TAMING THE SIGN PROBLEM IN LATTICE FIELD THEORY WITH DEFORMED PATH INTEGRAL CONTOURS

Abstract

In this thesis a generic method for taming the sign problem is developed. The sign problem is the name given to the difficult task of numerically integrating a highly oscillatory integral, and the sign problem inhibits our ability to understand the properties of a wide range of systems of interest in theoretical physics. Particularly notably for nuclear physics, the sign problem prevents the calculation of the properties of QCD at finite baryon density, thereby precluding an under- standing of the dense nuclear matter found in the center of a neutron star. The central idea developed in this thesis is to use the multidimensional generalization

of Cauchy’s Integral Theorem to deform the Feynman Path Integral of lattice fields theories into complexified field space to manifolds upon which the phase oscillations which cause the sign problem are gentle. Doing so allows calculations of theories with sign problems. Two practical manifold deformation methods, the holomorphic gradient flow and the sign-optimized manifold method, are developed. The holomorphic gradient flow, a generalization of the Lefschetz thimble method, continuously deforms the original path integration domain to a complex manifold via an evolution dictated by a complex first order differential equation. The sign-optimized manifold method is a way to generate a manifold with gentle phase oscillations by minimizing the sign problem in a parameterized family of manifolds through stochastic gradient ascent. With an eye toward QCD at finite density, the Cauchy’s Theorem approach is applied to relativistic quantum field theories of fermions at finite density throughout this thesis. Finally, these methods are general and can be applied to both bosonic and fermionic theories, as well as Minkowski path integrals describing real-time dynamics.