## PhD research topic - Gianluca Panati

#### Research areas

Mathematical Quantum Theory

#### Keywords

Quantum transport theory, topological insulators, Chern insulators, Fu-Kane-Mele index

#### Research topic

The understanding of transport properties of quantum systems out of equilibrium is a crucial challenge in Mathematical Physics. A long term goal is to explain the conductivity properties of solids starting from first principles, as e.g. from the many-body Schrödinger equation governing the dynamics of electrons and nuclei. While this general goal appears to be beyond the horizon of present research, some mathematical results have been obtained for specific models, in particular for independent electrons in a periodic or ergodic random background. In particular, a specific transport coefficient (the charge conductance of independent electrons in a 2-dimensional periodic background) turns out to be proportional to a topological invariant of the systems, namely the Chern number of the Bloch bundle associated to the occupied states. This astonishing connection between transport properties and topology, first noticed in [1], paved the way to the now flourishing field of Chern insulators [2], see [3] for a recent mathematical viewpoint on the subject. We expect a similar transport-topology correspondence to appear also in the context of the recently synthesized time-reversal symmetric (TRS) topological insulators. We conjecture that a similar correspondence will relate the Fu-Kane-Mele topological index [4-5] to the spin transport coefficients. The Ph.D. project consists in investigating some of the intriguing mathematical features of the conjectured correspondence, either on the operator-theoretic or on the differential-geometric side (the latter project involves a collaboration with D.Fiorenza).

#### Bibliography

[1] Thouless, D.J.; Kohmoto, M.; Nightingale, M.P.; den Nijs, M. : Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982).

[2] Haldane, F.D.M. : Model for a Quantum Hall effect without Landau levels: condensed matter realization of the “parity anomaly”. Phys. Rev. Lett. 61, 2017 (1988).

[3] Monaco, D.; Panati, G.; Pisante, A.; Teufel, S. : Optimal Decay of Wannier functions in Chern and Quantum Hall Insulators, Commun. Math. Phys. 359, 61–100 (2018).

[4] Graf, G.M.; Porta, M. : Bulk-edge correspondence for two-dimensional topological insulators, Commun. Math. Phys. 324, 851–895 (2013).

[5] Fiorenza, D.; Monaco, D.; Panati, G. : Z_2 invariants of topological insulators as geometric obstructions. Commun. Math. Phys. 343, 1115–1157 (2016).