Massimo Moscolari. Aalborg University

Title: Beyond Diophantine Wannier diagrams: gap labelling for Bloch-Landau Hamiltonians

Place: Pontificia Universidad Católica, Facultad de Matemáticas (Campus San Joaquin), Sala 5

Abstract:

In 1978 Wannier discovered a Diophantine relation expressing the integrated density of states of a gapped group of bands of the Hofstadter Hamiltonian as a linear function of the magnetic field flux with integer slope. I will show how to extend this relation to a gap labelling theorem for any 2D Bloch-Landau Hamiltonian operator and to certain non-covariant systems having slowly varying magnetic fields. The integer slope will be interpreted as the Chern character of the projection onto the space of occupied states. The talk is based on a joint work with H. Cornean and D. Monaco.

# Category: seminar FisMat

This is the regular Mathematical Physics Seminar at the Mathematics Department of the Pontificia Universidad Católica de Chile.

## FisMat Seminar, April 24, 15:45

Svetlana Jitomirskaya. University of California, Irvine

Title: Cantor spectrum of a model of graphene in magnetic field

Place: Pontificia Universidad Católica de Chile, Facultad de Matemáticas (Campus San Joaquin), Sala 5

Abstract:

We consider a quantum graph as a model of graphene in magnetic fields and give a complete analysis of the spectrum, for all constant fluxes. In particular, we show that if the reduced magnetic flux through a honeycomb is irrational, the continuous spectrum is an unbounded Cantor set of Lebesgue measure zero and Hausdorff dimension bounded by 1/2.

Based on joint works with S. Becker, R. Han, and also I. Krasovsky

## FisMat Seminar, April 17, 15:45

Walter de Siqueira Pedra. University of São Paulo

Thermodynamical Stability and Dynamics of Lattice Fermions with Mean-Field Interactions

Place: Pontificia Universidad Católica, Facultad Matemáticas (Campus San Joaquin) Sala 5

Abstract:

For lattice fermions we study the thermodynamic limit of the time evolution of observables when the corresponding finite-volume Hamiltonians contain mean-field terms (like, e.g., the BCS model). It is well-known that, in general, this limit does not exist in the sense of the norm of observables, but may exist in the strong operator topology associated to a well-chosen representation of the algebra of observables. We proved that this is always the case for any cyclic representation associated to an invariant minimizer of the free energy density, if the Hamiltonians are invariant under translations. Our proof uses previous results on the structure of states minimizing the free energy density of mean-field models along with Lieb-Robinson bounds for the corresponding families of finite-volume time evolutions. This is a joint work with Jean-Bernard Bru, Sébastien Breteaux and Rafael Miada.