Heat equation approach to geometric changes of the torus Laughlin-state

TL;DR

This paper introduces a two-body operator approach to describe how the torus Laughlin state evolves with geometry changes, connecting thin torus limits and Hall viscosity in fractional quantum Hall systems.

Contribution

It presents a novel two-body operator framework to generate the torus Laughlin state from the thin torus limit, linking geometry, state evolution, and Hall viscosity.

Findings

The guiding center degrees of freedom change with torus geometry via a two-body operator.

The method allows evolving the Laughlin state across different torus geometries.

The approach relates to the concept of Hall viscosity in fractional quantum Hall states.

Abstract

We study the second quantized -or guiding center- description of the torus Laughlin state. Our main focus is the change of the guiding center degrees of freedom with the torus geometry, which we show to be generated by a two-body operator. We demonstrate that this operator can be used to evolve the full torus Laughlin state at given modular parameter \tau\ from its simple (Slater-determinant) thin torus limit, thus giving rise to a new presentation of the torus Laughlin state in terms of its "root partition" and an exponential of a two-body operator. This operator therefore generates in particular the adiabatic evolution between Laughlin states on regular tori and the quasi-one-dimensional thin torus limit. We make contact with the recently introduced notion of a "Hall viscosity" for fractional quantum Hall states, to which our two-body operator is naturally related, and which serves as a demonstration of our method to generate the Laughlin state on the torus.