Energy-level statistics in strongly disordered systems with power-law hopping

Abstract

Motivated by neutral excitations in disordered electronic materials and systems of trapped ultracold particles with long-range interactions, we study energy-level statistics of quasiparticles with the power-law hopping Hamiltonian $\ensuremath{\propto}1/{r}^{\ensuremath{\alpha}}$ in a strong random potential. In solid-state systems such quasiparticles, which are exemplified by neutral dipolar excitations, lead to long-range correlations of local observables and may dominate energy transport. Focusing on the excitations in disordered electronic systems, we compute the energy-level correlation function ${R}_{2}(\ensuremath{\omega})$ in a finite system in the limit of sufficiently strong disorder. At small energy differences, the correlations exhibit Wigner-Dyson statistics. In particular, in the limit of very strong disorder the energy-level correlation function is given by ${R}_{2}(\ensuremath{\omega},V)={A}_{3}\frac{\ensuremath{\omega}}{{\ensuremath{\omega}}_{V}}$ for small frequencies $\ensuremath{\omega}\ensuremath{\ll}{\ensuremath{\omega}}_{V}$ and ${R}_{2}(\ensuremath{\omega},V)=1\ensuremath{-}(\ensuremath{\alpha}\ensuremath{-}d){A}_{1}{\left(\frac{{\ensuremath{\omega}}_{V}}{\ensuremath{\omega}}\right)}^{\frac{d}{\ensuremath{\alpha}}}\ensuremath{-}{A}_{2}{\left(\frac{{\ensuremath{\omega}}_{V}}{\ensuremath{\omega}}\right)}^{2}$ for large frequencies $\ensuremath{\omega}\ensuremath{\gg}{\ensuremath{\omega}}_{V}$, where ${\ensuremath{\omega}}_{V}\ensuremath{\propto}{V}^{\ensuremath{-}\frac{\ensuremath{\alpha}}{d}}$ is the characteristic matrix element of excitation hopping in a system of volume $V$, and ${A}_{1},{A}_{2}$, and ${A}_{3}$ are coefficients of order unity which depend on the shape of the system. The energy-level correlation function, which we study, allows for a direct experimental observation, for example, by measuring the correlations of the ac conductance of the system at different frequencies.