Citation: Henheik, Joscha, and Tom Wessel. "Response theory for locally gapped systems." arXiv preprint arXiv:2410.10809 (2024).
Web: https://arxiv.org/abs/2410.10809
Tags: Physical, Operator-algebras
In this paper, the authors introduce a notion of locally gapped system at use it to establish a version of response theory. Response theory is a sort of perturbation theory, but instead of looking at how the spectrum of the Hamiltonian changes under the perturbation one looks at how expectation values of observables change.
The main results of response theory are still conjectural in general; In a famous list of open problems, Barry Simon lits a justification of Kubo's formula (one of the key tools in response theory) as a challenge:
> Simon, Barry. "Fifteen problems in mathematical physics." Perspectives in mathematics, Birkhäuser, Basel 423 (1984).
In specific settings, in particular quantum many-body lattice systems with local Hamiltonians and global gaps, there is a proof. This paper shows that there results can be generalized to a setting in which one only assumes a local gap, which is preferable for several reasons, not least of which because perturbations can close gaps near critical points.
The notion of a local gap is very natural. Local gaps have been introduced and used productively already in the theory of Lie groups:
> Boutonnet, Rémi, Adrian Ioana, and Alireza Salehi Golsefidy. "Local spectral gap in simple Lie groups and applications." Inventiones mathematicae 208.3 (2017): 715-802.
The way that these authors define a local gap is natural from the point of view of response theory. In response theroy, gaps are important because they imply that the action of the Louisvillian on local operators is in a certain sense invertible. Requiring that such an inverse only holds locally (and approximately) is a good notion of a local gap.
The authors also introduce several other notions of local gap, and discuss the relationship between those notions.