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

Journal Article · · Journal of Mathematical Physics
DOI:https://doi.org/10.1063/1.5042618· OSTI ID:1540241
 [1]
  1. Princeton Univ., NJ (United States). Dept. of Physics; DOE/OSTI
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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.

Research Organization:
Princeton Univ., NJ (United States)
Sponsoring Organization:
USDOE Office of Science (SC)
Grant/Contract Number:
SC0002140
OSTI ID:
1540241
Alternate ID(s):
OSTI ID: 1459239
Journal Information:
Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 7 Vol. 59; ISSN 0022-2488
Publisher:
American Institute of Physics (AIP)Copyright Statement
Country of Publication:
United States
Language:
English

References (2)

A new construction of Eisenstein’s completion of the Weierstrass zeta function journal July 2015
Elliptic Functions According to Eisenstein and Kronecker: An Update journal January 2016

Cited By (9)

The origin of holomorphic states in Landau levels from non-commutative geometry and a new formula for their overlaps on the torus journal August 2018
Modular covariance properties of composite fermions on the torus journal February 2019
Lattice Monte Carlo for quantum Hall states on a torus journal March 2019
Berry Phase and Model Wave Function in the Half-Filled Landau Level journal October 2018
Dirac Fermion Hierarchy of Composite Fermi Liquids journal June 2019
Hall viscosity of composite fermions journal February 2020
Lattice Monte Carlo for Quantum Hall States on a Torus text January 2017
Dirac Fermion Hierarchy of Composite Fermi Liquids text January 2018
Hall Viscosity of Composite Fermions text January 2019

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Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
physics