Exact partition functions of the Ising model on M × N planar lattices with periodic-aperiodic boundary conditions

The Grassmann path integral approach is used to calculate exact partition functions of the Ising model on M × N square (sq), plane triangular (pt) and honeycomb (he) lattices with periodic-periodic (pp), periodic-antiperiodic (pa), antiperiodic-periodic (ap) and antiperiodic-antiperiodic (aa) boundary conditions. The partition functions are used to calculate and plot the specific heat, C/k_B, as a function of temperature, θ = k_BT/J. We find that for the N × N sq lattice, C/k_B for pa and ap boundary conditions are different from those for aa boundary conditions, but for the N × W pt and he lattices, C/k_B for ap, pa and aa boundary conditions have the same values. Our exact partition functions might also be useful for understanding the effects of lattice structures and boundary conditions on critical finite-size corrections of the Ising model.