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Electrons on a two-dimensional (2$d$) lattice which is exposed to a stronguniform magnetic field show intriguing physical phenomena. The spectrum of suchsystems exhibits a complex (multi-)band structure known as Hofstadter'sbutterfly. For fillings at which the system is a band insulator one observes aquantized integer-valued Hall conductivity $\sigma_{xy}$ corresponding to atopological invariant, the first Chern number $\mathcal{C}_1$. This is robustagainst many-body interactions as long as no changes in the gap structureoccur. Strictly speaking, this stability holds only at zero temperatures $T$while for $T>0$ correlation effects have to be taken into account. In thiswork, we address this question by presenting a dynamical mean field theory(DMFT) study of the Hubbard model in a uniform magnetic field. The inclusion oflocal correlations at finite temperature leads to (i) a shrinking of theinteger plateaus of $\sigma_{xy}$ as a function of the chemical potential and(ii) eventually to a deviation from these integer values. We demonstrate thatthese effects can be related to a correlation-driven narrowing and filling ofthe band gap, respectively.

Fate of a topological invariant for correlated lattice electrons at finite temperature