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Uniform-in-temperature locality estimates for weakly interacting quantum systems

Arka Adhikari, Joscha Henheik, Marius Lemm, Tom Wessel

TL;DR

The paper addresses uniform-in-temperature locality for Gibbs states of weakly interacting quantum lattice systems by proving exponential decay of correlations and local indistinguishability with decay lengths independent of $eta$. The authors fuse a low-temperature cluster expansion with a quantum swapping trick to achieve temperature-uniform bounds for DoC and LI, applicable even in disordered or non-translation-invariant settings. The results rely on a robust setup with a gapped on-site Hamiltonian $H^0$ perturbed by a small finite-range $V$ that is relatively form-bounded, and they extend to associated LPPL bounds for local perturbations. These findings have implications for the locality of temperature, efficient simulation and state preparation, and the development of low-temperature response theories in interacting quantum systems.

Abstract

The locality of thermal quantum states has emerged as a key input for applications to thermalization, response theory, and efficient simulability. Locality is either captured by the decay of correlations or by local indistinguishability, which allows to approximate local expectation values by those of local thermal states. Most techniques for deriving locality bounds deteriorate at small temperature, a physically highly relevant regime and so it is of interest to identify conditions for uniform-in-temperature bounds. Here we prove that a class of weakly interacting quantum Hamiltonians satisfies exponential decay of correlations and local indistinguishability uniformly in the temperature. The proof uses a low-temperature cluster expansion and a quantum version of a probabilistic swapping trick developed by the first author and Cao (Ann. Probab. 53, 2025) in the context of lattice gauge theories.

Uniform-in-temperature locality estimates for weakly interacting quantum systems

TL;DR

The paper addresses uniform-in-temperature locality for Gibbs states of weakly interacting quantum lattice systems by proving exponential decay of correlations and local indistinguishability with decay lengths independent of . The authors fuse a low-temperature cluster expansion with a quantum swapping trick to achieve temperature-uniform bounds for DoC and LI, applicable even in disordered or non-translation-invariant settings. The results rely on a robust setup with a gapped on-site Hamiltonian perturbed by a small finite-range that is relatively form-bounded, and they extend to associated LPPL bounds for local perturbations. These findings have implications for the locality of temperature, efficient simulation and state preparation, and the development of low-temperature response theories in interacting quantum systems.

Abstract

The locality of thermal quantum states has emerged as a key input for applications to thermalization, response theory, and efficient simulability. Locality is either captured by the decay of correlations or by local indistinguishability, which allows to approximate local expectation values by those of local thermal states. Most techniques for deriving locality bounds deteriorate at small temperature, a physically highly relevant regime and so it is of interest to identify conditions for uniform-in-temperature bounds. Here we prove that a class of weakly interacting quantum Hamiltonians satisfies exponential decay of correlations and local indistinguishability uniformly in the temperature. The proof uses a low-temperature cluster expansion and a quantum version of a probabilistic swapping trick developed by the first author and Cao (Ann. Probab. 53, 2025) in the context of lattice gauge theories.

Paper Structure

This paper contains 16 sections, 14 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Summary of main results
  3. Discussion
  4. Applications and outlook
  5. Setup and main results
  6. Mathematical setup
  7. Main results
  8. Proofs
  9. Proof strategy and the swapping trick
  10. Analyticity bound
  11. Cluster expansion
  12. Ratios of partition functions
  13. Proof of Theorem \ref{['thm:DoC']}
  14. Proof of Theorem \ref{['thm:LI']}
  15. Local perturbations perturb locally
  16. ...and 1 more sections

Key Result

theorem 1

Let $D$, $q$, $R\in \N$ and $C_{\mathup{int}}>0$. Then there exist $a\in \intervaloo{0,1}$ and $C_1$, $C_2$, $\xi>0$ such that the following holds. Consider the lattice $\Lambda \Subset \Z^D$ and a Hamiltonian $H^0_{\Lambda} + V_{\Lambda}$ as defined in Section sec:setup with $\norm{h}_\infty$, $\no for all $\conn$-connected sets $X$, $Y\subset \Lambda$ and observables $A \in \alg_X$ and $B \in \a

Theorems & Definitions (36)

  • remark 1: Relation between DoC and LI
  • remark 2
  • definition 1: $\conn$-connected sets
  • theorem 1: Decay of correlations
  • theorem 2: Local indistinguishability
  • remark 3: LI implies DoC
  • corollary 1: DoC and LI in the thermodynamic limit
  • proof
  • lemma 1
  • proof
  • ...and 26 more