Classical Dynamics Generated by Long-Range Interactions for Lattice Fermions and Quantum Spins

TL;DR

This paper develops a mathematical framework to analyze the classical macroscopic dynamics of lattice fermion and quantum-spin systems with long-range interactions, revealing complex behavior beyond previous models.

Contribution

It introduces a new framework for long-range macroscopic dynamics that combines classical and quantum aspects, focusing on the classical part in this work.

Findings

Classical dynamics derived from self-consistency equations

Dynamics governed by Liouville's equation

Framework applicable to fermion and quantum-spin systems

Abstract

We study the macroscopic dynamical properties of fermion and quantum-spin systems with long-range, or mean-field, interactions. The results obtained are far beyond previous ones and require the development of a mathematical framework to accommodate the macroscopic long-range dynamics, which corresponds to an intricate combination of classical and short-range quantum dynamics. In this paper we focus on the classical part of the long-range, or mean-field, macroscopic dynamics, but we already introduce the full framework. The quantum part of the macroscopic dynamics is studied in a subsequent paper. We show that the classical part of the macroscopic dynamics results from self-consistency equations within the (quantum) state space. As is usual, the classical dynamics is driven by Liouville's equation.