Phase transition in parameter (and not temperature) for a classical system

Question

Consider the $q$-state Potts model on $\mathbb{Z}^d$ for some integer $q$ - this also has an FK-representation for any real number $q$.

For $d = 2$ the model is exactly solvable and has a critical temperature at the point $p_c(q) = \frac{ \sqrt{q}}{1+\sqrt{q}}$. Further, it is know that for $q \leq 4$ this phase transition is continous. This for example means that with boundary conditions corresponding to all spins on the boundary pointing in one direction we have that the magnetisation at critical temperature $m_{q, p_c(q)}$ is 0. For $q > 4$ the magnetization at criticality $m_{q, p_c(q)}$ is strictly larger than 0.

Hence if I consider only the models at the thermal criticality then the magnetization is an order parameter for a phase transition in the variable $q$, i.e $m_{q, p_c(q)} = 0 $ for $q \leq q_c$ and $m_{q, p_c(q)} > 0 $ for $q > q_c$. In this case $q_c = 4$ in two dimensions.

Question 1: What are other examples of classical models which exhibit such a phase transition in a parameter and not in temperature?

Question 2: What is know for the Potts model (or FK/random cluster representations) in other dimensions than 2? Does one know that $m_{q, p_c(q)} \to 0 $ for $q \to q_c$ from above? Comment: One can see this from explicit computations for $d=2$.

Edit: One of the motivations for this equations is the study of the Kertész line for the random cluster model. The Kertész line is the percolation phase transition that occurs whenever one implements the random cluster model with a ghost and considers percolation of the model without using the ghost. Getting methods that can answer the questions above might help answer question about when the Kertész line phase transition is continuous as is elaborated on to some extend in our recent preprint: https://arxiv.org/abs/2206.07033

Answer 1

What are other examples of classical models which exhibit such a phase transition in a parameter and not in temperature?

Formulated like this, it is difficult to answer, because there are far too many examples (the simplest example would be the first-order phase transition in the Ising model below the critical temperature as the magnetic field crosses $0$; see, for instance, Chapter 3 in this book).

There exist, in fact, constructions allowing you to build models in which the order parameter is given by an essentially arbitrary finite collection $(f_1,\dots,f_n)$ of local functions! See Section 16.13 in Georgii's book for more on this.

Alternatively, if you write down a Hamiltonian depending on various parameters, whose set of ground states changes as these parameters are varied, then, under suitable assumptions, you can prove that the corresponding zero-temperature phase diagram is homeomorphic to the phase diagram at (small) positive temperatures. The standard way of doing this is via the Pirogov-Sinai theory (see Chapter 7 in this book).

The above results show how common phase transitions driven by a parameter different from the temperature actually are.

I feel, however, that you are interested in a more specific situation, but then you should be more explicit.

What is know for the Potts model (or FK/random cluster representations) in other dimensions than 2? Does one know that $m_{q,p_c(q)}\to 0$ for $q\to q_c$ from above?

Even for $q=1$ (Bernoulli percolation), it is not known in general that $m_{q,p_c(q)}=0$ (it is known when $d=2$ and when $d\geq 11$, I think). So, even the problem of determining the order of the phase transition for general values of $q$ is largely open above dimension 2. The only exceptions are $q=2$ (the Ising model, see this paper) and $q\gg 1$. In the latter case, the transition is known to be first order (the first proof is due to Kotecký and Shlosman).

Note that, when $d\geq 3$, the phase transition is expected to be of first order for all $q>2$. This has been proved (for integer $q$) for models with interactions of sufficiently long (but finite) range in this paper (see also this one); this even applies to the two-dimensional model, showing that the behavior of the planar (that is, nearest-neighbor) model is far from generic.