Microscopic Conductivity of Lattice Fermions at Equilibrium. Part II: Interacting Particles

Abstract

We apply Lieb-Robinson bounds for multi-commutators we recently derived [BP3] to study the (possibly non-linear) response of interacting fermions at thermal equilibrium to perturbations of the external electromagnetic field. This analysis leads to an extension of the results for quasi-free fermions of [BPH1, BPH2] to fermion systems on the lattice with shortrange interactions. More precisely, we investigate entropy production and charge transport properties of non-autonomous C * -dynamical systems associated with interacting lattice fermions within bounded static potentials and in presence of an electric field that is time-and space-dependent. We verify the 1st law of thermodynamics for the heat production of the system under consideration. In linear response theory, the latter is related with Ohm and Joule's laws. These laws are proven here to hold at the microscopic scale, uniformly with respect to the size of the (microscopic) region where the electric field is applied. An important outcome is the extension of the notion of conductivity measures to interacting fermions.

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

“…(iii) Linear response current density: Fix a direction w ∈ R d with w R d = 1 and a (time-dependent) continuous, compactly supported, electric field E ∈ C 0 0 (R; R d ), i.e., the external electric field is a continuous function t → E(t) ∈ R d of time t ∈ R with compact support. Then, as it is explained in Appendix C, [7,21] 1 shows that the space-averaged linear response current observable in the lattice region Λ and at time t = 0 in the direction w is equal to…”

“…In fact, the first term in the right-hand side of (19) corresponds to the paramagnetic coefficient, whereas the second one is the diamagnetic component. For more details, see [21,Theorem 3.7].…”

“…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.…”

Microscopic foundations of Ohm and Joule's laws – The relevance of thermodynamics

“…However, it is believed in theoretical physics that electric resistance of conductors should also result from the interactions between charge carriers, and not only from the presence of inhomogeneities (impurities). In [35,36,37], we succeed to extend our previous results to fermion systems with interactions.…”

“…However, there is a plethora [39,40] of other types of elementary excitations not covered by the Bogoliubov theory. We show [32,33,36] that the notion of conductivity measure we defined is nothing else than a spectral measure of the generator of dynamics in an appropriate representation.…”