Browsing by Author "Hetényi, Balázs"

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Open Access
Calculating the polarization in bipartite lattice models: application to an extended Su-Schrieffer-Heeger model
(American Physical Society, 2021-02-08) Hetényi, Balázs; Pulcu, Yetkin; Doğan, Serkan
We address the question of different representations of Bloch states for lattices with a basis, with a focus on topological systems. The representations differ in the relative phase of the Wannier functions corresponding to the diffferent basis members. We show that the phase can be chosen in such a way that the Wannier functions for the different sites in the basis both become eigenstates of the position operator in a particular band. A key step in showing this is the extension of the Brillouin zone. When the distance between sites within a unit cell is a rational number, p/q, the Brillouin extends by a factor of q. For irrational numbers, the Brillouin zone extends to infinity. In the case of rational distance, p/q, the Berry phase lives on a cyclic curve in the parameter space of the Hamiltonian, on the Brillouin zone extended by a factor of q. For irrational distances, the most stable way to calculate the polarization is to approximate the distance as a rational sequence and use the formulas derived here for rational numbers. The use of different bases are related to unitary transformations of the Hamiltonian, as such, the phase diagrams of topological systems are not altered, but each phase can acquire different topological characteristics when the basis is changed. In the example we use, an extended Su-Schrieffer-Heeger model, the use of the diagonal basis leads to toroidal knots in the Hamiltonian space, whose winding numbers give the polarization.
Open Access
Geometric cumulants associated with adiabatic cycles crossing degeneracy points: Application to finite size scaling of metal-insulator transitions in crystalline electronic systems
(American Physical Society, 2022-11-28) Hetényi, Balázs; Cengiz, Sertaç
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 gapclosure points or regions with closed gap (Luttinger liquid). We demonstrate this by model calculations for a onedimensional 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.
Open Access
Scaling and renormalization in the modern theory of polarization: application to disordered systems
(American Physical Society, 2021-12-15) Hetényi, Balázs; Parlak, Selçuk; Yahyavi, Mohammad
We develop a scaling theory and a renormalization technique in the context of the modern theory of polarization. The central idea is to use the characteristic function (also known as the polarization amplitude) in place of the free energy in the scaling theory and in place of the Boltzmann probability in a position-space renormalization scheme. We derive a scaling relation between critical exponents which we test in a variety of models in one and two dimensions. We then apply the renormalization to disordered systems. In one dimension, the renormalized disorder strength tends to infinity, indicating the entire absence of extended states. Zero (infinite) disorder is a repulsive (attractive) fixed point. In two and three dimensions, at small system sizes, two additional fixed points appear, both at finite disorder: Wa(Wr) is attractive (repulsive) such that Wa