Berry’s phase, chaos, and the deformations of Riemann surfaces

Abstract

Parametrized families of Landau Hamiltonians on compact Riemann surfaces corresponding to classically chaotic families of geodesic motion are investigated. The parameters describe deformations of such surfaces with genus $gg~1$. It is shown that the adiabatic curvature (responsible for Berry's phase) of the lowest Landau level for $gg1$ is the sum of two terms. The first term is proportional to the natural symplectic form on deformation space, and the second is a fluctuating term reflecting the chaos of the geodesic motion for $gg1$. For $g=1$ (integrable motion on the torus) we have no fluctuating term. We propose our result to be interpreted as a curvature analog of the well-known semiclassical trace formulas. Connections with the viscosity properties of quantum Hall fluids on such surfaces are also pointed out. An interesting possibility in this respect is the fractional quantization of certain components of the viscosity tensor of such fluids.