte Carlo simulations shows a good agreement between all these methods.Comment: 25 pages, 14 figure
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...
The standard lattice perturbation theory leads to the asymptotic series because of the incorrect interchange of the summation and integration. However, changing the initial approximation of the perturbation theory, one can generate the convergent series. We study the lattice φ 4 -model and compare the operator φ 2 n calculated using the convergent series and obtained by Monte Carlo simulations.
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