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Spin conductance and spin conductivity in topological insulators: analysis of kubo-like terms. (English) Zbl 07063411
Summary: 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 2$$d$$ time-reversal-symmetric topological insulators. Inspired by the Kubo theory of charge transport, and by using a proper definition of the spin current operator (Shi et al. in Phys Rev Lett 96:076604, 2006), 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. 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 its directional version.
##### MSC:
 47A General theory of linear operators 81T Quantum field theory; related classical field theories 81Q General mathematical topics and methods in quantum theory 81U Quantum scattering theory 47L Linear spaces and algebras of operators
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