Abstract
Monte Carlo simulation of gauge theories with a Ξ term is known to be extremely difficult due to the sign problem. Recently there has been major progress in solving this problem based on the idea of complexifying dynamical variables. Here we consider the complex Langevin method (CLM), which is a promising approach for its low computational cost. The drawback of this method, however, is the existence of a condition that has to be met in order for the results to be correct. As a first step, we apply the method to 2D U(1) gauge theory on a torus with a Ξ term, which can be solved analytically. We find that a naive implementation of the method fails because of the topological nature of the Ξ term. In order to circumvent this problem, we simulate the same theory on a punctured torus, which is equivalent to the original model in the infinite volume limit for |Ξ| < Ï. Rather surprisingly, we find that the CLM works and reproduces the exact results for a punctured torus even at large Ξ, where the link variables near the puncture become very far from being unitary.
Highlights
Which takes integer values on a compact space
We focus on the complex Langevin method (CLM), which can be applied to various physically interesting models with large system size in a straightforward manner. (See, for instance, refs. [35,36,37].) The only drawback of the method is that it can give wrong results depending on the system, the parameter region, and even on the choice of the dynamical variables
We show our results for the average plaquette w (Top), the topological charge (Middle) and the topological susceptibility (Bottom) against Ξ for (ÎČ, L) = (3, 10) and (12, 20) in the left and right columns, respectively, which correspond to a fixed physical volume Vphys ⥠L2/ÎČ = 102/3
Summary
A2(Fn,ΌΜ ) , n in the continuum limit up to an irrelevant constant with the identification ÎČ. Which gives an integer value even at finite a This can be proved by noting that n Pn = 1 since each link variable appears twice in this product with opposite directions. That the topological charge defined on the lattice in this way can take non-integer values in general before taking the continuum limit. We call this definition (2.12) the âsine definitionâ. Where Sg is given by (2.8) and SΞ is given by (2.3) with Q defined either by (2.11) or by (2.12) Since this theory is superrenormalizable, we can take the continuum limit a â 0 with fixed g, which is set to unity throughout this paper without loss of generality. We discuss how to apply the CLM to 2D U(1) gauge theory with a Ξ term and show some results, which suggest that a naive implementation of the method fails
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