MERCREDI 16 DECEMBRE 2009
14 heures
Salle Séminaire 5
Centre de Physique Théorique
Marseille-Luminy

Giuseppe de Nittis
Trieste

Title: Topological quantum numbers arising from symmetries, and
Quantum Hall Effect

Abstract: Topological quantum numbers play a prominent role in many
fields of quantum physics. In particular topological quantum numbers
provide a very elegant explanation for the Quantum Hall effect. In
dealing with topological quantum numbers, some aspects need a deeper
analysis: 1) There exists a general framework in which topological
quantum numbers emerge? 2) There exists a computable procedure which
enables us to derive the underlying topology associated to a quantum
system? Is this topology (and the related topological invariants)
unique for a given system?

Under general assumptions, a natural topological structure emerges
from the existence of a symmetry group. This topology can be
extracted through a procedure that generalizes the usual
Bloch-Floquet transform and it is a "fingerprint" for the physical
system.

This procedure shows that the physics of a Bloch electron in a very
weak magnetic field (QHE in Hofstadter regime) and that of a Bloch
electron in a very strong magnetic field (QHE in Harper regime) are
apparently similar but intimately different. Although the algebraic
structure is the same, these systems are not unitary equivalent since
they produce different Chern numbers.

In summary, the "abstract" algebraic structure does not contain all
the physical information. The "concrete" Hilbert space
representations provide additional structures (symmetries) which
define a topological content, thus yielding a richer description than
the simple algebraic structure.