Self-similarity in the classification of finite-size scaling functions for toroidal boundary conditions

The conventional periodic boundary conditions in two dimensions are extended to general boundary conditions, prescribed by primitive vector pairs that may not coincide with the coordinate axes. This extension is shown to be unambiguously specified by the twisting scheme. Equivalent relations between different twist settings are constructed explicitly. The classification of finite-size scaling functions is discussed based on the equivalent relations. A self-similar pattern for distinct classes of finite-size scaling functions is shown to appear on the plane that parametrizes the toroidal geometry.

Figure 1

Two primitive vectors on a planar square lattice: (a) general case; (b) the orthogonal case with the deviation from the coordinate axes of the underlying lattice by an angle $\chi$ called chirality.

Figure 2

(a) Twisted BC specified by the primitive vectors for the aspect ratio $A=N∕M$ and the twisting factor $\alpha =d∕M$. The vectors are rotated counterclockwise with the angle $\pi ∕2-\arctan \left(\alpha \right)$ to another set, shown in (b). By reversing one vector followed by $\alpha \to -\alpha$, we achieve the vectors shown in (c), which gives rise to the new aspect ratio $(1+{\alpha }^2)∕A$ and leaves $\alpha$ unaltered.

Figure 3

Binder parameter $g\left(m\right)$ versus $L(T-T_c)∕T_c$ for different lattices related by the transformation (a) $\alpha \to sA\pm \alpha$ with $A$ kept fixed and (b) $A\to (1+{\alpha }^2)∕A$ with $\alpha$ kept fixed. The left (right) vertical scale is for $q=2$ (3), the data are connected with a smooth solid line for $q=3$.

Figure 4

The same plot as Fig. 3 for lattice structures with $A=\left(\mathrm{a}\right)$ 1 and (b) 0.5 but different $\alpha$ values.

Figure 5

Equivalent geometric structures possess the same FSSFs as $A=1$ in the plane of $A$ vs $\alpha$. The circles mark the loci of fully helical tori. The gray regions are $R_2$ and the others are $R_1$ as discussed in the text.

Figure 6

Geometric structures in the plane of $u$ vs $v$ for two classes of FSSFs, with lines of crosses for $A=1$ and dotted lines for $A=0.5$. The circles mark the loci of fully helical tori.