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Response theory for locally gapped systems

Joscha Henheik, Tom Wessel

Abstract

We introduce a notion of a \emph{local gap} for interacting many-body quantum lattice systems and prove the validity of response theory and Kubo's formula for localized perturbations in such settings. On a high level, our result shows that the usual spectral gap condition, concerning the system as a whole, is not a necessary condition for understanding local properties of the system. More precisely, we say that an equilibrium state $ρ_0$ of a Hamiltonian $H_0$ is locally gapped in $Λ^{\mathrm{gap}} \subset Λ$, whenever the Liouvillian $- \mathrm{i} \, [H_0, \, \cdot \, ]$ is almost invertible on local observables supported in $Λ^{\mathrm{gap}}$ when tested in $ρ_0$. To put this into context, we provide other alternative notions of a local gap and discuss their relations. The validity of response theory is based on the construction of \emph{non-equilibrium almost stationary states} (NEASSs). By controlling locality properties of the NEASS construction, we show that response theory holds to any order, whenever the perturbation \(εV\) acts in a region which is further than $|\log ε|$ away from the non-gapped region $Λ\setminus Λ^{\mathrm{gap}}$.

Response theory for locally gapped systems

Abstract

We introduce a notion of a \emph{local gap} for interacting many-body quantum lattice systems and prove the validity of response theory and Kubo's formula for localized perturbations in such settings. On a high level, our result shows that the usual spectral gap condition, concerning the system as a whole, is not a necessary condition for understanding local properties of the system. More precisely, we say that an equilibrium state of a Hamiltonian is locally gapped in , whenever the Liouvillian is almost invertible on local observables supported in when tested in . To put this into context, we provide other alternative notions of a local gap and discuss their relations. The validity of response theory is based on the construction of \emph{non-equilibrium almost stationary states} (NEASSs). By controlling locality properties of the NEASS construction, we show that response theory holds to any order, whenever the perturbation acts in a region which is further than away from the non-gapped region .

Paper Structure

This paper contains 44 sections, 2 theorems, 173 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Response theory in many-body quantum systems
  3. A local dynamical gap condition
  4. Dynamical characterization of a spectral gap
  5. Verifying the local dynamical gap condition
  6. Discussion of our main result
  7. Structure of the paper
  8. Mathematical framework
  9. Spatial structure and algebra of observables
  10. Interactions and SLT operators
  11. Main result: Response theory for locally gapped systems
  12. Response theory
  13. Non-equilibrium almost stationary states and proof of Theorem \ref{['thm:linear-response']}
  14. How to formulate a local gap condition?
  15. Extrinsic local gap conditions
  16. ...and 29 more sections

Key Result

theorem 3

Fix $n,m \in \mathbb{N}$ and let $d\in \mathbb{N}$, $C_{\mathrm{vol}}>0$, $b>0$, $p\in \lparen0,1\rbrack$, $C_{\mathrm{int}}>0$ and $g > 0$, $C_\mathrm{gap}>0$, $\ell\in \mathbb{N} _0$, and $C_{\mathrm{switch}} > 0$. Take any $q \in (0, p)$. Then there exist a constant $C_{n,m}> 0$ (in particular Let $\rho^{\varepsilon, \eta, f}(t)$ be the solution of the time-dependent adiabatic Schrödinger eq

Figures (1)

  • Figure 1: Implications among the various local gap conditions from Sections \ref{['subsec:extrinsic']}--\ref{['subsec:intrinsic']}. The numbering refers to the precise statements in Proposition \ref{['prop:mechforlocalgap']}.

Theorems & Definitions (13)

  • theorem 3: Response theory to all orders
  • proof : Proof of Theorem \ref{['thm:linear-response']}
  • theorem 10: Time-dependent NEASS
  • proof : Proof of Proposition \ref{['prop:NEASS']}
  • proof : Proof of Proposition \ref{['prop:adswitch']}
  • proof : Proof of Proposition \ref{['prop:expansionNEASS']}
  • proof : Proof of Theorem \ref{['thm:time-dependent-NEASS']}
  • proof : Proof of Lemma \ref{['lem:stabilizer']}
  • proof
  • proof
  • ...and 3 more