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

Abstract

The Grassmann path integral approach is used to calculate exact partition functions of the Ising model on MxN square (sq), plane triangular (pt) and honeycomb (hc) 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 the temperature, $\theta =k_{B}T/J$. We find that for the NxN sq lattice, $C/k_B$ for pa and ap boundary conditions are different from those for aa boundary conditions, but for the NxN pt and hc 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.

17 pages, 13 Postscript figures, uses iopams.sty, submitted to J. Phys. A: Math. Gen

References in corpus (6)

• Exact Finite-Size Scaling and Corrections to Scaling in the Ising Model with Brascamp-Kunz Boundary Conditions
• Kronecker's Double Series and Exact Asymptotic Expansion for Free Models of Statistical Mechanics on Torus
• Finite-size scaling for the Ising model on the Moebius strip and the Klein bottle
• Exact finite-size corrections for the square lattice Ising model with Brascamp-Kunz boundary conditions
• Exact Finite-Size-Scaling Corrections to the Critical Two-Dimensional Ising Model on a Torus. II. Triangular and hexagonal lattices
• Universal scaling functions for bond percolation on planar random and square lattices with multiple percolating clusters

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