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On adiabatic theory for extended fermionic lattice systems

Joscha Henheik, Tom Wessel

Abstract

We review recent results on adiabatic theory for ground states of extended gapped fermionic lattice systems under several different assumptions. More precisely, we present generalized super-adiabatic theorems for extended but finite as well as infinite systems, assuming either a uniform gap or a gap in the bulk above the unperturbed ground state. The goal of this note is to provide an overview of these adiabatic theorems and briefly outline the main ideas and techniques required in their proofs.

On adiabatic theory for extended fermionic lattice systems

Abstract

We review recent results on adiabatic theory for ground states of extended gapped fermionic lattice systems under several different assumptions. More precisely, we present generalized super-adiabatic theorems for extended but finite as well as infinite systems, assuming either a uniform gap or a gap in the bulk above the unperturbed ground state. The goal of this note is to provide an overview of these adiabatic theorems and briefly outline the main ideas and techniques required in their proofs.
Paper Structure (27 sections, 5 theorems, 50 equations, 1 figure, 1 table)

This paper contains 27 sections, 5 theorems, 50 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Linear response and adiabatic theory
  3. Non-equilibrium almost-stationary states
  4. Brief statement of the results
  5. Mathematical Framework
  6. Algebra of observables
  7. Interactions and operator families
  8. Lipschitz potentials
  9. Adiabatic theorems for gapped quantum systems
  10. Systems with a uniform gap
  11. Extended but finite systems
  12. Infinite systems
  13. Systems with a gap in the bulk
  14. Infinite systems
  15. Extended but finite systems
  16. ...and 12 more sections

Key Result

Theorem I

(Adiabatic theorem for finite systems with a uniform gap teufel2020non) Under Assumptions ass:GAPunif and ass:INT1, there exists a sequence of near-identity Indeed, $\sup_{k\in \mathbb{N} } \bigl\lVert A - \beta^{\varepsilon,\eta,\Lambda_k}(t) \llbracket A \rrbracket \bigr\rVert \leq (\varepsilon+ are super-adiabatic NEASSs for the Heisenberg time-evolution $\mathfrak{U}_{t,t_0}^{\varepsilon, \e

Figures (1)

  • Figure 1: Let $H_0$ be a Hamiltonian with a gapped sector and a gap $g$. Perturbing with a Lipschitz potential $v(x)=\varepsilon\,x$, the gap gets closed (for large enough lattices). But, as indicated in the figure, a local gap persists and an electron at location $x_0$ would either need to overcome the gap (vertical arrow) or tunnel along the distance $g/\varepsilon$ (horizontal arrow) in order to make a transition from the gapped sector. teufel2020nonhenheikteufel2020

Theorems & Definitions (9)

  • Definition 1
  • Theorem I
  • Definition 2
  • Proposition 3
  • Theorem II
  • Remark 4
  • Theorem III
  • Definition 5
  • Theorem IV