Does this mean that gas and liquid are the same phases? Because in the phase diagram they are connected and they have the same symmetry(translation and rotation). If they are not the same phase, how to call the state in large pressure and large temperature? Liquid or gas?

Yes. From the modern point of view, liquid and gas are in the *same* phase. Because, as the asker has mentioned, they are continuously connected in the phase diagram through the "supercritical" regime. By definition, *two states of matter are in the same phase if they can be smoothly deformed to each other without going through phase transitions*. Historically, liquid and gas are named as different phases (by mistake) because people thought that "there must be one different phase at each side of the phase transition" (as argued in Diracology's answer). But this idea is wrong.

We can not declare different phases just by observing phase transitions. Otherwise, for example, in the following phase diagram on the left, we could have declared states A and B to be in different phases, simply because they *are* separated by phase transitions, as we can first go out of the blue phase and then reenter it. This way of separating phases is clearly stupid. Any reasonable person would agree that A and B should belong to the same phase in this case. Now we just deform the left phase diagram to the right by squeezing the intermediate red phase to a first-order transition line, then why we suddenly get confused about whether A and B are in the same phase or not? Definitely, they should still remain in the same phase! The liquid-gas transition is indeed a situation like this. So a logically consistent definition will have to define liquid and gas as a single phase.

Does this mean that above the critical point the transition from gas to liquid is not a phase transition, but below the critical point, the transition from gas to liquid is a phase transition?

Yes.

If answers to my first and second question are "yes", does this mean even in the same phase there still can have phase transition? This conclusion is so weird!

Given the example of the above phase diagrams, one will not feel weird about the fact that there can be (first-order) phase transitions inside a single phase. In fact, first-order transitions often appear by merging two second-order transitions together (this can be explained by Landau's theory). So going across a first-order transition is like going out of the phase and back again immediately, which definitely can happen inside a single phase. However, I am not aware of any example that continuous phase transitions can also happen within a single phase. So I conjecture that *if a phase transition happens inside a phase, it must be first-order*. (The conjecture is falsified by the recent discovery of "unnecessary" quantum criticalities in Bi, Senthil 2018, Jian, Xu 2019, Verresen, Bibo, Pollmann 2021. -- Edited 2021) The liquid-gas transition is one example of my conjecture.

From the Landau's paradigm, what's the symmetry breaking and order parameter in the gas-liquid phase transition? It seems that the symmetry is same in gas and liquid. Gas-liquid phase transition must be able to be explained by Landau's paradigm but Landau's paradigm says that there must be symmetry breaking in phase transition. There is an answer. I admit that from a modern point of view phase transition is not necessary due to symmetry breaking, but I don't think that gas-liquid transition has been beyond Landau's paradigm.

No symmetry breaking is associated with the liquid-gas transition. Landau's paradigm only says that there must be spontaneous symmetry breaking in *second-order* transitions, but not in *first-order* transitions. In fact, nothing can be said about first-order transitions, because first-order transitions can happen anywhere in the phase diagram without any reason. The liquid-gas transition is indeed a case like this.

Even though the liquid-gas transition is not a symmetry breaking transition, it can still be described within Landau's paradigm phenomenologically (who says that Landau's theory only applies to symmetry breaking transitions?). We can introduce the density $\rho$ of the fluid as the order parameter. Because no symmetry is acting on this order parameter, so there is no symmetry reason to forbid odd-order terms like $\rho, \rho^3, \cdots$ to appear in Landau's free energy. However the first order term can always be absorbed by redefining the order parameter with a shift $\rho\to\rho+\rho_0$, then the Landau free energy takes the general form of

$$F= F_0 + a \rho^2 + b \rho^3+ c \rho^4+\cdots.$$

First-order transition happens by driving the parameter $a$ to $a<9b^2/32c$. From this example, we can see that (within Landau's paradigm) *if a phase transition happens without breaking any symmetry, it must be first-order*. Again the liquid-gas transition is one such example.

So if given one phase, we firstly find the symmetry is same in this phase and then check several order parameters also same in this phase. However, how do you prove you must not be able to construct some weird order parameters such that in one part of this phase is zero and in another part of this phase is nonzero? For example, in a solid phase of water which has the same crystal structure, how to prove any order parameter that you can construct will not be zero in one part of the phase and nonzero in another part?

