Mathematics Colloquia and Seminars

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Recently, Anton Kapustin and Nikita Sopenko showed that a smooth parametrized family of groundstates of gapped Hamiltonians gives rise to a cohomology class in the parameter space (the higher Berry class). I will begin by summarizing the research program that motivates the study of these classes and their equivariant counterparts. I will then describe recent joint work with the above two authors concerning two important instances of the higher Berry phase: the case d=1 with no symmetries, and the case d=2 with U(1) symmetry. I will argue that the first invariant is integer-valued, and then show that these invariants are related by an argument analogous to Laughlin’s proof of Hall conductance quantization, which in particular leads to a proof that the second invariant is also quantized.