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?
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?
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).
@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.
I think even in the case of liquid-gas transition there is symmetry breaking. The order parameter in this system is the density, and the symmetry that the gas density respects is translational symmetry. On the other hand, the liquid does not have translational symmetry. There might be some confusion when thinking of a liquid as an incompressible fluid. Doesn't that imply that the density of the liquid phase is uniform too? Actually one needs to think about the whole system, which includes the container.
Lets look at it from the perspective of degenerate ground states. For a gas, there is only one ground state, the state where all the particles fill the container uniformly. For a liquid, there are many ground states (assuming the liquid has no surface energy). In a zero gravity environment, you can have the liquid fill the bottom half of the container, top half, or be broken up into many droplets. All have the same energy, thus a degeneracy of the ground state. This implies a form of symmetry breaking. This is very similar to the Ising ferromagnetic transition.
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).
