Why is there only one critical point in Ising model? While reading about Kramers-Wannier duality in "Statistical Field Theory" by Giuseppe Mussardo, I read that the hypothesis that there is only one critical point is fully justified from the physical point of view. However, I didn't find that kind of argument in the original paper by Kramers and Wannier "Statistics of the Two-Dimensional Ferromagnet. Part I". 
Could someone explain what this physical argument is about?
 A: Let me denote by $T_1$ the temperature such that there is long-range order for all $T<T_1$ but not for $T>T_1$. Then, let us denote by $T_2$ the temperature such that the 2-point function decays exponentially with the distance for all $T>T_2$, but not for $T<T_2$.
On $\mathbb{Z}^2$, it is easy to see that $T_1$ and $T_2$ are images of each other through the Kramers-Wannier duality transformation (for example, by observing that the low temperature surface tension is mapped by duality to the high temperature rate of exponential decay of the 2-point function). So what you want to show is that $T_1=T_2$, since this would then imply that both coincide with the self-dual temperature.
Now, observe first that it is impossible that $T_1>T_2$, since this would imply the existence of a phase with both long-range order and exponential decay of the 2-point function when $T_2<T<T_1$.
So, there remains to show that $T_1<T_2$ is also impossible (that is, there is no intermediate phase with no long-range order, but slow decay of the 2-point function). This is not easy.
There exist several proofs that there is no intermediate phase in the Ising model (actually, those work in any dimension); e.g., this one, this one or this one.
In principle, you can extract "physical arguments" from those, albeit not elementary ones.
Note, however, that there are generalizations of the Ising model for which this is not true, for example:


*

*the Ashkin-Teller model, in which the symmetry can be broken in two stages;

*the $q$-clock model with $q\geq 5$ for which there are two phase transitions as $T$ increases: a first one from a magnetized phase to a massless phase, and then a second one from this massless phase to a massive phase.


Such examples show that very generic "physical arguments" are unlikely to apply.
