变分离散Action理论的基础
Foundations of the Variational Discrete Action Theory
摘要：
变分波函数和格林函数是求解量子哈密顿量的两种重要方法，它们各有优点。在这里，我们详细的变分离散作用理论（VDAT），它利用这两种范式的优点，以近似解决量子哈密顿的基态。VDAT由两个核心部分组成：序贯乘积密度矩阵（SPD）和与SPD相关的离散动作。SPD是一个变分分析的启发，特罗特分解和其特征在于由一个整数$\mathcal{N}$，恢复许多著名的变分波函数，除了精确解$\mathcal{N}=\infty$。离散作用描述了关于SPD的有效整数时间演化的所有动力学信息。我们推广的路径积分到我们的整数时间形式主义，它转换成一个静态的关联函数在一个复合空间的动态关联函数在整数时间。我们还概括了通常的多体格林函数形式主义的整数时间，这导致类似的，但不同的数学结构，产生整数时间版本的生成功能，戴森方程，和Bethe-Salpeter方程。我们证明了在多带安德森杂质模型（AIM）中，通过对有限个图求和，可以精确地计算SPD。对于多带哈伯德模型，我们证明了动力学平均场理论的整数时间模拟--自洽正则离散作用量近似（SCDA）精确地计算了$d=\infty$的SPD. SCDA中的VDAT为捕获量子晶格模型的局域物理提供了一种有效而可靠的方法，这将在强关联电子材料中具有广泛的应用。更一般地说，VDAT应该在物理学中的各种多体问题中找到应用。
英文摘要：
Variational wave functions and Green's functions are two important paradigms for solving quantum Hamiltonians, each having their own advantages. Here we detail the Variational Discrete Action Theory (VDAT), which exploits the advantages of both paradigms in order to approximately solve the ground state of quantum Hamiltonians. VDAT consists of two central components: the sequential product density matrix (SPD) ansatz and a discrete action associated with the SPD. The SPD is a variational ansatz inspired by the Trotter decomposition and characterized by an integer $\mathcal{N}$, recovering many well known variational wave functions, in addition to the exact solution for $\mathcal{N}=\infty$. The discrete action describes all dynamical information of an effective integer time evolution with respect to the SPD. We generalize the path integral to our integer time formalism, which converts a dynamic correlation function in integer time to a static correlation function in a compound space. We also generalize the usual many-body Green's function formalism to integer time, which results in analogous but distinct mathematical structures, yielding integer time versions of the generating functional, Dyson equation, and Bethe-Salpeter equation. We prove that the SPD can be exactly evaluated in the multi-band Anderson impurity model (AIM) by summing a finite number of diagrams. For the multi-band Hubbard model, we prove that the self-consistent canonical discrete action approximation (SCDA), which is the integer time analogue of the dynamical mean-field theory, exactly evaluates the SPD for $d=\infty$. VDAT within the SCDA provides an efficient yet reliable method for capturing the local physics of quantum lattice models, which will have broad applications for strongly correlated electron materials. More generally, VDAT should find applications in various many-body problems in physics.
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