A New Approach to Transport Coefficients in the Quantum Spin Hall Effect
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Annales Henri Poincar´ e
A New Approach to Transport Coefficients in the Quantum Spin Hall Effect Giovanna Marcelli, Gianluca Panati
and Stefan Teufel
Abstract. We investigate some foundational issues in the quantum theory of spin transport, in the general case when the unperturbed Hamiltonian operator H0 does not commute with the spin operator in view of Rashba interactions, as in the typical models for the quantum spin Hall effect. A gapped periodic one-particle Hamiltonian H0 is perturbed by adding a constant electric field of intensity ε 1 in the j-th direction, and the linear response in terms of a S-current in the i-th direction is computed, where S is a generalized spin operator. We derive a general formula for the spin conductivity that covers both the choice of the conventional and of the proper spin current operator. We investigate the independence of the spin conductivity from the choice of the fundamental cell (unit cell consistency), and we isolate a subclass of discrete periodic models where the conventional and the proper S-conductivity agree, thus showing that the controversy about the choice of the spin current operator is immaterial as far as models in this class are concerned. As a consequence of the general theory, we obtain that whenever the spin is (almost) conserved, the spin conductivity is (approximately) equal to the spin-Chern number. The method relies on the characterization of a non-equilibrium almoststationary state (NEASS), which well approximates the physical state of the system (in the sense of space-adiabatic perturbation theory) and allows moreover to compute the response of the adiabatic S-current as the trace per unit volume of the S-current operator times the NEASS. This technique can be applied in a general framework, which includes both discrete and continuum models.
1. Introduction and Main Results The aim of this paper is to shed some light on the theory of spin transport in gapped (non-interacting) fermionic systems, a problem which is highly relevant to the research on topological insulators (see the end of Sect. 1.2).
G. Marcelli et al.
Ann. Henri Poincar´e
The theory of spin transport, as compared to charge transport, is still in a preliminary stage. First, despite two decades of scientific debate, no general consensus has been reached yet about the correct form of the operator representing the spin current density. Denoting by H0 the unperturbed Hamiltonian operator, by X = (X1 , . . . , Xd ) the position operator and by Sz the operator representing the z-component of the spin, one may consider1 (i) the “conventional” spin current operator 1 z := i[H0 , X] Sz + iSz [H0 , X] (1.1) JSconv 2 which has been used e.g. in [58,60,64]; (ii) the “proper” spin current operator z := i[H0 , XSz ] JSprop
(1.2)
proposed in [62,72]. Whenever [H0 , Sz ] = 0, the two above definitions agree and the theory of spin transport reduces to the theory of charge transport. However, in general [H0 , Sz ] = 0 in topological insulators, as it happens e.g. in the model proposed by Kane and
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