Energy relaxation rate and its mesoscopic fluctuations in quantum dots

TL;DR

This paper investigates the energy relaxation rate in quantum dots, analyzing its fluctuations and the conditions under which the Fermi-golden-rule approximation remains valid, using advanced theoretical techniques.

Contribution

It provides the first exact expression for mesoscopic fluctuations of the relaxation rate in quantum dots without assuming specific interaction matrix elements.

Findings

Fluctuations of the relaxation rate become large at energies around Δ√g.

The Fermi-golden-rule description breaks down when max{ε,T} ≈ Δ√g.

The approach uses non-perturbative supersymmetric sigma model techniques.

Abstract

We analyze the applicability of the Fermi-golden-rule description of quasiparticle relaxation in a closed diffusive quantum dot with electron-electron interaction. Assuming that single-particle levels are already resolved but the initial stage of quasiparticle disintegration can still be described by a simple exponential decay, we calculate the average inelastic energy relaxation rate of single-particle excitations and its mesoscopic fluctuations. The smallness of mesoscopic fluctuations can then be used as a criterion for the validity of the Fermi-golden-rule description. Technically, we implement the real-space Keldysh diagram technique, handling correlations in the quasi-discrete spectrum non-perturbatively by means of the non-linear supersymmetric sigma model. The unitary symmetry class is considered for simplicity. Our approach is complementary to the lattice-model analysis of Fock space: thought we are not able to describe many-body localization, we derive the exact lowest-order expression for mesoscopic fluctuations of the relaxation rate, making no assumptions on the matrix elements of the interaction. It is shown that for the quasiparticle with the energy $\varepsilon$ on top of the thermal state with the temperature $T$, fluctuations of its energy width become large and the Fermi-golden-rule description breaks down at $\max\{\varepsilon,T\}\sim\Delta\sqrt{g}$, where $\Delta$ is the mean level spacing in the quantum dot, and $g$ is its dimensionless conductance.