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

TL;DR

This paper introduces a modified Weierstrass sigma function that is modular-invariant and simplifies the construction of lowest-Landau-level wavefunctions on the torus, especially for high-symmetry lattices.

Contribution

It proposes a new modified sigma function with simpler quasiperiodicity, enhancing the modular-invariant formulation of wavefunctions on the torus.

Findings

Modified sigma function has simpler quasiperiodicity.

Coincides with original for square and hexagonal lattices.

Provides a more natural formulation before understanding quasi-modular forms.

Abstract

A "modified" variant of the Weierstrass sigma, zeta, and elliptic functions is proposed whereby the zeta function is redefined by $\zeta(z)$ $\mapsto$ $\tilde \zeta(z)$ $\equiv$ $\zeta(z) - \gamma_2z$, where $\gamma_2$ is a lattice invariant related to the almost-holomorphic modular invariant of the quasi-modular-invariant weight-2 Eisenstein series. If $\omega_i$ is a primitive half-period, $\tilde\zeta(\omega_i)$ = $\pi \omega_i^*/A$, where $A$ is the area of the primitive cell of the lattice. The quasiperiodicity of the modified sigma function is much simpler than that of the original, and it becomes the building block for the modular-invariant formulation of lowest-Landau-level wavefunctions on the torus. It is suggested that the "modified" sigma function is more natural than the original Weierstrass form, which was formulated before quasi-modular forms were understood. For the high-symmetry (square and hexagonal) lattices, the modified and original sigma functions coincide.