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

Joscha Henheik

https://orcid.org/0000-0003-1106-327X

Fachbereich Mathematik, Eberhard-Karls-Universität, Auf der Morgenstelle 10, 72076 Tübingen, Germany

E-mail Address: sven-joscha.henheik@student.uni-tuebingen.de

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and

Stefan Teufel

https://orcid.org/0000-0003-3296-4261

Fachbereich Mathematik, Eberhard-Karls-Universität, Auf der Morgenstelle 10, 72076 Tübingen, Germany

E-mail Address: stefan.teufel@uni-tuebingen.de

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https://doi.org/10.1142/S0129055X20600041Cited by:12 (Source: Crossref)

This article is part of the issue:

Special Issue on Mathematical Results in Quantum Mechanics

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.

Keywords:

AMSC: 81Q15, 81Q20, 81V70

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