论文标题
与绝热循环相关的几何累积物交叉脱落点：在晶体电子系统中的金属 - 绝缘体过渡的有限尺寸缩放点应用
Geometric cumulants associated with adiabatic cycles crossing degeneracy points: Application to finite size scaling of metal-insulator transitions in crystalline electronic systems
论文作者
论文摘要
在这项工作中，我们专注于两个问题。第一，我们补充机械以计算沿绝热循环的几何阶段，如下所示。几何阶段是沿绝热循环的一条线组成部分，如果循环环绕一个变性点，则该阶段将变得不平凡。如果循环越过退化点，则相位分歧。我们构造数量，这些数量是当路径越过堕落点时明确定义的。我们通过构建广义的Bargmann不变性来做到这一点，并指出它可以解释为累积的生成函数，而几何相是第一个累积液。我们表明，累积物的特定比率仍然是跨越一组孤立退化点的循环的有限比率。累积比以统计力学中有限尺寸缩放理论（我们将其命名为几何粘合剂累积物）所知。第二，我们表明，开发的机械可以应用于现代极化理论的背景下进行有限尺寸缩放。几何粘合剂累积物在间隙闭合点或带有封闭间隙（Luttinger液体）的区域的尺寸独立。我们通过模型计算为一维拓扑模型，几个二维模型和一维相关模型证明了这一点。在两个维度的情况下，我们将其分析到不同的情况下，其中一种情况是费米表面是一维（一条线），而两种情况为零尺寸（DIRAC点）。对于几何粘合剂累积物，即使在二维中，差距闭合点也可以通过一维缩放找到。作为技术点，我们强调累积物的某些有限差近似是适用的，因为并非所有近似方案都能够在封闭的间隙系统的情况下提取尺寸缩放信息。
In this work we focus on two questions. One, we complement the machinary to calculate geometric phases along adiabatic cycles as follows. The geometric phase is a line integral along an adiabatic cycle, and if the cycle encircles a degeneracy point, the phase becomes non-trivial. If the cycle crosses the degeneracy point the phase diverges. We construct quantities which are well-defined when the path crosses the degeneracy point. We do this by constructing a generalized Bargmann invariant, and noting that it can be interpreted as a cumulant generating function, with the geometric phase being the first cumulant. We show that particular ratios of cumulants remain finite for cycles crossing a set of isolated degeneracy points. The cumulant ratios take the form of the Binder cumulants known from the theory of finite size scaling in statistical mechanics (we name them geometric Binder cumulants). Two, we show that the machinery developed can be applied to perform finite size scaling in the context of the modern theory of polarization. The geometric Binder cumulants are size independent at gap closure points or regions with closed gap (Luttinger liquid). We demonstrate this by model calculations for a one-dimensional topological model, several two-dimensional models, and a one-dimensional correlated model. In the case of two dimensions we analyze to different situations, one in which the Fermi surface is one-dimensional (a line), and two cases in which it is zero dimensional (Dirac points). For the geometric Binder cumulants the gap closure points can be found by one dimensional scaling even in two dimensions. As a technical point we stress that only certain finite difference approximations for the cumulants are applicable, since not all approximation schemes are capable of extracting the size scaling information in the case of a closed gap system.
