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

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 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.

35 pages, 2 figures

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• Spin Hall conductivity in insulators with non-conserved spin
• A new approach to transport coefficients in the quantum spin Hall effect
• Localization of generalized Wannier bases implies Chern triviality in non-periodic insulators
• Approximate symmetries and conservation laws in topological insulators and associated $\mathbb{Z}$-invariants
• Improved energy estimates for a class of time-dependent perturbed Hamiltonians
• Středa formula for charge and spin currents
• Coherent electronic transport in periodic crystals
• From orbital magnetism to bulk-edge correspondence
• Adiabatic Evolution of Low-Temperature Many-Body Systems
• Purely linear response of the quantum Hall current to space-adiabatic perturbations
• Spin transport and lack of quantisation for time-reversal symmetric insulators on the honeycomb structure