[Paper Review] Complex Langevin analysis of 2D U(1) gauge theory on a torus with a $θ$ term

Mitsuaki Hirasawa, Akira Matsumoto+2 more|arXiv (Cornell University)|Apr 29, 2020

Theoretical and Computational Physics63 references4 citations

TL;DR

This paper applies the complex Langevin method (CLM) to 2D U(1) lattice gauge theory with a θ term on a punctured torus to circumvent the sign problem. Despite the topological nature of the θ term causing failure in standard CLM implementations, the method successfully reproduces exact results—even at large θ—by removing a single plaquette, which modifies the drift terms near the puncture and avoids topology freezing.

ABSTRACT

Monte Carlo simulation of gauge theories with a $θ$ 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 $θ$ term, which can be solved analytically. We find that a naive implementation of the method fails because of the topological nature of the $θ$ 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 $ |θ| < π$. Rather surprisingly, we find that the CLM works and reproduces the exact results for a punctured torus even at large $θ$, where the link variables near the puncture become very far from being unitary.

Motivation & Objective

• To investigate the applicability of the complex Langevin method (CLM) to 2D U(1) gauge theory with a θ term, a model known to suffer from the severe sign problem.
• To test whether CLM can correctly handle the nonperturbative, topological nature of the θ term, which classically does not affect the equations of motion.
• To explore whether modifying the topology via a puncture—removing one plaquette—can stabilize CLM and restore correct convergence.
• To compare the performance of CLM using two definitions of the topological charge: the logarithmic and sine-based definitions.
• To determine whether the CLM results on the punctured model converge to the exact analytical results in the continuum limit (large β, fixed physical volume).

Proposed method

• Apply the complex Langevin method (CLM) to 2D U(1) lattice gauge theory on a torus with a θ term, using complexified link variables to sample the complex action.
• Introduce a puncture by removing a single plaquette from the lattice, transforming the topology from a torus to a punctured torus, which alters the topological charge structure.
• Use two definitions of the topological charge: the logarithmic definition (based on log of plaquette variables) and the sine-based definition (using sin of plaquette angles), to test robustness.
• Compute drift terms in the complex Langevin equation that include contributions from both the gauge action (β term) and the θ term, with modified expressions near the puncture.
• Perform simulations at various β and L values, comparing CLM results with exact analytical results for the punctured model.
• Analyze the magnitude of the drift term and the distribution of the real part of the topological charge to assess the validity and convergence of CLM.

Experimental results

Research questions

• RQ1 Can the complex Langevin method (CLM) correctly reproduce exact results for 2D U(1) gauge theory with a θ term on a torus, despite the sign problem?
• RQ2 Why does a naive CLM implementation fail for the original torus model, and what topological or dynamical mechanism causes this failure?
• RQ3 Does removing a single plaquette (creating a punctured torus) restore the validity of CLM, and if so, why?
• RQ4 How do different definitions of the topological charge (log vs. sine) affect the convergence and accuracy of CLM in the presence of a θ term?
• RQ5 To what extent does the CLM performance depend on the lattice parameters β and L, and does it converge to the exact continuum limit?

Key findings

• The naive CLM implementation fails on the original torus due to topology freezing, where the topological charge becomes trapped in a single sector, invalidating the method.
• On the punctured torus, the CLM successfully reproduces the exact analytical results for the topological susceptibility and other observables, even at large θ = π.
• The drift term magnitude u in the CLM shows rapid fall-off for large β, indicating that the condition for correct convergence is satisfied, especially with the sine-based topological charge definition.
• For (β, L) = (12, 20), the real part of the topological charge density from CLM is widely distributed across −3 ≲ Re Q ≲ 3, confirming the absence of topology freezing.
• The results using the sine-based topological charge definition agree with those from the logarithmic definition in the large β limit, confirming consistency in the continuum limit.
• Even at small β = 3 with L = 10, the CLM on the punctured model yields results in complete agreement with exact values, indicating robustness across parameter regimes.