Laughlin states change under large geometry deformations and imaginary time Hamiltonian dynamics

TL;DR

This paper investigates how Laughlin quantum states evolve under significant geometric deformations of the sphere and plane, revealing concentration phenomena and connections to geometric quantization and Chern-Simons theory.

Contribution

It introduces a framework linking large geometric deformations to Laughlin state concentration and uses generalized coherent states and Chern-Simons theory for state lifting.

Findings

Laughlin states concentrate on Bohr-Sommerfeld orbits under large deformations.

Geometry of the sphere becomes a thin cigar shape in the limit.

Lifting of states involves generalized coherent state transforms and Chern-Simons parallel transport.

Abstract

We study the change of the Laughlin states under large deformations of the geometry of the sphere and the plane, associated with Mabuchi geodesics on the space of metrics with Hamiltonian $S^1$-symmetry. For geodesics associated with the square of the symmetry generator, as the geodesic time goes to infinity, the geometry of the sphere becomes that of a thin cigar collapsing to a line and the Laughlin states become concentrated on a discrete set of $S^1$--orbits, corresponding to Bohr-Sommerfeld orbits of geometric quantization. The lifting of the Mabuchi geodesics to the bundle of quantum states, to which the Laughlin states belong, is achieved via generalized coherent state transforms, which correspond to the KZ parallel transport of Chern-Simons theory.