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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
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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)∼-Aln|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)+Aln|1-T_c/T|∼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.
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