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. Focusing on the excitations in disordered electronic systems, we compute the energy-level correlation function $R_2\left(\omega \right)$ 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 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.

Figure 1

The correlation function $R_2\left(\omega \right)$ of the dipole energy levels in an electronic system as a function of frequency $\omega$ (in units ${\omega }_V)$, obtained from a numerical integration of Eq. (3.8). The small- and large-frequency asymptotic behaviors given by Eqs. (3.11), (3.12), and (3.14) are shown in red.