Geometric and Topological Phenomena in Condensed Matter Systems

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Technical SummaryThis award supports theoretical research and education on topological insulators. The PI will show whether crystal symmetries effect a refinement of the ten-fold way classification of the bulk bands. A large portion of the research efforts will be directed at exploring the analogies between the bulk-surface connection in topological insulators and their particle and gravitational physics analogs. In particular, the solid-state systems suggest new ways of thinking about chiral and gauge anomalies in quantum field theory. This will lead to an alternative method for calculation of transport coefficients in quark-gluon plasmas. In particular some curious relations between the coefficients in anomaly polynomials and those appearing in the well-known Sommerfeld's expansion will be explored. A key ingredient in this exploration is a set of tools developed for the description of the dynamics and quantum topology of a quantum spin as a classical mechanical system. This study will show whether these "classical" tools can be extended to derive the solid-state version of the gravitational anomaly which can be used to calculate the coefficient of the thermal analogue of the quantum Hall effect. The proposed research program will have impact beyond the field of theoretical condensed matter physics by providing guidelines for building a future generation of solid-state electronic devices that might lead to quantum computers and by allowing exploration of the speculative particle physics at a much lower cost than building large accelerators. The PI will develop a web portal to expose students and the public to recent developments in condensed matter physics. Non-Technical Summary This award supports theoretical research and education on topological insulators. A complete understanding of the universe is summarized in the "standard model" of particle physics. In this model the fundamental bits of matter interact through gauge fields which are generalizations of electricity and magnetism. Some of these interactions, the "weak" force, possess the curious feature that they only affect the left-handed parts of matter. A mathematically natural way for this handedness to arise is for matter to be trapped on a surface embedded in some higher dimensional space. This notion sounds as if it belongs in a science fiction story, but in recent years quite down-to-earth studies have shown that certain crystalline insulators containing heavy elements behave in exactly this way. These "topological insulators" are only insulators on their insides. On their surfaces, electrons are free to move and conduct electricity, but they do so in a manner that forms a precise analogue of the way that quarks interact. These conducting states are of great interest, partly for practical reasons in building new generations of solid-state electronic devices that might lead to quantum computers, but also because the analogy with particle physics allows exploration of speculative particle physics at a much lower cost than building large accelerators. The PI will develop a web portal to expose students and the public to recent developments in condensed matter physics.