Quantum Spin Chains Thermalize at All Temperatures

TL;DR

The work proves that 1D short-range quantum systems thermalize at all finite temperatures by establishing a system-size independent spectral gap for a quantum Gibbs sampler Lindbladian. The core method introduces a non-CP spectral conditional expectation built from an auxiliary generator $\\mathcal{K}$, proves a local spectral gap for $\\mathcal{K}$, and shows its quasi-local projections yield strong clustering; these results transfer to the physical Lindbladian via a Dirichlet-form comparison. Consequently, Gibbs states admit exponential clustering and can be prepared in polylogarithmic circuit depth through adiabatic (purified) state preparation, tying static correlation decay to dynamic mixing in noncommuting 1D systems. The analysis borrows and extends the Kastoryano-style spectral-gap recursion to non-CP maps, leveraging 1D complex-time evolution and Lieb-Robinson bounds to bootstrap from weak to strong clustering and to a constant gap. The findings illuminate the interplay between statics and dynamics in 1D quantum thermodynamics and provide a pathway toward sharp mixing-time results for non-commuting Hamiltonians, with implications for efficient thermal state preparation and quantum algorithm design.

Abstract

It is shown that every one-dimensional Hamiltonian with short-range interaction admits a quantum Gibbs sampler [CKG23] with a system-size independent spectral gap at all finite temperatures. Consequently, their Gibbs states can be prepared in polylogarithmic depth, and satisfy exponential clustering of correlations, generalizing [Ara69].

Figures (8)

• Figure 1: We consider a 1D Hamiltonian with nearest neighbour interaction; by taking large enough $q$ and rescaling, this captures all finite-range 1D models. We are interested in its rate of thermalization, when weakly coupled to a constant temperature bath on every site. Mathematically, we model the thermalization dynamics by an exactly detailed-balanced Lindbladian chen2023efficient, where each Lindbladian term corresponding to each system-bath coupling $\bm{A}^a$ is quasi-local.
• Figure 2: An outline of the proof of \ref{['thm:main']}, starting from the sub-exponential decay-of-correlations in the Gibbs state of arbitrary 1D Hamiltonians, at all finite temperatures, established by Kimura_2025.
• Figure 3: The quasi-locality of the spectral conditional expectation map. For any set $\mathsf{A}$, the spectral conditional expectation $\tilde{\mathbb{E}}_\mathsf{A}$ is well-approximated by that with a truncated Hamiltonian nearby $\mathsf{A}$. Quantitatively, the length of the "buffer zone" needs to be at least $\frac{\left\vert {\mathsf{\mathsf{A}}} \right\vert}{\log\left\vert {\mathsf{\mathsf{A}}} \right\vert}.$
• Figure 4: The Definition of Strong Clustering. The notion of strong clustering for our spectral conditional expectations is best described by a "gluing" property, see equation \ref{['eq:gluing_intro_TBE']}. Due to quasi-locality, we must take the size of the region $\mathsf{B}$ to scale at least with $\left\vert {\mathsf{A}\mathsf{C}} \right\vert/\log(\left\vert {\mathsf{A}\mathsf{C}} \right\vert)$.
• Figure 5: For any set $\mathsf{A}$, $\mathsf{C}$, the spectral conditional expectation is well-approximated by the composition of individual ones.

Theorems & Definitions (125)

• Theorem 1.1: System-size independent spectral gap
• Corollary 1.1: $\mathsf{polylog}$ depth adiabatic algorithms
• Corollary 1.2: Exponential clustering of correlations
• Definition 1.1: Conditional Expectation
• Example 1.1
• Definition 2.1: 1D Hamiltonians
• Lemma 2.1
• Lemma 2.2: Holder in KMS Norm, e.g., chen2025quantumMarkov
• proof
• Definition 2.2: KMS-Induced Superoperator Norm