A modular-invariant modified Weierstrass sigma-function as a building block for lowest-Landau-level 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 almost-holomorphic modular invariant of the quasi-modular-invariant weight-2 Eisenstein series. If ω i is a primitive half-period, $$\tilde{ζ}(ω_i) = πω^{*}_{i}/A$$, where A is the area of the primitive cell of the lattice. The quasiperi-odicity 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. Finally, for the high-symmetry (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 0022-2488
- 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 modular-invariant modified Weierstrass sigma-function as a building block for lowest-Landau-level wavefunctions on the torus. United States: N. p., 2018.
Web. doi:10.1063/1.5042618.
Haldane, F. D. M. A modular-invariant modified Weierstrass sigma-function as a building block for lowest-Landau-level wavefunctions on the torus. United States. doi:10.1063/1.5042618.
Haldane, F. D. M. Fri .
"A modular-invariant modified Weierstrass sigma-function as a building block for lowest-Landau-level wavefunctions on the torus". United States. doi:10.1063/1.5042618. https://www.osti.gov/servlets/purl/1540241.
@article{osti_1540241,
title = {A modular-invariant modified Weierstrass sigma-function as a building block for lowest-Landau-level 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 almost-holomorphic modular invariant of the quasi-modular-invariant weight-2 Eisenstein series. If ωi is a primitive half-period, $\tilde{ζ}(ω_i) = πω^{*}_{i}/A$, where A is the area of the primitive cell of the lattice. The quasiperi-odicity 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. Finally, for the high-symmetry (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
Works referenced in this record:
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