Kramers-Wannier duality high and low temperature expansions confusion I am reading the section on the 2D Ising model Krammer-Wannier duality in the book Exactly Solved Models in Statistical Mechanics (pg. ~76) by R.J. Baxter. I have two questions:

*

*What was the motivation behind studying this problem in the dual lattice? It seems very artificial to me.

*I understand how to get to the "low temperature" expression of the partition function
$$
\mathcal{Z}_L(N,L,K) = 2 \exp(M(K+L)) \sum_P \exp(-2(Lr+Ks)) 
$$
and to the "high temperature" expression
$$
\mathcal{Z}_H(N,\tilde{L},\tilde{K}) = 2^N(\cosh(\tilde{L})\cosh(\tilde{K}))^M\sum_P v^rw^s
$$
where $N$ is the no. of spins and $K \&L$ are the coupling constants with $\beta$ included inside them. As far as I understood, these expansions are exact in the thermodynamic limit, $N=M, N\rightarrow \infty$, (as the difference at the boundaries, if any, would vanish) so where does the low and high temperature regimes come in to play?

(Edited)
Following @YvanVelenik great help (as always!) if I let $v \equiv \tanh(\tilde{K})=e^{-2L}$ and $(K \leftrightarrow L)$ in $\mathcal{Z}_H$ hen one yields the following suggestive expression
$$
\mathcal{Z}_H(N,L,K) = 2^N(\cosh(\tanh^{-1}(e^{-2L}))\cosh(\tanh^{-1}(e^{-2K})))^M \sum_P \exp(-2(Lr+Ks)).
$$
If indeed we solve for $\sum_P \exp(-2(Lr+Ks))$ in $\mathcal{Z}_H$ and $\mathcal{Z}_L$ one yields
$$
\mathcal{Z}_H = g(M,N,K,L) \mathcal{Z}_L,
$$
where
$$
g(M,N,K,L) = \frac{2^N(\cosh(\tanh^{-1}(e^{-2L}))\cosh(\tanh^{-1}(e^{-2K})))^M}{2 \exp(M(K+L)) }
$$
is the claim that $g(M,N,K,L) \rightarrow 1$ in the high/low limits?
Thanks in advance!
 A: These expressions are valid in finite systems (in any simply connected box), but the duality transformation maps a model with $+$ boundary condition to a model with free boundary condition, which affects the finite-volume free energy. Of course, as you say, in the thermodynamic limit, the boundary condition becomes irrelevant and one obtains a nontrivial identity for the free energy density of the planar Ising model (equ. (6.2.15) in the book).
The reason they are called low-temperature and high-temperature is that $\mathcal{Z}_L$ can be seen as a polynomial in $e^{-\beta}$, which is a small quantity when $\beta$ is large, that is, at low temperatures, while $\mathcal{Z}_H$ can be seen as a polynomial in $\tanh(\beta)$, which is a small quantity when $\beta$ is small, that is, at high temperatures.
In particular, these two representations of the model lead to an expansion of the infinite-volume free energy density that is convergent at low, respectively high temperatures. (More information on these aspects can be found in Chapter 5 of this book; see Sections 5.7.3 and 5.7.4. The Kramers-Wannier duality itself is discussed in Chapter 3, Section 3.10.1.)
Moreover, the duality transformation sends the model at inverse temperature $\beta>\beta_{\rm c}$ to the model at inverse temperature $\beta<\beta_{\rm c}$ (and vice versa); the critical point $\beta_{\rm c}$ coincides with the fixed point of the transformation. In particular, this duality interchanges the low-temperature region and the high-temperature region.

In a comment, you ask why these expansions are necessary to obtain the value of the critical point. Let me briefly address this here (more information can be found in the book I mention above). For simplicity, I assume that all coupling constants are equal to $1$, so that I only have the inverse temperature $\beta$ to deal with, which simplifies a bit the expressions.
Let $\Lambda_N = \{-N, \dots, N\}^2$ and $\Lambda_N^* = \{-N-\frac12,-N+\frac12,-N+\frac32,\dots,N+\frac12\}^2$ be the dual box.
The main observation is that, up to simple explicit functions of the inverse temperature and $N$, the low-temperature expansion of $\mathcal{Z}_{\Lambda_N;\beta}^+$ and the high-temperature expansion of $\mathcal{Z}_{\Lambda_N^*;\beta^*}^{\rm free}$ coincide, provided you choose the dual inverse temperature $\beta^*$ in such a way that
$$
\tanh(\beta^*) = e^{-2\beta}.\tag{$\star$}
$$
Namely,
$$
2^{-4N^2 -8N -4}\cosh(\beta^*)^{-8N^2-10N-2}\mathcal{Z}_{\Lambda_N^*;\beta^*}^{\rm free} = e^{-\beta(8N^2+10N+2)} \mathcal{Z}_{\Lambda_N;\beta}^+.
$$
We are really interested in the free energy density in the thermodynamic limit. Therefore, let us take the logarithm and divide by $4N^2$ on both sides:
\begin{multline}
-\log(2) - 2 \log\cosh(\beta^*) + O(N^{-1}) + \frac{1}{(2N)^2} \log \mathcal{Z}_{\Lambda_N^*;\beta^*}^{\rm free}\\
=
-2\beta + O(N^{-1}) + \frac{1}{(2N)^2} \log \mathcal{Z}_{\Lambda_N;\beta}^+ .
\end{multline}
Now, we use the fact that
$$
\lim_{N\to\infty} \frac{1}{(2N)^2} \log \mathcal{Z}_{\Lambda_N^*;\beta^*}^{\rm free} = \phi(\beta^*)
$$
and
$$
\lim_{N\to\infty} \frac{1}{(2N)^2} \log \mathcal{Z}_{\Lambda_N;\beta}^+ = \phi(\beta),
$$
since the free energy density is independent of the boundary condition in the thermodynamic limit.
We thus obtain
\begin{align}
\phi(\beta)
&= 2\beta - \log(2) - 2\log\cosh(\beta^*) + \phi(\beta^*)\\
&= \phi(\beta^*) - \log \sinh(2\beta^*),
\end{align}
where we used the duality relation $(\star)$ to for the last identity.
Now, assuming that $\phi$ possesses exactly one singularity, located at the critical point $\beta_{\rm c}$, it follows from the above identity that $\beta_{\rm c} = \beta_{\rm sd}$, where $\beta_{\rm sd}$ is the self-dual point, that is, the unique value of $\beta$ such that
$$
\tanh(\beta) = e^{-2\beta}.
$$
Indeed, if $\beta_{\rm c} \neq \beta_{\rm sd}$, then the above identity implies that $\phi$ must have another singularity at
$$
\beta_{\rm c}^* = \mathrm{atanh}(e^{-2\beta_{\rm c}}),
$$
which would contradict the assumption that $\phi$ has only one singularity.
Finally, it is easy to check that $\beta_{\rm sd} = \frac12\log(1+\sqrt{2})$, which provides an explicit expression for the critical inverse temperature of the Ising model on $\mathbb{Z}^2$. This is actually the way the latter was first determined, by Kramers and Wannier, before the explicit computation of $\phi$ by Onsager.
