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 proportional to 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. Focusing 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) omega/omega(V) for small frequencies omega << omega(V) and R-2(omega, V) = 1 - (alpha - d)A(1) (omega(V)/omega)(d/alpha) - A(2) (omega(V)/omega)(2) for large frequencies omega << omega(V), where omega(V) proportional to V-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.