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

TL;DR

This paper derives exact partition functions for the Ising model on various planar lattices with different boundary conditions, revealing how these conditions affect thermodynamic properties like specific heat.

Contribution

It introduces a Grassmann path integral method to compute exact partition functions for the Ising model on square, triangular, and honeycomb lattices with multiple boundary conditions.

Findings

Partition functions calculated for different boundary conditions.

Specific heat varies with boundary conditions on square lattices.

Boundary conditions influence finite-size effects in the Ising model.

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, $\theta =k_BT/J$. 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.