Purely linear response of the quantum Hall current to space-adiabatic perturbations

Abstract

Using recently developed tools from space-adiabatic perturbation theory, in particular the construction of a non-equilibrium almost stationary state, we give a new proof that the Kubo formula for the Hall conductivity remains valid beyond the linear response regime. In particular, we prove that, in quantum Hall systems and Chern insulators, the transverse response current is quantized up to any order in the strength of the inducing electric field. The latter is introduced as a perturbation to a periodic, spectrally gapped equilibrium Hamiltonian by means of a linear potential; existing proofs of the exactness of Kubo formula rely instead on a time-dependent magnetic potential. The result applies to both continuum and discrete crystalline systems modelling the quantum (anomalous) Hall effect.

From charge to spin: analogies and differences in quantum transport coefficients

“…We therefore propose to avoid the time modulation of the perturbing field altogether, and follow ideas inspired by what is known in physics and chemistry as Rayleigh-Schrödinger perturbation theory [24], and which have been made rigorous in the mathematical physics community in the context of space-adiabatic perturbation theory (see [23,30] and references therein). This alternative method has been advanced by S. Teufel and collaborators in a series of works [20,31,9,10] and applied in the present context of quantum transport also in collaboration with the authors [18,15,19]. The approach allows to identify a so-called non-equilibrium almost stationary state (NEASS), which in the adiabatic regime well approximates the physical state once the dynamical switching drives the system out of equilibrium, but which can be defined without resorting to a time-dependent modulation of the perturbing field (see Theorem 2.1 for a precise stament).…”

“…On the other hand, the construction of the NEASS is fairly explicit, and allows in particular to expand it in powers of the strength ε of the external field around the unperturbed equilibrium state (which, under a spectral gap assumption, we take to be the Fermi projection onto the occupied energy bands of the reference system). For charge currents, the use of the NEASS allows in particular to recover the well-known Kubo formula (sometimes called Kubo-Chern formula or doublecommutator formula, DCF) for the Hall conductivity, and to prove its validity also beyond linear response (see [15] and Theorem 3.1 below).…”

“…Rashba spin-orbit coupling is in particular responsible for the presence of a spin torque, which prevents the spin current to satisfy a (sourceless) microscopic continuity equation [27]. As a further application of the construction of the NEASS to all orders developed in [15], and as an original contribution of this paper, we show that there is no mesoscopic spin torque in the NEASS up to any order in ε (that is, the trace per unit volume which measures the expectation of the spin torque operator vanishes in the NEASS faster than any power of ε). Finally, we comment in Section 4 on the relation between the intensive notion of conductivity versus the extensive notion of conductance, again for spin transport.…”

“…While we choose to adopt a class of one-particle, periodic model Hamiltonians throughout the paper, many of the mathematical objects we will describe have been studied and generalized in other frameworks: in particular, for example, quantum transport and the Hall effect have been investigated in presence of (ergodic) disorder, while the construction of the NEASS mentioned previously has been conducted also for models of interacting fermions on a lattice. At any rate, the reader is referred to [18,15] and references therein for a more complete comparison with the existing literature and for many details which we will skip over in order to avoid technicalities and leave the presentation fairly non-technical.…”

“…A possible answer to the above question in the context of quantum transport in gapped systems has been recently proposed by S. Teufel and collaborators [31,9,10], also jointly with the authors [20,18,15,19], in the form of the so-called nonequilibrium almost-stationary state (NEASS). The NEASS paradigm relies on ideas conceived in previous works on the space-adiabatic perturbation theory in quantum dynamics [22,23,30].…”