The XY model at finite chemical potential using complex Langevin dynamics

TL;DR

This paper investigates the effectiveness of complex Langevin dynamics in simulating the 3D XY model at finite chemical potential, introducing an adaptive stepsize method and diagnostic tests to assess convergence and identify limitations.

Contribution

The study implements an adaptive stepsize algorithm for complex Langevin dynamics and develops diagnostics to evaluate its correctness in the XY model.

Findings

Complex Langevin works well at larger beta values.

Fails in the disordered phase at smaller beta.

Failure is not solely due to the sign problem.

Abstract

The three-dimensional XY model is studied at finite chemical potential using complex Langevin dynamics. An adaptive stepsize algorithm is implemented to cure the problem of runaway solutions that appears when using a constant stepsize. The validity of complex Langevin dynamics is tested against calculations using imaginary chemical potential and the world line formalism. While complex Langevin dynamics is found to work correctly at larger beta, it fails for smaller beta, in the region of the phase diagram corresponding to the disordered phase. Diagnostic tests are developed to identify behaviour symptomatic of incorrect convergence. These indicate that the failure does not depend on the severeness of the sign problem, but has a different origin.