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 $f$ is presently available for some other planar lattices. But determining exactly the critical temperature $T_c$ only requires a partial derivation of $f$. It...
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2025-07-01
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| Series: | SciPost Physics |
| Online Access: | https://scipost.org/SciPostPhys.19.1.025 |
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| author | Laurent Pierre, Bernard Bernu, Laura Messio |
| author_facet | Laurent Pierre, Bernard Bernu, Laura Messio |
| author_sort | Laurent Pierre, Bernard Bernu, Laura Messio |
| collection | DOAJ |
| description | 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 determining exactly 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 significant 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. |
| format | Article |
| id | doaj-art-e4ae3deac5d649d4a18583f8d4606ee7 |
| institution | Kabale University |
| issn | 2542-4653 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | SciPost |
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| series | SciPost Physics |
| spelling | doaj-art-e4ae3deac5d649d4a18583f8d4606ee72025-08-20T03:28:09ZengSciPostSciPost Physics2542-46532025-07-0119102510.21468/SciPostPhys.19.1.025Derivation of free energy, entropy and specific heat for planar Ising models: Application to Archimedean lattices and their dualsLaurent Pierre, Bernard Bernu, Laura MessioThe 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 determining exactly 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 significant 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.https://scipost.org/SciPostPhys.19.1.025 |
| spellingShingle | Laurent Pierre, Bernard Bernu, Laura Messio Derivation of free energy, entropy and specific heat for planar Ising models: Application to Archimedean lattices and their duals SciPost Physics |
| title | Derivation of free energy, entropy and specific heat for planar Ising models: Application to Archimedean lattices and their duals |
| title_full | Derivation of free energy, entropy and specific heat for planar Ising models: Application to Archimedean lattices and their duals |
| title_fullStr | Derivation of free energy, entropy and specific heat for planar Ising models: Application to Archimedean lattices and their duals |
| title_full_unstemmed | Derivation of free energy, entropy and specific heat for planar Ising models: Application to Archimedean lattices and their duals |
| title_short | Derivation of free energy, entropy and specific heat for planar Ising models: Application to Archimedean lattices and their duals |
| title_sort | derivation of free energy entropy and specific heat for planar ising models application to archimedean lattices and their duals |
| url | https://scipost.org/SciPostPhys.19.1.025 |
| work_keys_str_mv | AT laurentpierrebernardbernulauramessio derivationoffreeenergyentropyandspecificheatforplanarisingmodelsapplicationtoarchimedeanlatticesandtheirduals |