AI 中文总结
本文提出双重局域化框架，结合几何与界面局域化证明量子吉布斯采样器谱间隙，应用于Kitaev有限群模型，对S3群在宽参数区间给出均匀正下界。
AI 中文摘要
我们引入双重局域化，这是一个通过两种互补操作证明量子吉布斯采样器谱间隙的框架：几何局域化，它选择在小空间区域中支持的更新；以及界面局域化，它聚焦于由模型定义的可观测量子空间，同时保持几何上的全局性。我们的抽象能隙定理将关于该子空间（称为界面）的全局界与在定量耦合和几何组装条件下的局部量子能隙估计相结合。由此产生的显式下界控制完整动力学的能隙，而不要求界面在生成元下保持不变。这种分离允许将全局估计（包括经典比较）与对剩余量子方向的局部控制相结合。我们将该框架应用于由Kitaev有限群量子双构造的顶点项（不含plaquette项）构建的哈密顿量，这些哈密顿量定义在围长至少为六的有限简单三正则图上。对于指定的局部Davies动力学，我们证明了对于每个固定的非平凡有限群和每个固定的逆温度$\beta$（满足$0\le\beta J<\log(5/3)$，其中$J>0$为耦合强度），存在一个不依赖于图大小的正常数作为无条件谱间隙下界。对于最小的非阿贝尔群$S_3$，双重局域化产生一个在图形大小和整个区间$0\le\beta J\le\log 3$上一致的正下界，提供了现有结果无法直接推导的保证。这展示了模型特定的有限计算如何扩展完整量子动力学的谱间隙保证。
英文摘要
We introduce double localization, a framework for proving spectral gaps of quantum Gibbs samplers through two complementary operations: geometric localization, which selects updates supported in small spatial regions, and interface localization, which focuses on a model-defined subspace of observables while remaining geometrically global. Our abstract gap theorem combines a global bound on this subspace, called the interface, with local quantum gap estimates under quantitative coupling and geometric assembly conditions. The resulting explicit lower bound controls the gap of the full dynamics without requiring the interface to be invariant under the generator. This separation allows global estimates, including classical comparison, to be combined with local control of the remaining quantum directions. We apply the framework to Hamiltonians built from the vertex terms of Kitaev's finite-group quantum double construction, without plaquette terms, on finite simple three-regular graphs of girth at least six. For the specified local Davies dynamics, we prove an unconditional spectral-gap lower bound by a positive constant independent of graph size for every fixed nontrivial finite group and each fixed inverse temperature $β$ with $0\leβJ<\log(5/3)$, where $J>0$ is the coupling strength. For the smallest non-Abelian group $S_3$, double localization yields a positive lower bound uniform in both graph size and the entire interval $0\leβJ\le\log 3$, providing a guarantee that does not follow directly from existing results. This demonstrates how model-specific finite calculations can extend spectral-gap guarantees for the full quantum dynamics.
Comments76 pages
