Localization and Diffusion in Open and Many Body Quantum Systems

Michigan State University, East Lansing MI

Investigators

Abstract

The main focus of this project is the study of the effects of disorder on quantum systems. In a broader context, the work draws inspiration from the basic scientific question: "What are the effects of disorder?" This is a fundamental question relevant to any scientific model, even one in which disorder is not explicitly included. After all, any real world system is subject to a small amount of noise, and experience shows that even weak disorder may have a profound effect on the behavior of the system. The equations studied in this project arise in the theory of disordered materials, but are of general interest because of the fundamental nature of both wave motion and disorder. Progress in understanding the solutions to these equations will improve basic understanding of models of theoretical physics and applied mathematics. In addition, a central goal of the research is pedagogical: to introduce students to a fundamental subject and convey to them that mathematical and theoretical physics are vibrant, growing fields. The project will proceed through a program of research on the effects of disorder in physical models. The main focus of the research is in understanding the evolution of quantum systems subject to intrinsic disorder. When disorder is weak, the expectation is that a diffusive evolution of the wave function results. At stronger disorder, it is known that localization of the wave function can occur, although the exact nature of localization in many body and open quantum systems is poorly understood. Two key goals are 1) to analyze the diffusion of waves in a weakly disordered medium over arbitrarily long times, and 2) to clarify the nature of localization in many body models. In recent years the PI and various post-doc and student collaborators have considered the problem of wave diffusion in time-dependent random media, with the time dependence generated by a Markov process. For such models the diffusive propagation, e.g., for the tight binding Schrodinger equation, can be established by spectral analysis. One aim of the present project is to approach the time independent equation as a perturbation of these time dependent equations, which have the virtue of sharing the expected qualitative behavior. In parallel, we will study disorder induced localization in polaron type models of a interacting systems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.