Non-Perturbative Simulations of Quantum Field Theories using Complex Langevin Dynamics

TL;DR

This paper explores the use of complex Langevin dynamics to perform non-perturbative simulations of quantum field theories with complex actions, addressing the sign problem in lattice formulations of such theories.

Contribution

It demonstrates the effectiveness of the complex Langevin method in simulating non-perturbative phenomena in supersymmetric and string-theoretic models with complex actions.

Findings

Successful simulation of supersymmetry breaking scenarios.

Investigation of PT-symmetric potentials.

Evidence of SO(10) symmetry breaking in matrix models.

Abstract

Non-perturbative formulations of field theories are essential to capture intriguing physical phenomena, including confinement in QCD, spontaneous supersymmetry breaking, and dynamical compactification in superstrings. Lattice regularization provides a robust framework to study these non-perturbative features through Euclidean path integrals. Conventionally, path integrals are numerically evaluated using Monte Carlo methods, where the Boltzmann factor is interpreted as a probability weight. However, complex actions in various physical systems render the Boltzmann factor complex, leading to the sign problem. The complex Langevin method overcomes the sign problem and can be used to evaluate complex integrals. This thesis employs the complex Langevin method to investigate various non-perturbative aspects of field-theoretic systems with complex actions. We probe the possibility of spontaneous supersymmetry breaking in the simplest realizations of supersymmetric field theories. These systems generally have complex actions arising from a complex determinant of the fermion operator. We studied various interesting classes of complex potentials, including those exhibiting PT-symmetry. Another exciting aspect explored is the dynamical compactification of extra dimensions in superstring theory. The IKKT matrix model, in the large-N limit, is a conjectured formulation for the 10D type IIB string theory. We employ the complex Langevin method to investigate the Euclidean version of this matrix model, which has an inherent complex Pfaffian, to probe the spontaneous breaking of SO(10) symmetry. The investigations performed in this thesis suggest that the complex Langevin method can successfully simulate non-perturbative aspects of quantum field theories by taming the associated sign problem.