Success and failure of the plasma analogy for Laughlin states on a torus

TL;DR

This paper examines the plasma analogy for Laughlin states on a torus, revealing that screening fails in the thin-torus limit and providing a method to approximate normalization across different geometries.

Contribution

It analytically and numerically investigates the plasma analogy's validity on a torus, especially in the thin-torus limit, and develops an approximation for the normalization of Laughlin states.

Findings

Plasma screens on a torus without short nontrivial paths.

Screening fails in the thin-torus limit.

An approximate normalization formula valid for all torus geometries.

Abstract

We investigate the nature of the plasma analogy for the Laughlin wave function on a torus describing the quantum Hall plateau at $\nu=\frac{1}{q}$. We first establish, as expected, that the plasma is screening if there are no short nontrivial paths around the torus. We also find that when one of the handles has a short circumference -- i.e. the thin-torus limit -- the plasma no longer screens. To quantify this we compute the normalization of the Laughlin state, both numerically and analytically. For the numerical calculation we expand the Laughlin state in a Fock basis of slater-determinants of single particle orbitals, and determine the Fock coefficients of the expansion as a function of torus geometry. In the thin torus limit only a few Fock configurations have non-zero coefficients, and their analytical forms simplify greatly. Using this simple limit, we can reconstruct the normalization and analytically extend it back into the 2D regime. We find that there are geometry dependent corrections to the normalization, and this in turn implies that the plasma in the plasma analogy is not screening when in the thin torus limit. Further we obtain an approximate normalization factor that gives a good description of the normalization for all tori, by extrapolating the thin torus normalization to the thick torus limit.