A modularinvariant modified Weierstrass sigmafunction as a building block for lowestLandaulevel wavefunctions on the torus
Abstract
A “modified” variant of the Weierstrass sigma, zeta, and elliptic functions is proposed whereby the zeta function is redefined by $$ζ(z) \mapsto \tilde{ζ}(z) ≡ ζ(z)  γ_2z$$, where γ_{2} is a lattice invariant related to the almostholomorphic modular invariant of the quasimodularinvariant weight2 Eisenstein series. If ω_{i} is a primitive halfperiod, $$\tilde{ζ}(ω_i) = πω^{*}_{i}/A$$, 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 buildingblock for the modularinvariant formulation of lowestLandaulevel wavefunctions on the torus. It is suggested that the “modified” sigma function is more natural than the original Weierstrass form, which was formulated before quasimodular forms were understood. Finally, for the highsymmetry (square and hexagonal) lattices, the modified and original sigma functions coincide.
 Authors:

 Princeton Univ., NJ (United States). Dept. of Physics
 Publication Date:
 Research Org.:
 Princeton Univ., NJ (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC)
 OSTI Identifier:
 1540241
 Alternate Identifier(s):
 OSTI ID: 1459239
 Grant/Contract Number:
 SC0002140
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Journal of Mathematical Physics
 Additional Journal Information:
 Journal Volume: 59; Journal Issue: 7; Journal ID: ISSN 00222488
 Publisher:
 American Institute of Physics (AIP)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; physics
Citation Formats
Haldane, F. D. M. A modularinvariant modified Weierstrass sigmafunction as a building block for lowestLandaulevel wavefunctions on the torus. United States: N. p., 2018.
Web. doi:10.1063/1.5042618.
Haldane, F. D. M. A modularinvariant modified Weierstrass sigmafunction as a building block for lowestLandaulevel wavefunctions on the torus. United States. doi:10.1063/1.5042618.
Haldane, F. D. M. Fri .
"A modularinvariant modified Weierstrass sigmafunction as a building block for lowestLandaulevel wavefunctions on the torus". United States. doi:10.1063/1.5042618. https://www.osti.gov/servlets/purl/1540241.
@article{osti_1540241,
title = {A modularinvariant modified Weierstrass sigmafunction as a building block for lowestLandaulevel wavefunctions on the torus},
author = {Haldane, F. D. M.},
abstractNote = {A “modified” variant of the Weierstrass sigma, zeta, and elliptic functions is proposed whereby the zeta function is redefined by $ζ(z) \mapsto \tilde{ζ}(z) ≡ ζ(z)  γ_2z$, where γ2 is a lattice invariant related to the almostholomorphic modular invariant of the quasimodularinvariant weight2 Eisenstein series. If ωi is a primitive halfperiod, $\tilde{ζ}(ω_i) = πω^{*}_{i}/A$, 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 buildingblock for the modularinvariant formulation of lowestLandaulevel wavefunctions on the torus. It is suggested that the “modified” sigma function is more natural than the original Weierstrass form, which was formulated before quasimodular forms were understood. Finally, for the highsymmetry (square and hexagonal) lattices, the modified and original sigma functions coincide.},
doi = {10.1063/1.5042618},
journal = {Journal of Mathematical Physics},
number = 7,
volume = 59,
place = {United States},
year = {2018},
month = {7}
}
Web of Science
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