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

作者：Haldane F D M^*

来源：Journal of Mathematical Physics, 2018, 59(7): 071901.

DOI：10.1063/1.5042618

A "modified" variant of the Weierstrass sigma, zeta, and elliptic functions is proposed whereby the zeta function is redefined by zeta(z) (sic) (zeta) over tilde (z) equivalent to zeta(z) - gamma(2)z, 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, (zeta) over tilde(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. Published by AIP Publishing.

• 出版日期2018-7