On the convergence of complex Langevin dynamics: the three-dimensional XY model at finite chemical potential

TL;DR

This paper investigates the effectiveness of complex Langevin dynamics in simulating the three-dimensional XY model at finite chemical potential, identifying conditions under which it converges correctly and diagnosing failures.

Contribution

The study provides diagnostic tools for detecting incorrect convergence in complex Langevin simulations and clarifies the method's limitations in the disordered phase.

Findings

Complex Langevin works at larger beta values.

Fails at smaller beta in the disordered phase.

Diagnostic tests can identify incorrect convergence.

Abstract

The three-dimensional XY model is studied at finite chemical potential using complex Langevin dynamics. The validity of the approach is probed at small chemical potential using imaginary chemical potential and continuity arguments, and at larger chemical potential by comparison with the world line method. While complex Langevin works for larger beta, we find that it fails for smaller beta, in the region of the phase diagram corresponding to the disordered phase. Diagnostic tests are developed to identify symptoms correlated with incorrect convergence. We argue that the erroneous behaviour at smaller beta is not due to the sign problem, but rather resembles dynamics observed in complex Langevin simulations of simple models with complex noise.