Spin Conductance and Spin Conductivity in Topological Insulators: Analysis of Kubo-Like Terms

  • Giovanna Marcelli
  • Gianluca PanatiEmail author
  • Clément Tauber


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 (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.


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We are indebted to Gian Michele Graf for sharing with us his insight into the mathematics of the QHE on the occasion of the Winter School “The Mathematics of Topological Insulators in Naples”, organized in the framework of the Cond-Math project (, and for pointing out to us some relevant references. We are grateful to Domenico Monaco and Stefan Teufel for many useful discussions, and to Massimo Moscolari for a careful reading of the manuscript.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica“La Sapienza” Università di RomaRomeItaly
  2. 2.Fachbereich MathematikEberhard Karls Universität TübingenTübingenGermany
  3. 3.Institute for Theoretical PhysicsETH ZürichZurichSwitzerland

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