We investigate some foundational issues in the quantum theory of spin transport, in the general case when the unperturbed Hamiltonian operator $$H_0$$ H 0 does not commute with the spin operator in view of Rashba interactions, as in the typical models for the quantum spin Hall effect. A gapped periodic one-particle Hamiltonian $$H_0$$ H 0 is perturbed by adding a constant electric field of intensity $$\varepsilon \ll 1$$ ε ≪ 1 in the j-th direction, and the linear response in terms of a S-current in the i-th direction is computed, where S is a generalized spin operator. We derive a general formula for the spin conductivity that covers both the choice of the conventional and of the proper spin current operator. We investigate the independence of the spin conductivity from the choice of the fundamental cell (unit cell consistency), and we isolate a subclass of discrete periodic models where the conventional and the proper S-conductivity agree, thus showing that the controversy about the choice of the spin current operator is immaterial as far as models in this class are concerned. As a consequence of the general theory, we obtain that whenever the spin is (almost) conserved, the spin conductivity is (approximately) equal to the spin-Chern number. The method relies on the characterization of a non-equilibrium almost-stationary state (NEASS), which well approximates the physical state of the system (in the sense of space-adiabatic perturbation theory) and allows moreover to compute the response of the adiabatic S-current as the trace per unit volume of the S-current operator times the NEASS. This technique can be applied in a general framework, which includes both discrete and continuum models.

We investigate spin transport in 2-dimensional insulators, with the long-term goal of establishing whether any of the transport coefficients corresponds to the Fu-Kane-Mele index which characterizes 2d time-reversal-symmetric topological insulators.Inspired by the Kubo theory of charge transport, and by using a proper definition of the spin current operator [SZXN], we define the Kubo-like spin conductance G sz K and spin conductivity σ sz K . We prove that for any gapped, periodic, near-sighted discrete Hamiltonian, the above quantities are mathematically welldefined and the equality G sz K = σ sz K holds true. Moreover, we argue that the physically relevant condition to obtain the equality above is the vanishing of the mesoscopic average of the spin-torque response, which holds true under our hypotheses on the Hamiltonian operator. This vanishing condition might be relevant in view of further extensions of the result, e. g. to ergodic random discrete Hamiltonians or to Schrödinger operators on the continuum. A central role in the proof is played by the trace per unit volume and by two generalizations of the trace, the principal value trace and it directional version.

Gapped periodic quantum systems exhibit an interesting Localization Dichotomy, which emerges when one looks at the localization of the optimally localized Wannier functions associated to the Bloch bands below the gap. As recently proved, either these Wannier functions are exponentially localized, as it happens whenever the Hamiltonian operator is time-reversal symmetric, or they are delocalized in the sense that the expectation value of |x| 2 diverges. Intermediate regimes are forbidden.Following the lesson of our Maestro, to whom this contribution is gratefully dedicated, we find useful to explain this subtle mathematical phenomenon in the simplest possible model, namely the discrete model proposed by Haldane [15]. We include a pedagogical introduction to the model and we explain its Localization Dichotomy by explicit analytical arguments. We then introduce the reader to the more general, model-independent version of the dichotomy proved in [25], and finally we announce further generalizations to non-periodic models.Date: September 6, 2019. Extended version of the paper published in Rend. Mat. Appl. 39, 307-327 (2018). In comparison with the published version, we added some details and the whole Chapter 5.2010 Mathematics Subject Classification. 81Q70, 81V70, 47A56, 47A10.

We investigate the relation between the localization of generalized Wannier bases and the topological properties of two-dimensional gapped quantum systems of independent electrons in a disordered background, including magnetic fields, as in the case of Chern insulators and quantum Hall systems. We prove that the existence of a well-localized generalized Wannier basis for the Fermi projection implies the vanishing of the Chern character, which is proportional to the Hall conductivity in the linear response regime. Moreover, we state a localization dichotomy conjecture for general non-periodic gapped quantum systems.