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Title: 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 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.

Authors:
 [1]
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  1. 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. https://doi.org/10.1063/1.5042618.
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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. https://doi.org/10.1063/1.5042618
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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. https://doi.org/10.1063/1.5042618. https://www.osti.gov/servlets/purl/1540241.
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@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}
}
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https://doi.org/10.1063/1.5042618
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Works referenced in this record:

Elliptic Functions According to Eisenstein and Kronecker: An Update
journal, January 2016

  • Charollois, Pierre; Sczech, Robert
  • EMS Newsletter, Vol. 2016-9, Issue 101
  • DOI: 10.4171/news/101/4

A new construction of Eisenstein’s completion of the Weierstrass zeta function
journal, July 2015

  • Rolen, Larry
  • Proceedings of the American Mathematical Society, Vol. 144, Issue 4
  • DOI: 10.1090/proc/12813
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    • Haldane, F. D. M.
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