Dynamical symmetry approach and topological field theory for path integrals of quantum spin systems

TL;DR

This paper introduces a novel dynamical symmetry approach to path integrals in quantum spin systems, linking stochastic differential equations and topological field theory to achieve exact solutions, including for time-dependent cases.

Contribution

It presents the first exact path integral solution for a many-body quantum system using a dynamical symmetry and topological field theory framework.

Findings

Derivation of stochastic differential equations on the group manifold.

Formulation of the supersymmetric effective action as a topological field theory.

Exact solutions for specific quantum many-body systems, including time-dependent parameters.

Abstract

We develop a dynamical symmetry approach to path integrals for general interacting quantum spin systems. The time-ordered exponential obtained after the Hubbard-Stratonovich transformation can be disentangled into the product of a finite number of the usual exponentials. This procedure leads to a set of stochastic differential equations on the group manifold, which can be further formulated in terms of the supersymmetric effective action. This action has the form of the Witten topological field theory in the continuum limit. As a consequence, we show how it can be used to obtain the exact results for a specific quantum many-body system which can be otherwise solved only by the Bethe ansatz. To our knowledge this represents the first example of a many-body system treated exactly using the path integral formulation. Moreover, our method can deal with time-dependent parameters, which we demonstrate explicitly.