发表机构
University of Cambridge; Universität Tübingen; Universidad Complutense de Madrid; Instituto de Ciencias Matemáticas (ICMAT); Instituto de Física Teórica UAM/CSIC（剑桥大学; 蒂宾根大学; 马德里康普顿斯大学; 数学科学研究所; UAM/CSIC理论物理研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对一维非对易哈密顿量，证明热浴动力学及其正则化版本的修正对数Sobolev不等式，实现快速混合，为首次非对易情形下的MLSI证明。
AI 中文摘要
对于有限自旋一维链上的有限范围非对易哈密顿量，我们证明了热浴动力学（一种准局域Gibbs采样器）以及正则化热浴动力学具有常数为$\Omega(1/\log n)$的修正对数Sobolev不等式。这意味着两种采样器在相对熵和迹范数意义下向Gibbs态的快速混合，混合时间阶为$\mathcal{O}(\log n\cdot \log(n/\epsilon))$。相互作用可以是空间非均匀的。证明结合了相对熵的弱拟因子化、衰减关联估计、一致条件局域能隙，以及条件熵与二次能量之间的缓冲比较。由此产生的有界熵缺陷通过全局能隙消除。据我们所知，这是非对易情形下MLSI的首次证明。
英文摘要
Given a finite-range, non-commuting Hamiltonian on a 1D chain of finite spins, we prove a modified logarithmic Sobolev inequality with constant $Ω(1/\log n)$ for the heat-bath dynamics, a quasi-local Gibbs sampler, and for the regularised heat-bath dynamics. This implies rapid mixing of both samplers towards the Gibbs state in relative entropy and trace norm, with mixing times of order $\mathcal{O}(\log n\cdot \log(n/ε))$. The interactions may be spatially inhomogeneous. The proof combines weak quasi-factorisation of relative entropy, decay-of-correlation estimates, a uniform conditional local gap, and a buffered comparison between conditional entropy and quadratic energy. The resulting bounded entropy defect is removed using the global gap. To the best of our knowledge, this is the first proof of MLSI in the non-commuting regime.
Comments56 pages, 10 figures
