The complex Langevin analysis of spontaneous symmetry breaking induced by complex fermion determinant

TL;DR

This paper introduces a deformation method within the complex Langevin approach to study spontaneous symmetry breaking caused by complex fermion determinants, successfully overcoming singular-drift issues in matrix models.

Contribution

The authors propose a deformation technique to avoid singular drifts in complex Langevin simulations, enabling accurate analysis of symmetry breaking in systems with complex fermion determinants.

Findings

Successfully applied to a matrix model showing SO(4) to SO(2) symmetry breaking.

Order parameters agree with Gaussian expansion predictions.

Demonstrated effectiveness over reweighting methods.

Abstract

In many interesting physical systems, the determinant which appears from integrating out fermions becomes complex, and its phase plays a crucial role in the determination of the vacuum. An example of this is QCD at low temperature and high density, where various exotic fermion condensates are conjectured to form. Another example is the Euclidean version of the type IIB matrix model for 10d superstring theory, where spontaneous breaking of the SO(10) rotational symmetry down to SO(4) is expected to occur. When one applies the complex Langevin method to these systems, one encounters the singular-drift problem associated with the appearance of nearly zero eigenvalues of the Dirac operator. Here we propose to avoid this problem by deforming the action with a fermion bilinear term. The results for the original system are obtained by extrapolations with respect to the deformation parameter. We demonstrate the power of this approach by applying it to a simple matrix model, in which spontaneous symmetry breaking from SO(4) to SO(2) is expected to occur due to the phase of the complex fermion determinant. Unlike previous work based on a reweighting-type method, we are able to determine the true vacuum by calculating the order parameters, which agree with the prediction by the Gaussian expansion method.