Vol. 2, No. 4, 2020

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Quantum fluctuations and large-deviation principle for microscopic currents of free fermions in disordered media

Jean-Bernard Bru, Walter de Siqueira Pedra and Antsa Ratsimanetrimanana

Vol. 2 (2020), No. 4, 943–971
Abstract

We extend the large-deviation results obtained by N. J. B. Aza and the present authors on atomic-scale conductivity theory of free lattice fermions in disordered media. Disorder is modeled by a random external potential, as in the celebrated Anderson model, and 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.

Keywords
quantum fluctuations, large deviations, fermionic charge transport, disordered media
Mathematical Subject Classification
Primary: 32A70, 60F10, 82C70
Milestones
Received: 16 June 2020
Accepted: 28 September 2020
Published: 25 February 2021
Authors
Jean-Bernard Bru
Departamento de Matemáticas
Facultad de Ciencia y Tecnología
Universidad del País Vasco
Bilbao
Spain
Basque Center for Applied Mathematics
Basque Foundation for Science
Bilbao
Spain
IKERBASQUE
Basque Foundation for Science
Bilbao
Spain
Walter de Siqueira Pedra
Instituto de Física, Departamento de Física Matemática
Universidade de Sao Paulo
Sao Paulo
Brazil
Antsa Ratsimanetrimanana
Basque Center for Applied Mathematics
Bilbao
Spain