Heat equation approach to geometric changes of the torus Laughlin state

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 $\ensuremath{\tau}$ 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.