模不变修正Weierstrass σ函数作为环面上最低朗道能级波函数的构造块
A modular-invariant modified Weierstrass sigma-function as a building block for lowest-Landau-level wavefunctions on the torus
摘要：
一个“修改”的变体维尔斯特拉斯西格玛，zeta，和椭圆函数的建议，其中zeta函数是重新定义的$ζ(z)$ $\mapsto$ $\tilde ζ(z)$ $\equiv$ $ζ(z) - γ_2z$，其中$γ_2$是一个晶格不变量有关的准模不变重量-2爱森斯坦系列的几乎全纯模不变量。如果$ω_i$是基元半周期，则$\tildeζ(ω_i)$ = $πω_i^*/A$，其中$A$是晶格的基元单元的面积。修改后的sigma函数的准周期性比原来的要简单得多，它成为环面上最低朗道能级波函数的模不变公式的基石。有人建议，“修改”的西格玛函数是更自然的比原来的维尔斯特拉斯形式，这是制定之前，准模形式的理解。对于高对称性（正方形和六边形）晶格，修改后的和原始的西格玛函数重合。
英文摘要：
A "modified" variant of the Weierstrass sigma, zeta, and elliptic functions is proposed whereby the zeta function is redefined by $ζ(z)$ $\mapsto$ $\tilde ζ(z)$ $\equiv$ $ζ(z) - γ_2z$, 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, $\tildeζ(ω_i)$ = $πω_i^*/A$, 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 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. For the high-symmetry (square and hexagonal) lattices, the modified and original sigma functions coincide.
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