Exact treatment of Ising model on the helical tori
  The exact closed forms of the partition functions of 2D Ising model on square lattices with twisted boundary conditions are given. The constructions of helical tori are unambiguously related to the twisted boundary conditions by virtue of the SL(2,Z) transforms. Numerical analyses reveal that the finite size effect is irrelevant to the chirality equipped with each helical boundary condition. 
Tsong-Ming Liaw, Ming-Chang Huang, Yen-Liang Chou, Simon C. Lin and   Feng-Yin Li