Exact heat capacity of the 2-dimensional Ising model

Question

The following is a section from the book

Newman, M., and G. Barkema. "Monte carlo methods in statistical physics" New York, USA (1999).

and then:

From those two quotes, it seems that there is a known exact analytical formula for the specific heat of the Ising model in the thermodynamic limit (probably connected to Onsager's solution), at least in 2 dimensions.

But I cannot find this expression. Two questions:

1. Is there a known exact analytical formula for the specific heat of the 2-dimensional Ising model in the thermodynamic limit?
2. If there is such a formula, what is it? Can it be derived from Onsager's solution?

Note that in 2. I am not asking for a full derivation of the result, since I guess that would involve Onsager's full tour de force. I am just asking for a derivation that connects it with Onsager's solution. For instance, can it be derived from Onsager's free energy (whose expression I know)?

Answer 1

Yes, it can be derived from Onsager's formula for the free energy $F$. Indeed, the internal energy $u$ can be obtained from $u=\frac{\partial\beta F}{\partial\beta}$ and then the specific heat from $c=\frac{\partial u}{\partial T}$.

The specific heat was derived many times, and is already present in Onsager's famous paper. The detailed computation can be found, for instance, in Section 3 of Chapter 5 of McCoy and Wu's book. The expression is rather complicated, so I don't think it makes sense to reproduce it here.

If you're interested in comparison with finite systems (for numerical simulations), I would also suggest that you have a look at the well-known paper by Ferdinand and Fisher. This paper contains the first detailed discussion of finite-size scaling for the specific heat (extending finite-system computations already present in Onsager's paper!).