Spin Conductance and Spin Conductivity in Topological Insulators: Analysis of Kubo-like terms

TL;DR

This paper rigorously analyzes spin conductance and conductivity in 2D topological insulators, establishing conditions under which these quantities are well-defined and equal, with implications for understanding topological invariants.

Contribution

It introduces a rigorous mathematical framework for spin transport coefficients in 2D insulators and proves their equality under specific conditions, linking them to topological properties.

Findings

G_K^{s_z} and rm-2_K^{s_z} are well-defined for certain Hamiltonians.

The equality G_K^{s_z} = rm-2_K^{s_z} holds under specific conditions.

Vanishing of the mesoscopic average of the spin-torque response is crucial for the equality.

Abstract

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, we define the Kubo-like spin conductance $G_K^{s_z}$ and spin conductivity $\sigma_K^{s_z}$. We prove that for any gapped, periodic, near-sighted discrete Hamiltonian, the above quantities are mathematically well-defined and the equality $G_K^{s_z} = \sigma_K^{s_z}$ 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 Schr\"odinger 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.