# Berry phases in condensed matter physics

#### Tuesday March 23 2021

**Conceptual Aspects of the Theory of Electric Polarization and Orbital Magnetization **

David Vanderbilt, Rutgers University

*In this talk I will take a conceptual approach, rather than a historical or mathematical one, to the "modern theory" of electric polarization and orbital magnetization. In particular, I will present a series of physical arguments that severely constrain the form of any such theory. For example, I will explain why we expect electric polarization to be well defined only up to a quantum, and discuss the theorem relating bulk polarization to surface charge in general terms. As for orbital magnetization, I will explain why this quantity is not subject to the same kind of quantum of indeterminacy that was found for the electric polarization. Only then will I introduce the mathematical concepts of Berry phase and Berry curvature, and briefly point the way to their utilization as the basis for the by-now well-established theories of electric polarization and orbital magnetization.*

**Electric Polarization, Orbital Magnetization, and Other Geometrical Observables**

Raffaele Resta, IOM-CNR Democritos, Trieste

*Electric polarization and orbital magnetization have a very similar definition at the textbook level; yet within quantum mechanics they are intensive geometrical observables of a very different nature. At the fundamental level polarization is essentially a one-dimensional quantity and orbital magnetization a two-dimensional one: the difference between them can be traced back to basic features of algebraic geometry in odd vs. even dimensions. One- and three-dimensional observables make sense in insulators only and are well defined only up to a quantum: the former case is polarization, the latter case occurs in magnetoelectric coupling. When a bounded sample is addressed, the quantum indeterminacy is fixed by the chosen termination. Two-dimensional observables exist in metals as well and are exempt from the quantum indeterminacy; their value does not depend on sample termination. Besides magnetization, I will discuss anomalous Hall conductivity in insulators and metals.*

**Introduction and Conceptual Aspects of the Theory of Electric Polarization and Orbital Magnetization D. Vanderbilt**

Rutgers University

**Electric Polarization, Orbital Magnetization, and Other Geometrical Observables R. Resta**

IOM-CNR Democritos, Trieste

**Interview and recollections D. Vanderbilt, R. Resta, N. Marzari**