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 expansions) in order to estimate time-dependent multi-commutators for interacting 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 observables of interacting fermions on lattices. We further show how bounds for multi-commutators of an order higher than two can be used to study linear and non-linear responses of interacting fermions to external perturbations. All results also apply to quantum spin systems, with obvious modifications. However, we only explain 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.

Universality of the Hall Conductivity in Interacting Electron Systems

“…The proof of this fact is a straightforward consequence of the existence of the infinite volume dynamics (see [19,49,50]) and of the existence of the β, L → ∞ limit of the Gibbs state. In appendix C, we reproduce this proof; that is, we prove that the limit lim β→∞ lim L→∞ J…”

“…The proof is a simple adaptation of [19,49], the only difference being the choice of boundary conditions (periodic, rather than free). We consider two bounded operators A, B on the fermionic Fock space, even in the fermionic operators, with supports X and Y , respectively, independent of L. We shall think the torus Λ L as a subset of Λ 'centered' at the barycenter of X and Y , to be denoted z 0 .…”

Accuracy of classical conductivity theory at atomic scales for free fermions in disordered media

“…Then, we apply on the quasi-free fermion system in disordered media some time-dependent electromagnetic fields and look at the linear response current density in the thermodynamic limit of macroscopic electromagnetic fields. This study is already done in great generality in [7,21,22] and we shortly explain it in Section 2.3, with complementary explanations postponed to Appendix C. Then, we will be in a position to state the main results of the paper about the exponential rate of convergence of current densities in the limit of macroscopic electromagnetic fields.…”

“…In Section 3, we explain how such a sequence and state naturally define an exponentially tight sequence of random variables on the real line X = R, via the Riesz-Markov theorem and functional calculus (cf. (22)).…”

“…Appendix C contains supplementary information on the mathematical framework and relevant physical concepts, in order to make unnecessary the use of further references for a clear understanding of the subject of the current paper. More precisely, Appendix C summaries some important results on linear response current of our papers [6,7,[18][19][20][21][22]. Appendix C.2 explains the origin of current observables in relation with the discrete continuity equation within the CAR algebra.…”