Periodic Mean-Field Solutions and the Spectra of Discrete Bosonic Fields: Trace Formula for Bose-Hubbard Models

TL;DR

This paper develops a semiclassical trace formula for the many-body spectra of bosonic lattice fields, linking quantum spectra to classical mean-field solutions and capturing quantum interference effects.

Contribution

It introduces a novel semiclassical approximation for many-body bosonic spectra that accounts for quantum effects without complexification of classical limits.

Findings

Derives a trace formula connecting quantum spectra to classical periodic solutions.

Validates the approach for both interacting and non-interacting bosonic systems.

Captures quantum interference effects beyond the Ehrenfest time.

Abstract

We consider the many-body spectra of interacting bosonic quantum fields on a lattice in the semiclassical limit of large particle number $N$. We show that the many-body density of states can be expressed as a coherent sum over oscillating long-wavelength contributions given by periodic, non-perturbative solutions of the, typically non-linear, wave equation of the classical (mean-field) limit. To this end we construct the semiclassical approximation for both the smooth and oscillatory part of the many-body density of states in terms of a trace formula starting from the exact path integral form of the propagator between many-body quadrature states. We therefore avoid the use of a complexified classical limit characteristic of the coherent state representation. While quantum effects like vacuum fluctuations and gauge invariance are exactly accounted for, our semiclassical approach captures quantum interference and therefore is valid well beyond the Ehrenfest time where naive quantum-classical correspondence breaks down. Remarkably, due to a special feature of harmonic systems with incommesurable frequencies, our formulas are generically valid also in the free-field case of non-interacting bosons.