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

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 $$d=2$$.

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 $$q\geq 3$$. This has been proved 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.

• Thanks a lot! I agree that I was too broad in the first question. Maybe it should be more like how does one show continuity of a phase transition in parameter which is not the temperature? For example in your Ising example with a magnetic field $h \to 0$ then that phase transition is continuous as long as $\beta \leq \beta_c$, but discontinuous for $\beta > \beta_c$ I know from physics. But how would I prove that with a general argument. (In this case it is clear since the limit of the magnetisation is the spontaneous magnetisation with wired boundary conditions -right?) – Frederik Ravn Klausen Aug 26 '20 at 17:08
• I am very happy that our book motivated you to work in this field :) . Concerning your first comment: (1) there is no phase transition in $h$ when $\beta<\beta_c$ (unique Gibbs state, analytic pressure), but there is a continuous phase transition when $\beta=\beta_c$. (2) As I briefly mention in my answer, the standard way to prove the existence of a first-order phase transition in perturbative regimes is through Peierls' argument and its generalization (in particular, the Pirogov-Sinai theory). In some models, this can then be made non-perturbative using suitable correlation inequalities. – Yvan Velenik Aug 26 '20 at 17:21
• (And you're correct: the spontaneous magnetisation is indeed equal to the probability of belonging to the infinite cluster under wired b.c.) Concerning your second comment: my point is only that one cannot even answer the simpler question of whether there is percolation at the critical point. So (in the absence of a miracle), it looks quite unlikely that one can prove continuity in $q$ at $p_c(q)$. Especially since there is no nice characterization of $p_c(q)$ similar to the one you have in the planar case thanks to duality. – Yvan Velenik Aug 26 '20 at 17:24
• Of course, there may be a very clever trick to obtain this information in FK. You should probably directly ask Hugo :) . I have no doubt that he thought about that. – Yvan Velenik Aug 26 '20 at 17:28
• I don't know of another one. But the fact that one does not have (and presumably cannot have), in the non-planar case, a nice characterization of the path $(q,p_c(q))$ along which the limit is taken makes it very unlikely to me that an independent argument can be made. – Yvan Velenik Aug 27 '20 at 5:31