Lieb–Robinson Bounds for Multi–Commutators and Applications to Response Theory

Abstract

We generalize to multi–commutators the usual Lieb–Robinson bounds for commutators. In the spirit of constructive QFT, this is done so as to allow the use of combinatorics of minimally connected graphs (tree expan- sions) in order to estimate time–dependent multi–commutators for interact- ing fermions. Lieb–Robinson bounds for multi–commutators are effective mathematical tools to handle analytic aspects of the dynamics of quantum particles with interactions which are non–vanishing in the whole space and possibly time–dependent. To illustrate this, we prove that the bounds for multi–commutators of order three yield existence of fundamental solutions for the corresponding non–autonomous initial value problems for observ- ables of interacting fermions on lattices. We further show how bounds for multi–commutators of order higher than two can be used to study linear and non–linear responses of interacting fermions to external perturbations. The results discussed here are also valid for quantum spins on lattices, with obvi- ous modifications. However, we only discuss the fermionic case in detail, in view of applications to microscopic quantum theory of electrical conduction discussed here and because this case is technically more involved.