Quantum Fluctuations and Large Deviation Principle for Microscopic Currents of Free Fermions in Disordered Media
Abstract
We contribute an extension of large-deviation results obtained in [N.J.B.
Aza, J.-B. Bru, W. de Siqueira Pedra, A. Ratsimanetrimanana, J. Math. Pures
Appl. 125 (2019) 209] on conductivity theory at atomic scale of free lattice
fermions in disordered media. Disorder is modeled by (i) a random external
potential, like in the celebrated Anderson model, and (ii) a
nearest-neighbor hopping term with random complex-valued amplitudes. In
accordance with experimental observations, via the large deviation
formalism, our previous paper showed in this case that quantum uncertainty
of microscopic electric current densities around their (classical)
macroscopic value is suppressed, exponentially fast with respect to the
volume of the region of the lattice where an external electric field is
applied. Here, the quantum fluctuations of linear response currents are
shown to exist in the thermodynamic limit and we mathematically prove that
they are related to the rate function of the large deviation principle
associated with current densities. We also demonstrate that, in general,
they do not vanish (in the thermodynamic limit) and the quantum uncertainty
around the macroscopic current density disappears exponentially fast with an
exponential rate proportional to the squared deviation of the current from
its macroscopic value and the inverse current fluctuation, with respect to
growing space (volume) scales.