This award supports mathematical research on problems in equilibrium and non-equilibrium quantum statistical mechanics with the primary focus on the dynamics of quantum lattice models, which include quantum spin systems and quantum oscillator lattice systems. In condensed matter physics these types of models are commonly used to describe three types of degrees of freedom (separately or in combination): atomic oscillations in a crystal lattice, spin magnetic moments, and itinerant electrons. The same type of system, defined on a graph, is currently studied extensively in the theory of quantum information and computation. In that context, the degrees of freedom are qubits (or, more generally, qudits), which are the basic entities carrying quantum information, on which quantum algorithms are implemented as a sequence of unitary transformations. The project focuses on studying the dynamics (Hamiltonian or irreversible) of these systems and will lead to fundamental results such as propagation estimates of Lieb-Robinson type for new classes of systems. In the course of this project the investigator and his graduate students will also analyze the low-lying spectrum of quantum lattice models and investigate important properties such as the behavior of the spectral gap above the ground state and the ordering of the low-lying excitations in Heisenberg-type models. The mathematical results will furthermore be applied to the analysis of transfer-matrix-like objects that appear in other contexts, such as techniques of network analysis currently employed in biological applications.
This is fundamental research on a range of mathematical models for the dynamics of systems that involve a large number of interacting components. In this project, the main focus is on systems in which these components are microscopic and described by quantum mechanics. The investigator will work with graduate students, who in the process of their research will acquire a working knowledge of the state of the art in analytic and algebraic methods for quantum lattice systems. As fundamental research and technology advances, the results and techniques developed in this project are expected to have an impact in technological applications such as quantum information processing, quantum cryptography, quantum computing, and the modeling of electronic or spintronic devices that exploit the quantum properties of novel materials. Some of the mathematical techniques developed for this project will also be useful in the analysis of structured networks such as arise, for example, in systems biology.
The behavior of matter at low and very low temperatures is often dominated by quantum effects. Physicists have discovered new states of matter, sometimes called quantum states of matter, that not only can reveal to us new information about the fundamental laws governing the physical universe, but also have potential for interesting applications in new technologies. The prospect of building a reliable and large scale quantum memory using quantum states with topolgical order is a major driver for research on the low energy states of quantum spin models and other quantum many-body sytems. The main overarching goal of this project was to develop precise mathematical tools to study such systems. Under very general conditions, the speed of propagation of information and disturbances in a quantum lattice system satisfies an upper bound, ususally refered to as a Lieb-Robinson bound. In this project we proved Lieb-Robinson bounds for a more general class of systems than previously possible, including oscillator lattice systems. We then used these bounds to study the dynamics of quantum lattice systems and to derive mathematical consequences of a commonly used notion of gapped quantum phase. We found a new invariant of symmetry protected gapped phases in space one dimension. We also used Lieb-Robinson bounds to show that low-energy excitations of gapped quantum systems have a quasi-particle structure that may be helpful in the design of new, more powerful computaional methods. In this project we also applied some of the expertise and methods we originally developed to study quantum lattice systems, to the design of a new computational approach to analyzing gene expression networks. The research in this project was carried out with the participation of junior researchers, i.e., students and postdocs. They successfully contributed to the research of this project which in turn provided them with valuable experience and skills for their own careers as researchers in industry and academia.
