Topological phases of matter and disordered systems

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NONTECHNICAL SUMMARYThis award supports theoretical research and education to study and understand the properties and possible phases of matter, particularly matter made of electrons in materials and in engineered materials structures, under different conditions. Topological phases of matter are theoretically expected to occur in various materials and materials systems. Important examples include a two-dimensional liquid of electrons trapped at engineered semiconductor interfaces in a high magnetic field, specific insulators known as topological insulators, and in some types of superconductors. Superconductors are a state of electrons in some materials that have distinctive quantum mechanical properties and can transport electricity without loss. Topological phases are characterized by properties that are robust, so they cannot be easily destroyed by imperfections and defects, but they are not connected with symmetries as would be the case in more traditional phases. Understanding topological phases requires the theory of quantum mechanics in an essential way. Experiments have discovered or confirmed the existence of some topological phases. There is great potential for these phases of matter to lead or contribute to new technologies, including spintronic devices which exploit not only on the electron's charge but also its intrinsic magnetic properties for their operation, and in future computer technologies based on the manipulation of quantum mechanical states. The PI will build on his previous work and further develop the theory of these phases, making theoretical predictions that can be tested in experiments or make it possible to identify topological phases as they occur. The specific areas of focus include methods for modeling topological phases as networks, and an unusual form of viscosity that arises in some of these systems.In other phases of matter, disorder, or the effects of random impurities or defects that are endemic in solids, can lead again to distinct phases of matter. This project will advance the theoretical understanding of these, especially "spin glasses." In any given example of one of these systems it is very difficult to guess what the lowest energy state is, because there are many low-energy alternatives. Quantifying the number of these is a topic of research. In a fundamental way, these glassy systems are related to many other applications of networks, and to optimization in computer programming. TECHNICAL SUMMARYThis award supports theoretical research and education on topological phases and disordered materials. Fundamental aspects of topological phases will be considered, as well as applications and diagnostic tools. The PI will investigate entanglement in the context of topological phases. The PI plans to continue work on tensor network states for chiral topological phases of matter; tensor networks have been fruitful in, for example, classifying topological phases in one dimension. He aims to engage the problem of transport phenomena that are related to basic topological properties such as the "central charge," and the possibility of measuring the Hall viscosity in quantum Hall systems. On disordered systems: the PI will use replica symmetry breaking theory to produce results on the number of pure states in the spin-glass phase at low temperatures in the short-range Ising spin glass model, and others. This relates to basic conceptual issues in spin glass theory concerning the number and organization of pure states, and is a subject of controversy.