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From decay of correlations to locality and stability of the Gibbs state
Mathematical Physics · Submitted 13 Oct 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | result 0.94 | 0.95 | 0.93 | asserted 1.00 |
| s1 | 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. | method 0.75 | 0.82 | 0.11 | asserted 1.00 |
| s2 | 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. | contribution 0.75 | 0.88 | 0.04 | asserted 0.99 |
| s3 | 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. | result 0.50 | 0.45 | 0.04 | asserted 0.93 |
| s4 | 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. | result 0.86 | 0.95 | 0.05 | asserted 1.00 |
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Original 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.
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6 arXiv papers cite this in 10 sentences
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We note that decay of correlation for Gibbs states were previously only proved for a limited number of settings, typically for 1D systems or for high temperatures [ref] , whereas our result is not constrained by the geometry of the lattice or by the temperature, but is instead constrained by the non-quadratic interaction strength.
Fast mixing of weakly interacting fermionic systems at any temperature · Introduction · 0.98
Cites it as background 2 papers
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This is often stated in terms of connected correlation functions [ref] , but also with other quantifiers such as the mutual information [ref] (although they are often equivalent [ref] ), as well as measures of quantum entanglement [ref] .
Conditional Independence of 1D Gibbs States with Applications to Efficient Learning · Introduction · 0.94
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See [ref] for relevant discussion.
Clustering theorem in 1D long-range interacting systems at arbitrary temperatures · Main results · 0.82
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