Two-particle spectral function and ac conductivity of an amorphous system far below the mobility edge: A problem of interacting instantons

We use a variational approach to calculate the two-particle spectral function $S_2(x_1, x_1, x_2, x_2, E_1{E}_2)$ of a Gaussian-disordered electron system in the limit of deeply localized states and small energy difference $\omega =E_1-E_2$. The solution of the variational equations yields a two-center potential, each center in lowest order being determined by the square of an instanton function. The two instantons interact via the constraint that the Hamiltonian has to have lowest eigenvalues $E_1$, $E_2$. As the two centers approach the minimum distance allowed for given $\omega$ by the tunnel effect, we are confronted with a problem of confluent saddle points, which forces us to introduce an additional constraint. Our method is rigorous in the limit of weak disorder $|E_1+E_2|\to \infty$, $\frac{\omega }{|E_1+E_2|}=\mathrm{const}\ll 1$. We also apply it to the hydrodynamic limit $\frac{\omega }{|E_1+E_2|}\to 0, |E_1+E_2|$ large. It is found that these limits cannot be interchanged. In both limits we evaluated the ac conductivity. The result $\sigma \left(\omega \right)\sim {\omega }^2{\left(\ln \omega \right)}^{d+1\mathrm{}}$ is found in the hydrodynamic limit.