Justifying Kubo's formula for gapped systems at zero temperature: a brief review and some new results

Abstract

We first review the problem of a rigorous justification of Kubo's formula for transport coefficients in gapped extended Hamiltonian quantum systems at zero temperature. In particular, the theoretical understanding of the quantum Hall effect rests on the validity of Kubo's formula for such systems, a connection that we review briefly as well. We then highlight an approach to linear response theory based on non-equilibrium almost-stationary states (NEASS) and on a corresponding adiabatic theorem for such systems that was recently proposed and worked out by one of us in [51] for interacting fermionic systems on finite lattices. In the second part of our paper we show how to lift the results of [51] to infinite systems by taking a thermodynamic limit.

25 pages, 1 figure; final version; Contribution to the proceedings of QMath14 in Aarhus 2019

References in corpus (3)

• Linear response theory for magnetic Schroedinger operators in disordered media
• Lieb-Robinson Bounds for Multi-Commutators and Applications to Response Theory
• Improved energy estimates for a class of time-dependent perturbed Hamiltonians

Cited by in corpus (9)

• Inhomogeneous conformal field theory out of equilibrium
• A new approach to transport coefficients in the quantum spin Hall effect
• Adiabatic theorem in the thermodynamic limit: Systems with a uniform gap
• From charge to spin: analogies and differences in quantum transport coefficients
• Improved energy estimates for a class of time-dependent perturbed Hamiltonians
• Adiabatic Evolution of Low-Temperature Many-Body Systems
• Extension of the Adiabatic Theorem
• Purely linear response of the quantum Hall current to space-adiabatic perturbations
• On adiabatic theory for extended fermionic lattice systems