Středa formula for charge and spin currents

Abstract

We consider a 2-dimensional Bloch--Landau--Pauli Hamiltonian for a spinful electron in a constant magnetic field subject to a periodic background potential. Assuming that the $z$-component of the spin operator is conserved, we compute the linear response of the associated spin density of states to a small change in the magnetic field, and identify it with the spin Hall conductivity. This response is in the form of a spin Chern marker, which is in general quantized to a half-integer, and to an integer under the further assumption of time-reversal symmetry. Our result is thus a generalization to the context of the quantum spin Hall effect to the well-known formula by Středa, which is formulated instead for charge transport.

arXiv comment10 pages. Improved exposition, references added and minor corrections. Final version matches published paper on Reviews in Mathematical Physics

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Cited by in corpus (3)

• From charge to spin: analogies and differences in quantum transport coefficients
• From orbital magnetism to bulk-edge correspondence
• Incipient quantum spin Hall insulator under strong correlations