# Symmetry Breaking And Phase transition

1. Is every phase transition associated with a symmetry breaking? If yes, what is the symmetry that a gaseous phase have but the liquid phase does not?

2. What is the extra symmetry that normal $\bf He$ has but superfluid $\bf He$ does not? Is the symmetry breaking, in this case, a gauge symmetry breaking?

Update Unlike gases, liquids have short-range order. Does it not mean that during the gas-to-liquid transition, the short-range order of liquids breaks the translation symmetry? At least locally?

• 1- the order parameter is related to the difference of the densities in the two phases, though I don't remember the details. 2- It is the U(1) global symmetry (not really a gauge symmetry, since there is no gauge field).
– Adam
Mar 26, 2014 at 13:41
• for the liquid-gas transition, short range correlation is not important (though it has of course important physical consequences). The order parameter is something like $\rho_A-\rho_B$ where $\rho_A$ is the density in the current phase, and $\rho_B$ the other. It plays the same role than the magnetisation in the Ising model (the symmetry is $Z_2$).
– Adam
Mar 26, 2014 at 14:23
• @YvanVelenik - Then the conclusion is: All phase transitions are not associated with symmetry breaking.
– SRS
Mar 26, 2014 at 17:32
• @Roopam : yes, of course they are not always related to symmetry breaking. Mar 26, 2014 at 20:27
• @YvanVelenik: I never said that all transition is associated with a symmetry breaking, which is definitely not the case. I'm just arguing that saying that there is no symmetry breaking in the liquid-gas transition is too crude an answer.
– Adam
Mar 28, 2014 at 13:55

## 3 Answers

Let me answer your first question: Phase transitions do not necessarily imply a symmetry breaking. This is clear in the example your are mentioning : The liquid-gas transition is characterized by a first order phase transition but there is no symmetry breaking. Indeed, liquid and gas share the same symmetry (translation and rotation invariance) and may be continuously connected in the high temperature/pressure regime. In quantum systems at zero-temperature, one may also encounter transition in between quantum spin-liquid states for which there is also no symmetry breaking. Yet another example is the case of the 2D XY model where there is a continuous phase transition but there is no symmetry breaking (Kosterlitz-Thouless transition).

• KT and spin liquids are nice example, but the liquid-gas case is more subtle. It can be mapped onto an Ising model, with the associated order parameter and symmetry broken phase.
– Adam
Mar 27, 2014 at 19:38
• Note that it is easy to construct models with no symmetry in which 1st order phase transitions occur (actually this is the generic situation! You have to work harder to construct models with symmetry breaking). There is a mathematical theory devoted to this problem: the Pirogov-Sinai theory. Mar 28, 2014 at 7:53
• @Adam The liquid-gas maps to the Z2 Ising model(0T) with all spin up represents liquid, say, and all spin down to gas. I see no symmetry breaking during this transition. The finite temperature case is similar with the spin up sites decrease while spin down increase. No symmetry breaking occurs. Apr 4, 2014 at 6:07
• As far as I know symmetry breaking happens when we pass through some critical point. So liquid and gas need to share the same symmetry. Is the supercritical liquid that have symmetries broken by the gas-liquid regime. Nov 8, 2015 at 20:30
• So Landau's paradigm cannot explain liquid-gas phase transition? Or generally Landau's paradigm cannot explain 1st order phase transition? Feb 22, 2017 at 23:20

@VanillaSpinIce I agree most part of the answer from VanillaSpinIce, instead "The liquid-gas transition is characterized by a first order phase transition but there is no symmetry breaking."

Below the critical point,when a gas-liquid phase transition happens, an interface form between the gas and the liguid(since they have different density), thus a discrete refleciton symmetry (between gas and liquid) is broken.

The classical situation with no symmetry breaking is the case of the, so-called, isostructural transitions. The word "isostructural" is misleading, since what is meant is "isosymetric". However, historically the term emerged. There is a number of examples of such transiotions. One is the alpha-alpha' transitions in the hydrogen-metal systems, another is phase separations in fluids and polymer solutions, the coil-globule transition in polymers. Such a transition in a solid phase has been reported for SmS. In the case of the solid phase the crystal lattice changes its volume, but preserves its structure (this gave rise to its name).

• This doesn't appear to actually answer the questions asked, though it is tangentially related. Apr 15, 2014 at 14:16
• This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post - you can always comment on your own posts, and once you have sufficient reputation you will be able to comment on any post. Apr 15, 2014 at 15:36
• @ Kyle This answers the first part of the question. You simply cannot recognize the answer. It indicates the class of transition without the symmetry (or more generally, the structural) change, and gives examples of materials, where such transuitions are observed. Read it once more. I do not answer the second part of the question, where I am not a specialist. May 2, 2014 at 7:34
• @DavePhD this indeed give the answer to the question. Read my comment to Kyle and my answer. In contrast the answers above along with discussion contain serious mistakes. May 2, 2014 at 7:39