Geometric and electromagnetic response and anomalies in topological phases of matter

topological phases;quantum Hall;SPT phases;Hall viscosity;anomalies

Lapa, Matthew F.

关键词: topological phases;    quantum Hall;    SPT phases;    Hall viscosity;    anomalies;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/100928/LAPA-DISSERTATION-2018.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

In this thesis we characterize various topological phases of matter by studying their response to external perturbations. In the first half of the thesis we study incompressible fluid phases with odd, or Hall, viscosity. This kind of viscosity was originally discovered in quantum Hall fluids, where it can be computed by studying the stress response of the quantum Hall fluid to time-dependent area-preserving deformations, which are a particular example of a geometric perturbation. In Chapter 2 we study classical two-dimensional fluids with Hall viscosity in their own right. In particular, we study the physics of a swimmer immersed in such a fluid, and in the low Reynolds number regime in which the effects of conventional viscous forces outweigh the effects of inertial forces. There we find that the Hall viscosity leads to a number of striking effects on the swimmer's motion, for example the swimmer can rotate itself only by changing its area. In Chapter 3 we study Hall viscosity directly in the context of the quantum Hall effect. There we compute the Hall viscosity in the Chern-Simons matrix model of the Laughlin states, which is a certain regularization of the noncommutative Chern-Simons theory of these states proposed by Susskind. Our calculations show that these noncommutative theories are able to describe the most important contribution, namely the guiding center contribution, to the Hall viscosity (and other geometric response properties) of the Laughlin states.In the second half of the thesis we study electromagnetic response and anomalies in two families of bosonic symmetry-protected topological phases. These are the bosonic integer quantum Hall (BIQH) and bosonic topological insulator (BTI) phases. Although these phases were originally defined in three and four spacetime dimensions, respectively, we generalize them to all higher spacetime dimensions (odd dimensions for BIQH and even for BTI). We then study the bulk electromagnetic response of these theories as well as the anomalies in quantum field theories which can describe the boundaries of these phases. Our results include the discovery of an interesting quantization of the response coefficients (analogous to Hall conductance) for these phases, which depends explicitly on the spacetime dimension. We also study perturbative and global anomalies in the boundary theories for the BIQH and BTI phases, and we prove that the anomalies we compute are robust to a large set of deformations of the boundary theories which preserve the symmetry of the BIQH and BTI phases. We provide an introduction to perturbative and global anomalies in Chapter 1 of the thesis so that readers can follow the discussion of anomalies for the BIQH and BTI states in Chapter 5.