Numerical Integrations of Some Gauge Invariant Properties

This talk will provide an introduction to the gauge invariance property of some physically measurable quantities in condensed matter physics, and how to preserve such property in computations. We will first review some key concepts such as the Bloch wave and Berry phase. Then we will focus on numerical methods, in which we'll see polarization as an integral of differential 1-form in 1-space, Chern number as 1-form in 2-space, and the integral of abelian Chern-Simon 3-form in 3-space. Finally, comes the open problem: what’s the gauge-invariant discretization of the non-abelian Chern-Simons 3-from?