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

TL;DR

The paper establishes that for quantum Gibbs states, decay of correlations implies local stability under local perturbations and local indistinguishability, and that these locality notions are in fact equivalent under locality assumptions. The authors develop a rigorous framework based on Lieb-Robinson bounds and quantum belief propagation (QBP) to translate decay-of-correlations information into quantitative LPPL and LI bounds, valid for finite-range, short-range, and long-range interactions in any dimension. They provide concrete applications to one-dimensional and higher-dimensional spin systems, including translation-invariant 1D short-range chains, 1D long-range chains, and high-temperature Gibbs states, deriving explicit decay rates and constants. A key methodological contribution is the use of QBP to derive differential equations for Gibbs states and to obtain locality properties of the generator and the exponential, enabling stability results and a circle of implications among the three locality notions. The work also discusses stability under sum-of-local-terms perturbations and outlines precise short/long-range decay bounds that underpin the theoretical connections, with potential impact on quantum simulation and thermalization analyses.

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.

Figures (6)

• Figure 1: The diagram shows the main implications discussed in this work for short-range interactions. In particular, we show “equivalence” of the three concepts in the picture. Note, that the formulas are mainly illustrative for the concepts and in particular the constants change, see Remark \ref{['remark:decay-of-correlations-after-one-circle']}. A crucial ingredient in all the implications is quantum belief propagation (QBP) coupled with Lieb-Robinson bounds. For precise statements we refer to the Theorems. In certain physical dimensions and temperature regimes, exponential decay of correlations is known to hold by earlier results, for which all three properties are thus satisfied.
• Figure 2: Depicted is the main idea for the proof of local indistinguishability from uniform LPPL. The idea is to remove all points $x\in \varLambda \mathbin{ \setminus }\varLambda'$ one by one. Therefore, we first apply LPPL to the sum of all interaction terms connecting $x$ with its $R$-neighbourhood $B_x(R)$. For short-range interactions, the remaining interaction terms including $x$ are exponentially small in $R$ and can be removed using QBP. Furthermore, the points $x$ are grouped into shells $S_q \ni x$ according to their distance $q:=\dist{x,Y}$ to $Y$. We then choose the parameter $R$ depending on $q$, so that the error for operators $B\in \mathcal{A}_Y$ introduced by removing all points in $S_q$ decays exponentially in $q$. This allows to sum the error terms introduced by removing all shells with $q$ and still obtain exponential decay in the distance $\dist{Y,\varLambda \mathbin{ \setminus }\varLambda'}$.
• Figure 3: Depicted is the situation from the proof of Theorem \ref{['thm:local-indistinguishability-implies-decay-of-correlations']}. By local indistinguishability, the covariance of the Gibbs state on $\varLambda$ and $\varLambda'=X_\ell \cup Y_\ell$ are similar. The remaining distance $\dist{X_\ell,Y_\ell}$ must be chosen so large that the remaining interactions coupling both regions are small. In the case of finite-range interactions, the distance must be chosen larger than the interaction range, so that the regions completely decouple.
• Figure 4: The diagram shows the main implications for one-dimensional (translation-invariant) spin chains, which are discussed in this section. Here, $I \subset \mathbb{Z}$ is a finite interval, $X\subset I$ a subinterval and $Y\subset I$ a union of two intervals. In particular, we show “equivalence” of the four concepts in the picture. Note that the constants are not the same, and we refer to the Theorems for precise statements. A crucial ingredient in all the implications is quantum belief propagation (QBP) coupled with Lieb-Robinson bounds. For finite-range or exponentially decaying interactions, exponential decay of correlations is known to hold by earlier results for the infinite-chain regime at every positive or high enough temperature, respectively, for which all four properties are thus satisfied.
• Figure 5: Representation of an interval $I$ with subintervals $X,Y_1,Y_2 \subset I$. An example of a perturbation $V \in \mathcal{A}_X$ such that the distance between $X_r$ and $Y = Y_1 \cup Y_2$ is at least $r$. Here, $r = \lfloor d(X,Y)/2\rfloor = 2$.

Theorems & Definitions (65)

• Definition 1: (Uniform) decay of correlations
• Definition 2: (Uniform) local perturbations perturb locally (LPPL)
• Definition 3: Local indistinguishability
• Remark 4
• Remark 5
• Theorem 6
• Corollary 7
• Theorem 8: [Theorem 1]KK2024
• Corollary 9
• Theorem 10: [Theorem 1]KK2024