Hyperdeterminantal computation for the Laughlin wave function

TL;DR

This paper introduces a novel hyperdeterminantal approach to compute coefficients in the Laughlin wave function expansion, enabling efficient calculation of specific terms without full decomposition, demonstrated up to eleven variables.

Contribution

The paper presents the first expression of Laughlin wave function coefficients as hyperdeterminants of sparse tensors, along with an algorithm for targeted coefficient computation.

Findings

Successfully computed coefficients for up to eleven variables

Established a new hyperdeterminantal framework for Laughlin wave functions

Provided an efficient algorithm for partial wave function expansion

Abstract

The decomposition of the Laughlin wave function in the Slater orthogonal basis appears in the discussion on the second-quantized form of the Laughlin states and is straightforwardly equivalent to the decomposition of the even powers of the Vandermonde determinants in the Schur basis. Such a computation is notoriously difficult and the coefficients of the expansion have not yet been interpreted. In our paper, we give an expression of these coefficients in terms of hyperdeterminants of sparse tensors. We use this result to construct an algorithm allowing to compute one coefficient of the development without computing the others. Thanks to a program in {\tt C}, we performed the calculation for the square of the Vandermonde up to an alphabet of eleven lettres.