Topological spectral correlations in 2D disordered systems

It is shown that the tail in the two-level spectral correlation function R(s) for 2D disordered systems depends on the system geometry and the boundary conditions. In particular, for closed surfaces (with no boundary), R(s) = -chi/(6 pi(2)beta s(2)), where beta = 1, 2 or 4 for the orthogonal, unitary, and symplectic ensembles, respectively, and chi = 2(1 - p) is the Euler characteristic of the surface with p "handles" (holes). The result is valid for g much less than s much less than g(2) for beta = I, 4 and for g much less than s much less than g(3) for beta = 2, where g much greater than 1 is the dimensionless conductance. [S0031-9007(98)08123-X].