We define a Banach space M 1 of models for fermions or quantum spins in the lattice with long range interactions and explicit the structure of (generalized) equilibrium states for any m ∈ M 1 . In particular, we give a first answer to an old open problem in mathematical physics -first addressed by Ginibre in 1968 within a different context -about the validity of the so-called Bogoliubov approximation on the level of states. Depending on the model m ∈ M 1 , our method provides a systematic way to study all its correlation functions and can thus be used to analyze the physics of long range interactions. Furthermore, we show that the thermodynamics of long range models m ∈ M 1 is governed by the non-cooperative equilibria of a zero-sum game, called here the thermodynamic game.

We consider free lattice fermions subjected to a static bounded potential and a time-and space-dependent electric field. For any bounded convex region R ⊂ R d (d ≥ 1) of space, electric fields E within R drive currents. At leading order, uniformly with respect to the volume |R| of R and the particular choice of the static potential, the dependency on E of the current is linear and described by a conductivity (tempered, operator-valued) distribution. Because of the positivity of the heat production, the real part of its Fourier transform is a positive measure, named here (microscopic) conductivity measure of R, in accordance with Ohm's law in Fourier space. This finite measure is the Fourier transform of a time-correlation function of current fluctuations, i.e., the conductivity distribution satisfies Green-Kubo relations. We additionally show that this measure can also be seen as the boundary value of the Laplace-Fourier transform of a so-called quantum current viscosity. The real and imaginary parts of conductivity distributions are related to each other via the Hilbert transform, i.e., they satisfy Kramers-Kronig relations. At leading order, uniformly with respect to parameters, the heat production is the classical work performed by electric fields on the system in presence of currents. The conductivity measure is uniformly bounded with respect to parameters of the system and it is never the trivial measure 0 dν. Therefore, electric fields generally produce heat in such systems. In fact, the conductivity measure defines a quadratic form in the space of Schwartz functions, the Legendre-Fenchel transform of which describes the resistivity of the system. This leads to Joule's law, i.e., the heat produced by currents is proportional to the resistivity and the square of currents.

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.

Electric resistance in conducting media is related to heat (or entropy) production in the presence of electric fields. In this paper, by using Araki's relative entropy for states, we mathematically define and analyze the heat production of free fermions within external potentials. More precisely, we investigate the heat production of the nonautonomous C -dynamical system obtained from the fermionic second quantization of a discrete Schrödinger operator with bounded static potential in the presence of an electric field that is time-and spacedependent. It is a first preliminary step towards a mathematical description of transport properties of fermions from thermal considerations. This program will be carried out in several papers. The regime of small and slowly varying in space electric fields is important in this context and is studied the present paper. We use tree-decay bounds of the n-point, n 2 2N, correlations of the many-fermion system to analyze this regime. We verify below the first law of thermodynamics for the system under consideration. The latter implies, for systems doing no work, that the heat produced by the electromagnetic field is exactly the increase of the internal energy resulting from the modification of the (infinite volume) state of the fermion system. The identification of heat production with an energy increment is, among otheg things, technically convenient. We initially focus our study on noninteracting (or free) fermions, but our approach will be later applied to weakly interacting fermions.

The thermodynamic impact of the Coulomb repulsion on s-wave superconductors is analyzed via a rigorous study of equilibrium and ground states of the strong coupling BCS-Hubbard Hamiltonian. We show that the one-site electron repulsion can favor superconductivity at fixed chemical potential by increasing the critical temperature and/or the Cooper pair condensate density. If the one-site repulsion is not too large, a first or a second order superconducting phase transition can appear at low temperatures. The Meißner effect is shown to be rather generic but coexistence of superconducting and ferromagnetic phases is also shown to be feasible, for instance near half-filling and at strong repulsion. Our proof of a superconductorMott insulator phase transition implies a rigorous explanation of the necessity of doping insulators to create superconductors. These mathematical results are consequences of "quantum large deviation" arguments combined with an adaptation of the proof of Størmer's theorem [1] to even states on the CAR algebra.