Consider Onsager's exact solution of two-dimensional Ising model for square lattices with nearest neighbour interaction energy ‘J ‘being equal in the horizontal and vertical directions. At the critical temperature, the partition function estimated by numerical integration (e.g using eqn 115 of Onsager’s original paper) is approximately 0.92969. In a few Journal articles, it is mentioned that the integrand appearing in the double integral of Onsager’s exact solution leads to a singularity of the free energy at the critical temperature. Can this be clarified?
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$\begingroup$ As already noted in Onsager's famous paper, the specific heat diverges as $-\log|T-T_c|$ as $T$ approaches the critical temperature $T_c$. $\endgroup$– Yvan VelenikCommented Nov 13, 2023 at 13:46
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$\begingroup$ While the divergence of specific heat with logarithmic singularity from Onsager's exact solution is well-known, whether the free energy arising from the double integral also becomes infinity at the critical temperature seems unclear. $\endgroup$– sangaraCommented Nov 14, 2023 at 10:23
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$\begingroup$ The free energy density does not diverge at $T_c$: it is finite for all $T$… You need to consider second order derivatives of this quantity to observe singular behavior. $\endgroup$– Yvan VelenikCommented Nov 14, 2023 at 10:47
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$\begingroup$ Note that you don’t need the exact solution to see that: existence and finiteness of the free energy density is an extremely general (and rather soft) result. $\endgroup$– Yvan VelenikCommented Nov 14, 2023 at 10:52
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$\begingroup$ Thanks for the complete clarity . $\endgroup$– sangaraCommented Nov 15, 2023 at 7:44
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