# Isn't the transition at the critical point always a continuous phase transition?

At page 145 of Chaikin and Lubensky's Principles of Condensed matter physics, there are two figures 4.0.1(a) and 4.0.1(b). Figure (a) shows that at $T=T_c$, there is a continuous transition (the order parameter changes continuously) while figure (b) shows that at $T=T_c$ there is a discontinuous transition (the order parameter jumps discontinuously).

But isn't the transition at the critical point always continuous? How would the second situation as described in diagram (b) (the order parameter suddenly jumping to a nonzero value from a zero value) arise at the critical point?

Simple example: Water-ice transition. If you take the density to be your order parameter, the density drops discontinuously at $0^\text{o}$ C.

A second simple example: An Ising model in an external field at $T=0$. If the external field, $H$, is positive, the magnetization, $M$, is $+1$. If the external field is negative, $M=-1$. Clearly $M$ changes discontinuously at $H=0$.

A third, more complex example: A 2D Ising model below the critical temperature. If you don't like the fact that the above transition takes place at $T=0$, in 2D the Ising model spontaneously magnetizes even at small nonzero temperatures. In this case, just above $H=0$, $M=M_0$, and just below $H=0$, $M=-M_0$. $M$ jumps discontinuously.

In general, a critical point is where any phase transition occurs, continuous or discontinuous. It refers to the combination of parameters that sit on the boundary between one phase and another. So a critical temperature is the temperature that separates two phases (Any phases! Any type of transition!) and a critical field is the field that separates two phases, etc.

If you think about Landau theory, a first-order (discontinuous) phase transition happens when the Landau free energy has two local minima, and the global minima changes from one value to another. Consider the following graph, where $\phi$ is the order parameter and $\mathcal{L}$ is the Landau free energy.

Here, just above $T_c$, the Landau free energy has two local minima, at $\phi=0$ and $\phi>0$. Above $T_c$, $\phi=0$ is the global minimum. At $T_c$, the two minima have the same value for the Landau free energy. Below $T_c$, the minimum with $\phi>0$ has the lowest Landau free energy. Thus, above $T_c$, the system will have $\phi=0$, and below $T_c$ the system will have $\phi>0$ discontinuously.

• But in the phase diagram of water, there is a critical point only on the liquid-gas coexistence curve. There is no other critical point. $0^0$ is not the critical point. @JahanClaes – SRS Jul 27 '17 at 18:27
• @SRS If you plot the density as a function of temperature at fixed pressure, you will find that the density jumps discontinuously at $0^\text{o}$. It's true that this point is not "the critical point of water", since people usually use the phrase "critical point" to refer to the end of the liquid-gas coexistence curve when talking about water. – Jahan Claes Jul 27 '17 at 18:37
• @SRS But you can't deny that a first-order phase transition is happening at $T=0$. – Jahan Claes Jul 27 '17 at 18:37
• @SRS And more generally, people use the phrase "critical point" to refer to any point where a phase transition is happening. The term is overloaded. – Jahan Claes Jul 27 '17 at 18:38
• @JMac Critical points and triple points are different. See the diagram here – Jahan Claes Jul 27 '17 at 20:17