Macroscopic behavior of non-relativistic interacting many fermion systems

Abstract

We aim to derive macroscopic universal behavior of cold matter from first principles for interacting many fermion systems, in a mathematically rigorous way. [1] contains first results in this direction for the case of the electric conductivity and states that the linear charge transport of interacting lattice fermions in disordered media with non-vanishing short-range interparticle interactions is well-defined at macroscopic scales and is governed by a Lévy (conductivity). The latter quantifies the production of heat in presence of electric fields, in the spirit of the original works of Joule. Moreover, the conductivity measures for microscopic regions converge to the one related to the macroscopic scale, in the limit of large regions. Motivated by the validity of Ohm's law at the microscopic scale, an unexpected property experimentally verified in 2012 [2], we will investigate large deviation principles behind this convergence. In [3] we proved that free fermions in disordered media have non-vanishing conductivity measures and we intend now to prove the same for system without disorder, but with non-vanishing interparticle interactions. Indeed, it is believed in theoretical physics that electric resistance of usual conductors should also result from interactions between charge carriers. From the technical point of view, it is a challenging issue to mathematically verify physical conjectures for interacting particles. Rigorous methods of constructive quantum field theory, as Grassmann-Berezin integration, Brydges-Kennedy tree expansions and determinant bounds [4], will be pivotal in this context. Space decay of interactions is frequently used to obtain convergence of such expansions schemes. Hence, the lack of such a decay for long range interactions (like, mean field interactions, for instance) represents a serious technical problem in various cases of physical interest. We intend to use the idea of the Bogolioubov approximation method to convert the problem of construction of correlation functions for interactions containing mean field terms in their interactions in a problem of constructing such correlation functions for a convenient family of short range interactions [5], each of which can be studied via well-known methods of the constructive theory. Such an enlargement of the domain of application of constructive methods will allow us to use them in the scope of the microscopic theory of superconductors, like the BCS theory. In this context, special attention will be paid to "exotic" phenomena in high-Tc superconductors, like d-wave electron pairing and density waves [6,7].[1] J.-B. Bru and WdSP, From the 2nd Law of Thermodynamics to the AC-Conductivity..., M3AS (2015).[2] B. Weber et al., Ohm's Law Survives to the Atomic Scale, Science 335 (2012).[3] JBB, WdSP and C. Kurig, Macroscopic Conductivity of Free Fermions in Disordered Media, Rev. Math. Phys. 26(5) (2014).[4] JBB and WdSP, Universal Bounds for Large Determinants..., mp_arc 16-16.[5] JBB and WdSP, Non-cooperative Equilibria..., Memoirs of the AMS (2013).[6] D. A. Bonn, Are high-temperature superconductors exotic?, Nature 2 (2006).[7] JBB, A. Delgado and WdSP, d-Wave Pairing..., J. Stat. Mech. (2015). (AU)