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

doi:10.1063/1.5042618

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 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:

Haldane, F. D. M. ^[1]

Princeton Univ., NJ (United States). Dept. of Physics

Publication Date:: 2018-07-06

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.

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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.  doi: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.  doi: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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DOI: 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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