Topological and Disordered Phases of Matter

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NONTECHNICAL SUMMARYThe award supports theoretical research and education on profound questions that arise in the study of modern materials. In a solid material, there are many electrons, say one for each atom in the material, and they may remain active at low temperatures. In the quantum mechanical world of the electron, new phenomena can arise as a consequence of interactions between electrons that lead to correlations in their motion. They can also lead to the formation of a state of electronic matter with no analog in the familiar world governed by classical mechanics. A feature of this state is quantum entanglement, meaning very long-range correlations between electrons. The quantum phases of matter under investigation, known as topological phases of matter, involve non-trivial long-range entanglement.Various ways of theoretically studying such states of matter are known. Some involve techniques called tensor networks. The PI will explore the intrinsic limitations of these methods. Other approaches involve methods from quantum field theory and from the branch of mathematics known as algebraic topology. The PI will bring his experience with these to bear on the problems. Research on these topics may contribute to quantum information science, where there are proposals that utilize topological phases of matter to perform quantum computation. There is also the possibility that these unusual states of matter could be applied as novel materials in other technological applications.The PI will also study another distinct area of research, disordered systems. This means that the degrees of freedom, for example atoms in glass, that interact with one another do so with strengths that differ from one place to another, and are modeled as random numbers. This aspect, which models the lack of perfection intrinsic to real materials, leads to some very difficult questions; a particular class of examples are called "spin glasses". Finding the lowest energy state may be a computationally hard problem. Even at a statistical level, it may be very hard to characterize the properties of a spin glass. The PI will continue the use of rigorous mathematical approaches to advance understanding of this problem. TECHNICAL SUMMARYThe award supports theoretical research and education to study low-temperature phenomena in both classical and quantum condensed matter systems, usually in lattice systems.On the quantum side, the goal of these studies is to understand the topological phases of matter in these systems in greater depth. An approach of particular interest is tensor network states. The PI will investigate whether these methods can be successfully applied to topological phases of matter in more than one dimension, or whether they possess intrinsic (topological) limitations that make these applications impossible. The theoretical techniques that will be used include concepts from algebraic topology, such as K-theory, and quantum information theory, as well as many-body and quantum-field theory. The PI will also study disordered systems such as classical spin glasses. Outstanding controversial issues include the basic question of whether or not there are many pure states in the low temperature phase in the limit of an infinite system; this is the question of replica symmetry breaking. The use of rigorous mathematical analysis may be the only approach that can sidestep disputes about interpretation that have dogged non-rigorous and computational approaches. The PI uses Newman and Stein's metastate approach to provide a rigorous framework for attacking the problem, but may also uses replica methods to gain insight and to make connections with non-rigorous methods or experiments.In both areas, while the emphasis is on fundamental theoretical issues, experimental relevance is always in view.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.