# Geometry and topology in condensed-matter physics.

Resta R.

Relazione generale
II - Fisica della materia
Aula GSSI Rettorato - Auditorium - Mercoledì 25 h 14:30 - 15:30
The paradigmatic geometrical observable is the macroscopic electric polarization $P$ of a crystalline insulator: since the early 1990s it is known that $P$ is a geometric phase (Berry phase) of the ground-state wave function. The geometrical nature of several other observables has been elucidated over the years: most notably orbital magnetization and anomalous Hall conductivity. In some special cases a geometrical observable is quantized, and becomes therefore topological: extremely robust with respect to perturbations, and measurable in principle with infinite precision. In this contribution I will start explaining what geometrical means in quantum mechanics. Then I will address $P$ giving a nontechnical outline of the modern theory, and explaining its most counterintuitive feature: when expressed in dimensionless units, $P$ is a phase angle and is therefore defined modulo $2\pi$. At variance with $P$. orbital magnetization and anomalous Hall conductivity are exempt from any $2\pi$ ambiguity; geometrical observables in this class also admit a dual formulation in coordinate space and become local: they can be defined and computed even for inhomogeneous systems ($e.g.$ hereojunctions) in a spatially resolved form. A similar approach is impossible for $P$: one can define a local magnetization, but not a local polarization.