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

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 $\propto 1/r^\alpha$ 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. Focussing on the excitations in disordered electronic systems, we compute the energy-level correlation function $R_2(\omega)$ 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 $R_2(\omega,V)=A_3\frac{\omega}{\omega_V}$ for small frequencies $\omega\ll\omega_V$ and $R_2(\omega,V)=1-(\alpha-d)A_{1}\left(\frac{\omega_V}{\omega}\right)^\frac{d}{\alpha} -A_2\left(\frac{\omega_V}{\omega}\right)^2$ for large frequencies $\omega\gg\omega_V$, where $\omega_V\propto V^{-\frac{\alpha}{d}}$ is the characteristic matrix element of excitation hopping in a system of volume $V$, and $A_1$, $A_2$ and $A_3$ are coefficient 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.