# Spontaneous symmetry breaking in fluids

BACKGROUND:

One can think of solids as spontaneously breaking translational symmetries in the sense that each atom in a lattice has to pick a particular position. Yet, as with everything in our universe it still must respect Poincare symmetry. Thus the ground state of the solid is not invariant under translations even as the laws of physics are invariant. From this perspective, we can interpret the 3 phonons (1 longitudinal and 2 transverse) as the Goldstone modes corresponding to broken translational symmetry.

The density matrix describing a solid, $$\rho_{solid}$$, should therefore satisfy $$[\rho_{solid},\hat P_i]\neq0,$$ (where $$\hat P_i$$ are the momentum operators) since translations are spontaneously broken.

If you melt a solid and obtain a fluid, you still have three phonons (even though the speed of sound of the transverse phonons goes to zero). However, in a fluid, the positions of the atoms are not fixed and the density matrix should be translationally invariant, that is $$[\rho_{fluid},\hat P_i]=0.$$ But this seems to indicate that translations are not spontaneously broken.

QUESTIONS:

1. Do fluids spontaneously break translations?

2. Can we understand the existence of fluid phonons as arising from spontaneous symmetry breaking of translations?

3. If so, how do we define spontaneous symmetry breaking of $$\hat P_i$$ when $$[\rho_{fluid},\hat P_i]=0$$? If not, where do the gapless phonon modes come from?

Many papers and books seem to indicate that fluids do break translations and that the resulting Goldstones correspond to phonons, for example https://arxiv.org/abs/1107.0731 and https://arxiv.org/abs/1211.6461 and https://www.amazon.com/Hydrodynamic-Fluctuations-Symmetry-Correlation-Functions/dp/0201410494. So I am inclined to believe that fluid phonons can be understood as Goldstones.

There is one simple case in which sound is a Goldstone mode, which is the case of a superfluid. There is a spontaneously broken $$U(1)$$ symmetry, which leads to a single longitudinal sound mode. This is described in standard text books, like Forster. What is more subtle is the fact that at $$T\neq 0$$ there is a second sound mode (called second sound) which is not so easily related to a symmetry.
In an ordinary fluid the relationship between the existence of a sound mode and a spontaneously broken symmetry is more subtle, and it is not discussed in the classic literature (like Landau, Forster, Kadanoff and Martin, etc.). In any piece of matter there is a preferred frame, the local rest frame, in which the momentum density of the matter vanishes. But this is not the same as ordinary breaking of translational invariance, and it is not sufficient for the existence of a sound mode. What we need is local thermal equilibrium, so that the stress tensor in the matter rest frame takes the form $$\Pi_{ij}=P\delta_{ij}$$, with $$P$$ fixed by the local conserved charges (in a simple fluid energy density and matter density).
The proposal in the recent literature (some of which you mention, see also Haehl et al, "The fluid manifesto", and Gloriso and Liu, "Lectures on non-equilibrium effective field theories") is that there is a spontaneously broken diffeomorphism invariance. A general state is described by a density matrix (or operators on a Keldysh contour), but a locally thermal state satisfies a KMS condition (operators live on a thermal circle). This corresponds to breaking the symmetry from $${\it diff}(M)^2$$ to just $${\it diff}(M)$$.