标题： 基于机器学习的量子多体物理压缩：顶点函数的PCA和自编码器表示 显示英文标题

标题： Machine learning-based compression of quantum many body physics: PCA and autoencoder representation of the vertex function

摘要： 表征物质的复杂多体相一直是量子物理几十年来的核心问题。 围绕重整化群（RG）流方程的近似方法为最初的问题提供了可靠且可系统改进的答案——是什么简单的物理驱动着量子有序和无序？ 这些流方程是一组高维的耦合非线性方程，其解是两粒子顶点函数，这是一个由三个连续动量描述的函数，用于描述粒子-粒子散射，并包含了许多低能物理信息，包括系统是否表现出各种形式的长程有序。 在本工作中，我们采用一种简单且可解释的数据驱动方法来解决压缩两粒子顶点的开放性问题。 我们使用主成分分析（PCA）和自编码神经网络，为格点上的相互作用费米子情况推导出紧凑的、低维的底层物理表示。 我们通过多种度量标准量化了表示中的误差，并表明简单的线性PCA比名义上更表达能力强的非线性模型提供了更多的物理洞察力和更好的分布外（零样本）泛化能力。 即使主成分数量适中（10-20），我们在相图上也实现了顶点函数的良好重构。 这一结果表明，许多其他多体函数可能同样可以被压缩，从而可能实现可观测量的高效计算。 最后，我们识别出在已知相之间共享的主成分子空间，提供了新的物理见解。 显示英文摘要

摘要： Characterizing complex many-body phases of matter has been a central question in quantum physics for decades. Numerical methods built around approximations of the renormalization group (RG) flow equations have offered reliable and systematically improvable answers to the initial question -- what simple physics drives quantum order and disorder? The flow equations are a very high dimensional set of coupled nonlinear equations whose solution is the two particle vertex function, a function of three continuous momenta that describes particle-particle scattering and encodes much of the low energy physics including whether the system exhibits various forms of long ranged order. In this work, we take a simple and interpretable data-driven approach to the open question of compressing the two-particle vertex. We use PCA and an autoencoder neural network to derive compact, low-dimensional representations of underlying physics for the case of interacting fermions on a lattice. We quantify errors in the representations by multiple metrics and show that a simple linear PCA offers more physical insight and better out-of-distribution (zero-shot) generalization than the nominally more expressive nonlinear models. Even with a modest number of principal components (10 - 20), we find excellent reconstruction of vertex functions across the phase diagram. This result suggests that many other many-body functions may be similarly compressible, potentially allowing for efficient computation of observables. Finally, we identify principal component subspaces that are shared between known phases, offering new physical insight.