From decay of correlations to locality and stability of the Gibbs state

Abstract

We show that whenever the Gibbs state of a quantum spin system satisfies decay of correlations, then it is stable, in the sense that local perturbations affect the Gibbs state only locally, and it satisfies local indistinguishability, i.e. it exhibits local insensitivity to system size. These implications hold in any dimension, require only locality of the Hamiltonian, and are based on Lieb-Robinson bounds and on a detailed analysis of the locality properties of the quantum belief propagation for Gibbs states. To demonstrate the versatility of our approach, we explicitly apply our results to several physically relevant models in which the decay of correlations is either known to hold or is proved by us. These include Gibbs states of one-dimensional spin chains with polynomially decaying interactions at any temperature, and high-temperature Gibbs states of quantum spin systems with finite-range interactions in any dimension. We also prove exponential decay of correlations above a threshold temperature for Gibbs states of one-dimensional finite spin chains with translation-invariant and exponentially decaying interactions, and then apply our general results.

54 pages, 6 figures; v2: added section about SLT perturbations, updated and added references, fixed typos; v3: added results using DC from Kimura and Kuwahara (arXiv:2403.11431), streamlined presentation, updated references, fixed typos (final version corresponding to the published article)

References in corpus (10)

• Sample-efficient learning of quantum many-body systems
• Quantum Belief Propagation
• Hierarchy of linear light cones with long-range interactions
• Improved thermal area law and quasi-linear time algorithm for quantum Gibbs states
• Some Properties of Correlations of Quantum Lattice Systems in Thermal Equilibrium
• Exponential decay of mutual information for Gibbs states of local Hamiltonians
• Quantum many-body systems in thermal equilibrium
• Local Perturbations Perturb -Exponentially- Locally
• Local stability of ground states in locally gapped and weakly interacting quantum spin systems
• Stability of invertible, frustration-free ground states against large perturbations

Cited by in corpus (4)

• Clustering theorem in 1D long-range interacting systems at arbitrary temperatures
• Enhanced Lieb-Robinson bounds for commuting long-range interactions
• Strong decay of correlations for Gibbs states in any dimension
• Stability of thermal equilibrium in long-range quantum systems