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

Joscha Henheik, Stefan Teufel

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.

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

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.

Paper Structure

This paper contains 18 sections, 26 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Mathematical framework
  3. The lattice and algebras of local observables
  4. Interactions and associated operator families
  5. Lipschitz potentials
  6. Interactions and potentials with a thermodynamic limit
  7. States
  8. Main results
  9. Assumptions
  10. Adiabatic theorem and linear response in the thermodynamic limit
  11. General space-time adiabatic theorem
  12. Proofs of the main results
  13. Time-dependent super-adiabatic state for finite systems
  14. Infinite volume dynamics
  15. Conditional expectation for fermionic systems
  16. ...and 3 more sections

Key Result

Proposition 2.2

(Thermodynamic limit of evolution operators) Let $H_0 \in \mathcal{L}_{I, \zeta, 0 }$ and $v \in \mathcal{V}_I$ both have a thermodynamic limit, i.e. $\Phi_{H_0}\stackrel{\rm t.d.}{\rightarrow} \Psi_{H_0}$ and $v\stackrel{\rm t.d.}{\rightarrow} v_\infty$. Let $H = H_0 + V_v$ and let $U^{\eta,{\Lambd with $U^{\eta,{\Lambda_k}}(t_0,t_0) = \mathrm{id}$. Then there exists a co-cycle of automorphisms $

Theorems & Definitions (50)

  • Definition 2.1
  • Proposition 2.2
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • Theorem 3.5
  • Proposition 3.6
  • ...and 40 more