Dynamics, Ground States, and Elementary Excitations of Quantum Many-Body Systems

University Of California-Davis, Davis CA

Investigators

Abstract

Miniaturization of electronics has been the driving force of the ever increasing power of computers, whether expressed as units of performance per unit of volume, or per unit of energy consumed, or per dollar expended. Miniaturization will reach its ultimate limit when the size of individual components in electronic devices approaches the size of a single atom. Unless we succeed in implementing a new paradigm of computation, called Quantum Computation, the growth in computational efficiency, which is often referred to as Moore's Law, will come to an end. One of the most promising research directions to implement the ideas of Quantum Computation is the development of topological materials. Topological materials have the capacity to store quantum information, which under ordinary conditions is subject to fast deterioration due to a process called decoherence. In this project the PI and his collaborators will study mathematical models of topological materials. The main goal is to better understand the factors that determine the robustness of a quantum memory based on topological materials. The project will address three questions that are crucial for the possible application of topologically ordered materials to the development of reliable quantum computation devices. First, what determines the rate with which the spectral gap at critical points vanishes as the system size increases? Second, under what conditions is the anyonic excitation spectrum of systems with topologically ordered ground states stable under sufficiently small perturbations? Third, what is the effect of randomness (such as occurs in doped materials) on the structure of the ground state and low-lying excitations of such systems? The project will also investigate the dynamical behavior of the model systems. Questions regarding the energy of low-lying excitations above the ground state, i.e., the spectral gap, are central to many issues in quantum many-body physics. The goal of the project is to significantly extend the range of applicability of methods to estimate the spectral gap in at least two directions. One is the situation where the gap vanishes with increasing system size (critical points or regions). The second aim is to develop methods to analyze the gap of systems in two or more dimensions, which has so far only been achieved in a few special cases. Further, in the case of disordered systems the spectral gap may close, but a so-called mobility gap may play a very similar role and another goal of the project is to extend current methods to situations where the spectral gap is replaced by a mobility gap. Techniques from representation theory, analysis, and probability will be used when available and new methods will be developed when needed.