Quantum Hall States response to toroidal geometry deformation

Abstract

In this paper, we apply techniques of geometric quantization to study the response of the integer and fractional quantum Hall effects to toroidal geometry deformation. The main method is that of using complex time Hamiltonian evolution to induce the geometry change and then the associated generalized coherent state transforms (gCST) to find the evolution of the Laughlin states. We consider two kinds of deformations. The first are flat toroidal deformations. Although Laughlin states for all flat toroidal geometries have been thoroughly studied before, we believe that our approach via the gCST is novel. It also serves as a testing ground to study the non-flat K\"ahler deformations. The Hamiltonians used in the flat deformations are quadratic in the generators of translations and therefore non periodic. The second kind of deformations involve nonflat K\"ahler toroidal deformations, generated by global, thus bi-periodic, Hamiltonians on the torus. The corresponding imaginary time flows are (elliptic curve modulus) $\tau$-preserving Mabuchi geodesics in the space of K\"ahler metrics on the torus, hitting a curvature singularity in finite imaginary time. By restricting to $S^1$-invariant deformations we find explicit analytic expressions for the evolution of the toroidal geometry and of the Laughlin states all the way to the singularity.