This CAREER award supports theoretical research aimed at improving our understanding of topological phases of matter. Topological phases, such as fractional quantum Hall states and topological insulators, pose a fundamental conceptual challenge because they cannot be characterized using order parameters or symmetry breaking. Consequently, new concepts and tools need to be developed to understand these systems. The research supported by this award will focus on two aspects of these novel phases. First, the PI will investigate the general relationship between bulk and boundary properties of topological phases of matter. This 'bulk-boundary correspondence' is important both for theory and experiment since one of the easiest ways to probe these systems is via transport measurements at the boundary. Second, the PI will investigate concrete microscopic models for these phases, with the dual aims of (a) identifying experimental realizations and (b) understanding the deeper structure of these states and the limitations of numerical approaches. Some of the main topics that will be explored include:
1) Conditions for protected edge modes for 2D topological phases,
2) Exactly soluble lattice models for 2D topological phases and their limitations,
3) Quasi-realistic models for bosonic integer quantum Hall states,
4) Bulk and surface properties of 3D bosonic symmetry-protected topological phases.
In addition to research, this award supports education in two ways. First, the PI will create a new graduate class on 'Topology and Physics.' Importantly, this class will cover topological phenomena in both condensed matter and high energy physics with the goal of attracting a mixture of students from both groups and thereby facilitating interactions between the two subfields. Second, the PI will supervise summer research projects for undergraduates. Other benefits of this award will come from outreach activities, including several talks on quantum computing and superconductivity that will be given to the undergraduate students participating in the summer REU program at the University of Chicago.
NONTECHNICAL SUMMARY
This CAREER award supports theoretical research aimed at improving our understanding of exotic phases of matter known as 'topological phases.' Just as a collection of water molecules can behave like a solid, a liquid, or a gas depending on the temperature and pressure, the electrons in a material can organize into metals, insulators, superconductors and other phases depending on various parameters. One of the basic goals of condensed matter physics is to characterize and analyze the different phases that can form in these many-electron or many-particle systems. More than 70 years ago, Lev Landau developed a powerful framework for answering these types of questions. His theory, based on the concept of symmetry, was able to explain a vast array of phases -- everything from superconductors to magnets. Despite these successes it is now clear that, in some materials, quantum effects can lead to new phases of matter that are fundamentally beyond the Landau paradigm. The most dramatic examples are known as 'topological phases.' These phases have a rich internal structure, but unlike conventional phases like magnets or superconductors, this structure has nothing to do with symmetry. Instead, the defining features of these phases have a topological character. As a result, entirely new concepts and tools need to be developed to understand these systems. The research supported by this award aims to develop this machinery with the ultimate goal of building a general theoretical framework, analogous to Landau theory, for analyzing these novel phases of matter.
In addition to research, this award supports education in two ways. First, the PI will create a new graduate class on 'Topology and Physics.' Importantly, this class will cover topological phenomena in both condensed matter and high energy physics with the goal of attracting a mixture of students from both groups and thereby facilitating interactions between the two subfields. Second, the PI will supervise summer research projects for undergraduates. Other benefits of this proposal will come from outreach activities, including several talks on quantum computing and superconductivity that will be given to the undergraduate students participating in the summer REU program at the University of Chicago.
