Abstract: The last few decades witnessed an increasing interest, among solid state physicists, for physical phenomena having a topological origin. This interest traces back to the milestone paper by Thouless, Kohmoto, Nightingale and den Nijson the Quantum Hall Effect (QHE), and involves the seminal papers by Fu, Kane and Mele concerning the Quantum Spin Hall Effect (QSHE) to further developments in the flourishing field of topological insulators.
As well known, in the QHE a topological invariant (Chern number) is related to an observable quantity, the charge (Hall) conductance. By analogy, in the context of the QSHE, one would like to connect the relevant topological invariant (Fu-Kane-Mele index) to a macroscopically observable quantity. The natural candidates are spin conductance and spin conductivity, which in general are not equivalent. As a paradigmatic case, we will analyse Kubo-like terms for spin conductance and spin conductivity in a discrete two-dimensional model. In view of the continuity equation for spin transport, derived from the first principles of Quantum Mechanics, our physical intuition suggests that spin conductance equals the spin conductivity whenever the spin torque mesoscopic mass vanishes. Indeed, we will prove the previous statement, as far as Kubo-like terms are concerned. To achieve the goal we first introduce the definition of the principal value trace and of the jj-principal value trace (for j∈{1,2}j∈{1,2}), and then develop a suitable machinery to compute them.
The seminar is based on joint work with Gianluca Panati and Clément Tauber.