Interplay of quasi-periodic potential and disorder in 3D Anderson model

Abstract

Localization physics studies how waves stop propagating under certain conditions. The relevant physical concepts are quite broad, ranging from classical waves to quantum waves. That is why Anderson localization was first discovered in the study of wave-function of spin but later became important in nano- and meso-scale technology, acoustic waves and so on. Chaos physics and disorder system are interesting topics not only in physics but also in many mathematical research works. Disorder is one of the reasons causing the localization of wave-function. The critical behaviors and universal properties near the localization transition have been heavily investigated in recent years.

In this thesis, we investigate the localization physics in the presence of both disorder and quasi-periodic potential. As is known, both random disorder potential and quasi-periodic potential can localize quantum mechanical wave functions. We investigate how the two types of potentials interplay in the localization phenomenon by studying a modified 3D Anderson lattice model. Specifically, we consider an anisotropic quasi-periodic potential along the Z direction, while disorder remains the usual isotropic white-noise-type random potential. The quasi-periodic potential resembles the one in the 1D Aubry-Andre model.

We study the model numerically with the transfer matrix method and finite-size scaling technique. We obtain a qualitative phase diagram and analyze the critical behaviors near phase transitions. Our numerical data shows interesting results of the model. Firstly, we identify an interesting phase, which we call Directional Metal. In this phase, the wave function is localized along the Z direction, but extended in the X(Y) directions. In other words, it is insulating along Z direction while metallic along X(Y) directions. The system can be thought of as "a random stack of 2D metals" along the Z direction. In contrast to a standalone 2D metal in the orthogonal symmetry class, which is localized by infinitesimal disorder strength, the "random stack of 2D metals" remains metallic in X(Y) direction until the disorder is too strong. Secondly, we observe that, at certain strength of the quasi-periodic potential, the model displays a two-stage phase transition along Z direction as the disorder strength increases: it is localized at small disorder strength, then becomes delocalized, and eventually enters the localized phase again. Therefore, disorder delocalizes, instead of localizes, the state when it is small compared to the quasi-periodic potential, manifesting an interesting interplay between the two potentials. Thirdly, we also estimate the phase boundaries and the critical exponents. In particular, we observe that the conventional Anderson transition critical point becomes tricritical in our model.