Laughlin wave function, Berry Phase and Quantization

TL;DR

This paper links the Laughlin wave function to a Hamiltonian framework using Kirchhoff equations and Berry connection, proposing a Hermitian model for fractional quantum Hall effect and emphasizing topological quantization.

Contribution

It introduces a self-adjoint Hamiltonian model for the fractional quantum Hall effect based on Kirchhoff equations and superymmetric quantum mechanics, connecting topology with quantization.

Findings

Laughlin wave function viewed as a Hamiltonian

Berry connection is non-Hermitian, leading to a new Hermitian model

Quantization is topologically driven through singularities

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.