What symmetries distinguish liquids, crystals, glasses, and isotropic solids?

 One of the most important ideas in condensed matter physics is that different states of matter are associated with different symmetries. These different symmetries result in different types of elementary excitations such as the Goldstone bosons associated with continuous symmetry breaking. The symmetries of the low-lying excited states reflect the symmetries of the ground state.

For example, consider the transition from a liquid to a cubic crystal. The continuous rotational and translational symmetry of the liquid is broken to the discrete rotational and translational symmetry of the crystal. Long-wavelength sound waves reflect these changes in symmetry. In the crystal, there are three distinct sound waves: one longitudinal and two shear modes. In contrast, in the liquid, there are only longitudinal modes. 

An isotropic solid, such as studied in elasticity theory, supports two types of distortions: compression and shear. Consequently, there are three types of sound waves (longitudinal and transverse phonons. The latter can have two different polarisations). The isotropic solid has continuous, not discrete, rotational and translational symmetries. A glass is an example.

This leads to a fundamental question:

What is the difference between liquids and solids at the level of fundamental symmetries?

In different words, what is the order parameter for the liquid-solid transition? A possible answer is the shear modulus G, which vanishes in the liquid state.

A related question is: What is the fate of the transverse phonons upon transitioning from the solid state to the liquid state?

I would have thought that these questions would have been settled decades ago. However, they have not. Just two years ago, Physical Review E published a 22 page article that aims to address the questions above.

Deformations, relaxation, and broken symmetries in liquids, solids, and glasses: A unified topological field theory

Matteo Baggioli, Michael Landry, and Alessio Zaccone

The paper immediately drew a Comment claiming the paper

"contradicts the known hydrodynamic theory of classical liquids." The authors have a Reply.

I do not have the expertise to give insight on the subtle technical issues in this debate. My only comment is that it is amazing how we are struggling to answer such basic questions.

I thank Jean-Noel Fuchs for getting me interested in these subtle questions. This happened when he kindly pointed out an error in Condensed Matter Physics: A Very Short Introduction. On page 40, I erroneously stated that shear sound waves exist in a liquid. This was part of a confused discussion about how sound waves can be used to distinguish different states of matter.  I have drafted a corrected paragraph and inserted it in my post listing the errors in my book.

I welcome any comments about the issues discussed above.