Kramers-Wannier duality high and low temperature expansions confusion

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.

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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 $$ tanh(beta^*) = e^{-2beta}.tag{$star$} $$ 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 logcosh(beta^*) + O(N^{-1}) + frac{1}{(2N)^2} log mathcal{Z}_{Lambda_N^*;beta^*}^{rm free} = -2beta + O(N^{-1}) + frac{1}{(2N)^2} log mathcal{Z}_{Lambda_N;beta}^+ . end{multline} Now, we use the fact that $$ lim_{Ntoinfty} frac{1}{(2N)^2} log mathcal{Z}_{Lambda_N^*;beta^*}^{rm free} = phi(beta^*) $$ and $$ lim_{Ntoinfty} frac{1}{(2N)^2} log mathcal{Z}_{Lambda_N;beta}^+ = phi(beta), $$ since the free energy density is independent of the boundary condition in the thermodynamic limit.

We thus obtain begin{align} phi(beta) &= 2beta - log(2) - 2logcosh(beta^*) + phi(beta^*) &= phi(beta^*) - log sinh(2beta^*), 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 $$ tanh(beta) = e^{-2beta}. $$ Indeed, if $beta_{rm c} neq beta_{rm sd}$, then the above identity implies that $phi$ must have another singularity at $$ beta_{rm c}^* = mathrm{atanh}(e^{-2beta_{rm c}}), $$ which would contradict the assumption that $phi$ has only one singularity.

Finally, it is easy to check that $beta_{rm sd} = frac12log(1+sqrt{2})$, 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.