Geometric cumulants associated with adiabatic cycles crossing degeneracy points: Application to finite size scaling of metal-insulator transitions in crystalline electronic systems

In this work, we focus on two questions. One, we complement the machinery 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 nontrivial. 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 developed machinery 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.

Figure 1

$A_n$ [defined in Eq. (59)] as a function of $n$ (upper panel). The Binder cumulant as a function of $n$ for a gapless system.

Figure 2

Binder cumulants $U_L^{\left(n\right)}$ approximated to different orders $\left[O\right(L^{-2n}$)] for the one-dimensional SSH model [Eq. (60)]. The phase transition point occurs at $\delta J=0$. The different curves in each panel are for different system sizes: $L=50,100,200,500$.

Figure 3

Variance as a function of the order of approximation, $n$, one-dimensional SSH model [Eq. (60)], two cases, $\delta J=0.2$ and $\delta J=0.1$, calculated via two methods: the Resta-Sorella approximation and the moment based approximation (MBA) developed in this work.

Figure 4

Geometric Binder cumulants for two-dimensional models. Upper panel shows the results for a tight-binding model with an on-site potential (strength $\mathrm{\Delta }$) on a square lattice, the middle panel shows the results for a tight-binding model with an on-site potential (strength $\mathrm{\Delta }$) on a hexagonal lattice, and the lower panel shows results for the topological Haldane model on a hexagonal lattice.

Figure 5

Variance for the hexagonal lattice (upper panel) and the Haldane model (lower panel), both at gap closure, averaged in the transverse direction, shown on a log-log plot. For all calculations, $L_x$ is held fixed and $L_y$ is varied. The values of $L_y$ are shown above each data point. The straight dashed line is the log-log plot of a curve of the form $f\left(L_y\right)=a{L}_y^2$. For the hexagonal model (upper panel), $a=0.000593473$, and for the Haldane model (lower panel), $a=0.000325063$.

Figure 6

Binder cumulant for the $t-V$ one-dimensional interacting model, as a function of $V/t$. The upper panel shows results for the Resta-Sorella approximation, and the lower one for the moment based approximation.