Quantum-to-Classical Correspondence and Hubbard-Stratonovich Dynamical Systems, a Lie-Algebraic Approach

TL;DR

This paper introduces a Lie-algebraic duality framework to analyze quantum and classical dynamical systems, providing new equations and dual descriptions for many-body lattice models, bridging quantum and classical dynamics.

Contribution

It develops a novel Lie-algebraic duality approach, deriving dual Schrödinger-Bloch equations and extending the method to interacting lattice models with new non-perturbative dual descriptions.

Findings

Dual Schrödinger-Bloch equations derived and analyzed.

Quantum-to-classical correspondence established via symmetry group actions.

New non-perturbative dual descriptions for Bose-Hubbard and spin models obtained.

Abstract

We propose a Lie-algebraic duality approach to analyze non-equilibrium evolution of closed dynamical systems and thermodynamics of interacting quantum lattice models (formulated in terms of Hubbard-Stratonovich dynamical systems). The first part of the paper utilizes a geometric Hilbert-space-invariant formulation of unitary time-evolution, where a quantum Hamiltonian is viewed as a trajectory in an abstract Lie algebra, while the sought-after evolution operator is a trajectory in a dynamic group, generated by the algebra via exponentiation. The evolution operator is uniquely determined by the time-dependent dual generators that satisfy a system of differential equations, dubbed here dual Schrodinger-Bloch equations, which represent a viable alternative to the conventional Schrodinger formulation. These dual Schrodinger-Bloch equations are derived and analyzed on a number of specific examples. It is shown that deterministic dynamics of a closed classical dynamical system occurs as action of a symmetry group on a classical manifold and is driven by the same dual generators as in the corresponding quantum problem. This represents quantum-to-classical correspondence. In the second part of the paper, we further extend the Lie algebraic approach to a wide class of interacting many-particle lattice models. A generalized Hubbard-Stratonovich transform is proposed and it is used to show that the thermodynamic partition function of a generic many-body quantum lattice model can be expressed in terms of traces of single-particle evolution operators governed by the dynamic Hubbard-Stratonovich fields. Finally, we derive Hubbard-Stratonovich dynamical systems for the Bose-Hubbard model and a quantum spin model and use the Lie-algebraic approach to obtain new non-perturbative dual descriptions of these theories.