The origin of holomorphic states in Landau levels from non-commutative geometry, and a new formula for their overlaps on the torus

Abstract

Holomorphic functions that characterize states in a two-dimensional Landau level been central to key developments such as the Laughlin state. Their origin has historically been attributed to a special property of "Schrödinger wavefunctions" of states in the "lowest Landau level". It is shown here that they instead arise in any Landau level as a generic mathematical property of the Heisenberg description of the non-commutative geometry of guiding centers. When quasiperiodic boundary conditions are applied to compactify the system on a torus, a new formula for the overlap between holomorphic states, in the form of a discrete sum rather than an integral, is obtained. The new formula is unexpected from the previous "lowest-Landau level Schrödinger wavefunction" interpretation.

8 pages, no figures. corrected small error in equation (31); added publication info

References in corpus (1)

• A modular-invariant modified Weierstrass sigma-function as a building block for lowest-Landau-level wavefunctions on the torus

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