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

The 2d ferromagnetic Ising model was solved by Onsager on the square lattice in 1944, and an explicit expression of the free energy density

is presently available for some other planar lattices. But determining exactly the critical temperature

T_cTc

only requires a partial derivation of

ff

. 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_cTc

:

c_V(T)\sim -A\ln|1-T_c/T|cV(T)∼−Aln|1−Tc/T|

, in all the ferromagnetic and some antiferromagnetic cases. While the non-universal weight

AA

of the leading term has often been evaluated, this is not the case for the sub-leading order term

BB

such that

c_V(T)+A\ln|1-T_c/T|\sim BcV(T)+Aln|1−Tc/T|∼B

, despite its significant impact on the

c_V(T)cV(T)

values in the vicinity of

T_cTc

, particularly important in experimental measurements. Explicit values of

T_cTc

,

AA

,

BB

and other thermodynamic quantities are given for the Archimedean lattices and their duals for both ferromagnetic and antiferromagnetic interactions.