The Berry phase is a well-known phenomenon in quantum mechanics with many profound implications. It describes the response of the phase of the wavefunction to the adiabatic evolution of system parameters, defining a U(1) connection on the parameter space. In many-body systems described by quantum field theory, we may also allow the parameters to vary in space, and we find a higher group connection generalizing the Berry phase. This connection also describes phenomena such as the Thouless pump and its generalizations. It allows us to constrain the global structure of phase diagrams by probing non-contractible cycles in the space of quantum field theories. In a typical phase diagram drawn in R^n, these cycles surround topologically-protected critical loci called diabolical points, in analogy to the quantum mechanical singularities which act as "monopoles" for the Berry connection. I will discuss these concepts in more detail, as well as a bulk-boundary correspondence and some recent applications to phase diagrams of topologically ordered systems. This talk is based on https://arxiv.org/abs/2004.10758 w/ Po-Shen Hsin and Anton Kapustin https://arxiv.org/abs/2110.07599 and its sequel, 2110.xxxx w/ Nathanan Tantivasadakarn, Ashvin Vishwanath, and Ruben Verresen.
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Vasily Pestun & Slava Rychkov