Characterization of the Quasi-Stationary State of an Impurity Driven by Monochromatic Light II: Microscopic Foundations

From quantum mechanical first principles only, we rigorously study the time-evolution of a N-level atom (impurity) interacting with an external monochromatic light source within an infinite system of free electrons at thermal equilibrium (reservoir). In particular, we establish the relation between the full dynamics of the compound system and the effective dynamics for the N-level atom, which is studied in detail in Bru et al. (Ann Henri Poincaré 13(6):1305–1370, 2012). Together with Bru et al. (Ann Henri Poincaré 13(6):1305–1370, 2012) the present paper yields a purely microscopic theory of optical pumping in laser physics. The model we consider is general enough to describe gauge invariant atom–reservoir interactions

Entanglement of classical and quantum short-range dynamics in mean-field systems

The relationship between classical and quantum mechanics is usually understood via the limit ħ→0. This is the underlying idea behind the quantization of classical objects. The apparent incompatibility of general relativity with quantum mechanics and quantum field theory has challenged for many decades this basic idea. We recently showed (Bru and de Siqueira Pedra, 0000; Bru and de Siqueira Pedra, 2021 [46,47]) the emergence of classical dynamics for very general quantum lattice systems with mean-field interactions, without (complete) suppression of its quantum features, in the infinite volume limit. This leads to a theoretical framework in which the classical and quantum worlds are entangled. Such an entanglement is noteworthy and is a consequence of the highly non-local character of mean-field interactions. Therefore, this phenomenon should not be restricted to systems with mean-field interactions only, but should also appear in presence of interactions that are sufficiently long-range, yielding effective, classical background fields, in the spirit of the Higgs mechanism of quantum field theory. In order to present the result in a less abstract way than in its original version, here we apply it to a concrete, physically relevant, example and discuss, by this means, various important aspects of our general approach. The model we consider is not exactly solvable and the particular results obtained are new.CNPq (309723/2020-5), FAPESP (2017/22340- 9), as well as by the Basque Government through the grant IT641-13 and by the Spanish Ministry of Science, Innovation and Universities: MTM2017-82160-C2-2-P

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

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.FAPESP under Grant 2013/13215-5 Basque Government through the grant IT641-13 SEV-2013-0323 MTM2014-5385

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

We apply Lieb–Robinson bounds for multi-commutators we recently derived (Bru and de Siqueira Pedra, Lieb–Robinson bounds for multi-commutators and applications to response theory, 2015) 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 (Bru et al. Commun Pure Appl Math 68(6):964–1013, 2015; Bru et al. J Math Phys 56:051901-1–051901-51, 2015) to fermion systems on the lattice with short-range 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

Weak* Hypertopologies with Application to Genericity of Convex Sets

We propose a new class of hypertopologies, called here weak$^{\ast }$ hypertopologies, on the dual space $\mathcal{X}^{\ast }$ of a real or complex topological vector space $\mathcal{X}$. The most well-studied and well-known hypertopology is the one associated with the Hausdorff metric for closed sets in a complete metric space. Therefore, we study in detail its corresponding weak$^{\ast }$ hypertopology, constructed from the Hausdorff distance on the field (i.e. $\mathbb{R}$ or $\mathbb{C}$) of the vector space $\mathcal{X}$ and named here the weak$^{\ast }$-Hausdorff hypertopology. It has not been considered so far and we show that it can have very interesting mathematical connections with other mathematical fields, in particular with mathematical logics. We explicitly demonstrate that weak$^{\ast }$ hypertopologies are very useful and natural structures\ by using again the weak$^{\ast }$-Hausdorff hypertopology in order to study generic convex weak$^{\ast }$-compact sets in great generality. We show that convex weak$^{\ast }$-compact sets have generically weak$^{\ast }$-dense set of extreme points in infinite dimensions. An extension of the well-known Straszewicz theorem to Gateaux-differentiability (non necessarily Banach) spaces is also proven in the scope of this application.FAPESP (2017/22340-9) CNPq (309723/2020-5) by the Basque Government through the grant IT641-13 MTM2017-82160-C2-2-P

Macroscopic Dynamics of the Strong-Coupling BCS-Hubbard Model

The aim of the current paper is to illustrate, in a simple example, our recent, very general, rigorous results on the dynamical properties of fermions and quantum-spin systems with long-range, or mean-field, interactions, in infinite volume. We consider here the strong-coupling BCS-Hubbard model, because this example is very pedagogical and, at the same time, physically relevant for it highlights the impact of the (screened) Coulomb repulsion on ($s$-wave) superconductivity.FAPESP (2017/22340-9); CNPq (309723/2020-5); by the Basque Government through the grant IT641-13; MTM2017-82160-C2-2-