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

Laurent Pierre, Bernard Bernu, Laura Messio

Published 2025-01-30Version 1

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. An exact derivation of the critical temperature $T_c$ only requires a partial derivation of $f$ and has been performed on many lattices, including the 11 Archimedean lattices. We give general expressions of the free energy, energy, entropy and specific heat for planar lattices with a single type of non-crossing links. 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 $c_V(T)$ values in the vicinity of $T_c$, particularly important in experimental measurements. Explicit values of these thermodynamic quantities and of $A$ and $B$ are given for the Archimedean lattices and their dual for both ferromagnetic and antiferromagnetic interactions.