Dynamical and Quantum Phase Transitions with Tridiagonal Matrices

Abstract

The purpose of this research is to better understand the static and dynamical properties of quantum systems. What makes a quantum system a metal, a material in which particles can move easily? What makes it an insulator, a material in which particles are trapped to confined regions of space? What triggers the transition from a metallic phase to an insulating phase? To answer these questions, we consider quantum systems described by Hamiltonian matrices that have a tridiagonal form. A tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the first diagonal below the main diagonal, and the first diagonal above the main diagonal. We study how the eigenvalues and eigenstates of these matrices depend on the parameters of the Hamiltonian and use these results to predict the dynamical behavior of the system. The advantage of using tridiagonal matrices is that they are easy to study numerically and in some cases can even lead to analytical solutions. Despite their simplicity, they model realistic systems of nature and provide a pedagogical framework to learn about concepts and tools of quantum mechanics which include: quantum spins, quantum superposition, delocalization measures, the Schrödinger equation, and quantum phase transitions.