Laughlin wave function, Berry Phase and Quantization

Abstract

In this paper, we show that the Laughlin wave function is a Hamiltonian and its associated Berry connection as the Schr\"odinger equation by transforming the Schr\"odinger equation into the Kirchhoff equation which describes the evolution of $n$ point vortices in Hydrodynamics. This helps us to view the Berry connection associated with Laughlin wave function or Schr\"odinger equation is not Hermitian, therefore we propose a self adjoint model Hamiltonian for the fractional quantum Hall effect, from the study of Kirchhoff equation, using superymmetric quantum mechanics whose solutions are complex Hermite polynomials. The Schr\"odinger equation as Berry connection allows us to formulate the problem of quantisation in terms of topology, as the quantum numbers are topological invariant which arise due to singularities in the Kirchhoff equation. Quantisation arises as it continuously connects the topologically inequivalent Hamiltonians in the Hilbert space.