Hyperdeterminantal computation for the Laughlin wave function

Adrien Boussicault (1) , Christophe Tollu (2) , Jean-Gabriel Luque (3)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge
2 LIPN - Laboratoire d'Informatique de Paris-Nord
3 LITIS - Laboratoire d'Informatique, de Traitement de l'Information et des Systèmes

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.

Mots clés: Hyperdeterminant; Laughlin wave function; even powers of the Vandermonde

Domaines: Combinatoire [math.CO]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Adrien Boussicault, Christophe Tollu, Jean-Gabriel Luque. Hyperdeterminantal computation for the Laughlin wave function. Journal of Physics A: Mathematical and Theoretical, 2009, 42 (14), pp.145301.1-145301.13. ⟨10.1088/1751-8113/42/14/145301⟩. ⟨hal-00326880⟩