Universal finite-size scaling functions for critical systems with tilted boundary conditions

Abstract

We calculate finite-size scaling functions (FSSF's) of Binder parameter g and magnetization distribution function $p(m)$ for the Ising model on ${L}_{1}\ifmmode\times\else\texttimes\fi{}{L}_{2}$ square lattices with periodic boundary conditions in the horizontal ${L}_{1}$ direction and tilted boundary conditions in the vertical ${L}_{2}$ direction such that the $i\mathrm{th}$ site in the first row is connected with the $\mathrm{mod}{(i+cL}_{1}{,L}_{1})\mathrm{th}$ site in the ${L}_{2}$ row of the lattice, where $1l~il~{L}_{1}.$ For fixed sets of $(a,c)$ with ${a=L}_{1}{/L}_{2},$ the FSSF's of g and $p(m)$ are universal and in such cases ${a/(c}^{2}{a}^{2}+1)$ is an invariant. For percolation on lattices with fixed a, the FSSF of the existence probability (also called spanning probability) is not affected by c.