Adiabatic theorem in the thermodynamic limit: Systems with a uniform gap

TL;DR

This paper extends adiabatic theorems to infinite interacting many-body systems with a uniform spectral gap, providing a rigorous foundation for response theory and adiabatic approximations in the thermodynamic limit.

Contribution

It generalizes super-adiabatic theorems to infinite systems with a uniform gap, including certain gap-closing perturbations, and introduces a resummation technique for the adiabatic expansion.

Findings

Validates adiabatic approximation in the thermodynamic limit

Establishes response theory for infinite gapped systems

Allows for non-local observables in adiabatic analysis

Abstract

We show that recent results on adiabatic theory for interacting gapped many-body systems on finite lattices remain valid in the thermodynamic limit. More precisely, we prove a generalised super-adiabatic theorem for the automorphism group describing the infinite volume dynamics on the quasi-local algebra of observables. The key assumption is the existence of a sequence of gapped finite volume Hamiltonians which generates the same infinite volume dynamics in the thermodynamic limit. Our adiabatic theorem holds also for certain perturbations of gapped ground states that close the spectral gap (so it is an adiabatic theorem also for resonances and in this sense `generalised'), and it provides an adiabatic approximation to all orders in the adiabatic parameter (a property often called `super-adiabatic'). In addition to existing results for finite lattices, we also perform a resummation of the adiabatic expansion and allow for observables that are not strictly local. Finally, as an application, we prove the validity of linear and higher order response theory for our class of perturbations also for infinite systems. While we consider the result and its proof as new and interesting in itself, they also lay the foundation for the proof of an adiabatic theorem for systems with a gap only in the bulk, which will be presented in a follow-up article.