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)0ex0ex=0ex0ex-χ/(6π²βs²), where β0ex0ex=0ex0ex1, 2 or $4$ for the orthogonal, unitary, and symplectic ensembles, respectively, and χ0ex0ex=0ex0ex2(1-p) is the Euler characteristic of the surface with p ``handles'' (holes). The result is valid for gsg² for β0ex0ex=0ex0ex1, 4 and for gsg³ for β0ex0ex=0ex0ex2, where g1 is the dimensionless conductance.