Derivation of the free energy, entropy and specific heat for planar Ising models: Application to Archimedean lattices and their duals

TL;DR

This paper derives explicit formulas for free energy, entropy, and specific heat for planar Ising models on Archimedean lattices and their duals, including critical behavior and sub-leading terms near phase transitions.

Contribution

It provides general expressions for thermodynamic quantities of planar Ising models with non-crossing links, including sub-leading terms in specific heat near critical temperature.

Findings

Explicit critical temperatures and thermodynamic parameters for Archimedean lattices.

Detailed analysis of specific heat singularity and sub-leading terms.

Application to both ferromagnetic and antiferromagnetic cases.

Abstract

The 2d ferromagnetic Ising model was solved by Onsager on the square lattice in 1944, and an explicit expression of the free energy density $f$ is presently available for some other planar lattices. But an exact derivation of the critical temperature $T_c$ only requires a partial derivation of $f$. It has been performed on many lattices, including the 11 Archimedean lattices. In this article, we give general expressions of the free energy, energy, entropy and specific heat for planar lattices with a single type of non-crossing links. It is known that the specific heat exhibits a logarithmic singularity at $T_c$: $c_V(T)\sim -A\ln|1-T_c/T|$, in all the ferromagnetic and some antiferromagnetic cases. While the non-universal weight $A$ of the leading term has often been evaluated, this is not the case for the sub-leading order term $B$ such that $c_V(T)+A\ln|1-T_c/T|\sim B$, despite its strong impact on the $c_V(T)$ values in the vicinity of $T_c$, particularly important in experimental measurements. Explicit values of $T_c$, $A$, $B$ and other thermodynamic quantities are given for the Archimedean lattices and their duals for both ferromagnetic and antiferromagnetic interactions.