Spin-projection orientations in the plane square-lattice Ising model   with periodic boundary conditions
  The periodic boundary conditions changed the plane square-lattice Ising model to the torus-lattice system which restricts the spin-projection orientations. Only two of the three important spin-projection orientations, parallel to the x-axis or to the y-axis, are suited to the torus-lattice system. The infinitesimal difference of the free-energies of the systems between the two systems mentioned above makes their critical temperatures infinitely close to each other, but their topological fundamental groups are distinct. 
You-gang Feng