Indeed, you can never rule out the possibility that some weird order parameter is hiding there to further divide the phase into more phases. That is actually why the solid phase of water is divided into so many different crystal phases. Each crystal structure is associated with a different symmetry-breaking pattern. Sometimes the symmetries are just so complicated that you may miss one or two of them if you are not careful enough. In that case, you will also miss the order parameters associated with the missing symmetry, until you see a specific heat anomaly in the experiment where you did not expect, then you start to realize that oh there is a missing order parameter that actually changes across this transition, and one needs to add some additional symmetry to explain it. This is actually the typical way that physicists work every day, they never figure out the full classification of phases until they see the evidence for new phases and phase transitions. I think this is also the fun part of condensed matter physics: *there are always new phases of matter waiting for us to discover*.

There are two different ideas that are mixed up in this question. Let me try to separate them.

The first has to do with the form of a thermodynamic phase diagram. Consider, for example, the phase diagram of a magnet in the plane of temperature T and external magnetic field H. This is a 2-dimensional space. As GGphys points out, there is a line in this space across which the magnetization changes discontinuously. For a symmetric magnet, this line is on the $H=0$ axis, from $T=0$ to some higher temperature $T=T_c$. If we cross this line, say from $H >0$ to $H<0$, there is a discontinuity in $M$, and we call this a first-order phase transition. At very high temperature, $M = 0$ when $H = 0$. In practice, on the line of discontinuity, $M$ decreases and goes to zero at $T = T_c$. Above $T_c$, there is no discontinuity. We call the behavior at $T_c$ a second-order phase transition, and we call the point $H=0, T = T_c$ the "critical point".

The vicinity of the point $H=0$, $T=0$ is a special place with long-range correlations of the magnetization and, singularities in thermodynamic functions. For example, the magnetic susceptibility $\partial M/\partial H$ goes to infinity at $H=0$, $T=0$. So a special, more sophisticated, theory is needed to describe this region.

This is all for a 2-dimensional phase diagram. If we add more thermodynamic variables, all of this extends to higher dimensions. An example is an antiferromagnet in an external magnetic field or a liquid-gas system with a varying density of a solute in the liquid. Maybe the example of the antiferromagnet is most clear. We can imagine a "staggered" magnetic field $H_s$ that turns on the antiferromagnetic order. The variables are then $H_s$, a uniform field $H$, and the temperature. This is a 3-dimensional thermodynamic space. The line of discontinuities becomes a plane of discontinuities in the plane $H_s$ = 0, and the second-order phase transition becomes a line of critical points at $H_s = 0$, $T = T_c(H)$. There can also be other features. If you make the uniform $H$ field very large, all of the spins will line up uniformly. So if you start from the antiferromagnetic phase and raise $H$, eventually you will find a transition (usually, discontinuous or first-order) above which there is a uniform magnetization. It is hard to make a staggered magnetic field in the lab, so when we measure the phases of an antiferromagnet we are restricted to the $H_s = 0$ plane. This has a (vertical) line of second-order phase transitions at $T = T_c(H)$ and a (horizontal) line of first-order phase transitions at $H = H_f(T)$ that meet and end at an even more special point called the "tricritical" point. As you add more thermodynamic variables, more complicated behaviors are possible.

So far, I have not talked about Landau theory. Landau theory is an approximation to the free energy in which you assume that $M$ is small and write the Gibbs free energy as a polynomial in $M$. This gives an approximate description in the vicinity of the critical point. Even away from the critical point, Landau theory gives a qualitative description of the phase diagram. For example, for the antiferromagnet above, the Landau theory would have a polynomial in $M$ and the staggered magnetization $M_s$, and it would show the various phase transitions that I have described.

Landau theory is then useful to get a qualitative picture of the phase diagram in however many dimensions you have. It turns out that it is also useful as a starting point for a quantitative theory of the thermodynamics near a critical point.

## Best Answer

Good point, I think, but when we look at your progression, from (ii) -> (iii), you break two unrelated symmetries (lattice rotation, then lattice translation). This is fine, and you would have two distinct second order transitions.

Butwhen you try to bring the two critical points together, you correctly notice that this is going to require fine tuning of the parameters. Now, that's not to say such points don'texist.I think they tend to go under the name "multicritical points," and I seem to recall work, possibly by Dagotto, arguing that CMR transitions were governed by some sort of bicriticality (not my expertise, unfortunately.)The point though, and I think the point Landau is making, is that that sort of fine tuning is unlikely to be relevant for real phase transitions, with only a handful of knobs (That is, you write down the action density and it's something like $a \phi^2+b \psi^2+$interactions etc. but for reasons beyond our control, $a(g)$ and $b(g)$ must both go to zero at the same value of $g$... On the other hand, some folks believe there are "deconfined"

quantumcritical points, where no fine-tuning is necessary to drive two unrelated order parameters to zero at the same "tuning" parameter point.)