AI 中文总结
本文提出一种统计方法，通过独立采样小方格并利用完整性和角点重加权，解决二维格点SU(N)规范理论的采样难题，在SU(2)和SU(3)上实现高接受率与有效样本量。
AI 中文摘要
在格点规范理论的Wilson表述中，基本自由度是群值链接变量，而作用量是对小方格（最小的Wilson环）迹的求和。对于诸如归一化流之类的生成采样方法，这构成了挑战：单个小方格的分布易于建模，但将采样的小方格映射到链接是障碍。我们在二维空间中为具有Wilson小方格作用量的SU(N)规范理论探索了一种统计方法。作用量可以用完整变量（holonomy）来表示，一旦扩展格点的四个角满足一致性条件，这些变量就决定了链接。使用仅影响完整性的边界条件，我们将此一致性条件转化为条件采样问题，该问题简化为在给定群交换子Z=XYX†Y†的情况下采样X,Y∈SU(N)，其中Z∈SU(N)取决于四个角。小方格独立采样，由于额外的条件采样，生成的构型带有权重。我们获得了Z的密度（决定这些权重）的闭式表达式，适用于SU(2)和SU(3)。归一化流为SU(2)和SU(3)的单小方格分布建模；对于群交换子问题，我们使用SU(2)的闭式采样器和经过训练的SU(3)归一化流。在我们对2×2和32×32格点各两个耦合的测试中，每个小方格的接受率超过99%，最终权重的有效样本量适中，通常超过一半。
英文摘要
In the Wilson formulation of lattice gauge theory, the fundamental degrees of freedom are group-valued link variables, while the action is a sum over the trace of the plaquettes, the smallest Wilson loops. For generative sampling methods such as normalizing flows, this poses a challenge: the distribution of an individual plaquette is easy to model, but mapping sampled plaquettes to the links is the obstruction. We explore a statistical way around this in two dimensions for the $\mathrm{SU}(N)$ gauge theory with the Wilson plaquette action. The action can be written in terms of holonomy variables, which determine the links once a consistency condition at the four corners of an extended lattice is satisfied. Using a boundary condition that only affects the holonomies, we trade this consistency condition for a conditional sampling problem, which reduces to sampling $X,Y\in\mathrm{SU}(N)$ given the group commutator $Z=XYX^\dagger Y^\dagger$, where $Z\in\mathrm{SU}(N)$ depends on the four corners. Plaquettes are sampled independently, and the resulting configurations carry weights due to the additional conditional sampling. We obtain the density of $Z$, which determines these weights, in closed form for $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$. A normalizing flow models the single-plaquette distribution for $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$; for the group-commutator problem we use a closed-form sampler for $\mathrm{SU}(2)$ and a trained normalizing flow for $\mathrm{SU}(3)$. In our tests on $2\times2$ and $32\times32$ lattices at two couplings each, per-plaquette acceptance rates exceed $99\%$ and the effective sample size of the final weights is moderate, typically above one half.
Comments21 pages, 7 figures
