Historically, many people seemed to lean towards defining a phase of matter by its (broken) symmetries. For instance, a ferromagnet has broken rotational symmetry and time-reversal symmetry, a solid has broken translational and rotational symmetry, etc. In light of the discoveries of the Quantum Hall Effect and topological insulators, it seems like this symmetry classification does not encompass all states of matter.
The symmetry classification is largely a theoretical construct, however. I would think that one defines a state of matter by particular experimental properties that it exhibits. For example, one could define a superconductor by requiring it to exhibit the following:
- Zero Resistivity
- Meissner Effect
- Zero Peltier Coefficient
Put another way, to verify that one has discovered a superconductor, these three criteria must be satisfied.
Let us take another example: a simple metal. The criterion that must be satisfied for this case is the existence of a Fermi surface. This can be measured by quantum oscillation measurements, angle-resolved photoemission, or a few other probes.
Yet another example: a 2D topological insulator. What one must observe is:
- The Fermi energy intersects an odd number of topologically protected edge states in half the edge Brillouin zone (which was shown by transport in this classic paper)
- The existence of a spin-polarization associated with the edge states
While these three examples were chosen because they were simple, I have remaining doubts. Are these observations necessary and sufficient to define these states of matter? Are there cases where one can better define a state of matter theoretically?
For instance, a theorist may define a 2D topological insulator by the existence of a non-trivial topological number, which seems like a perfectly valid criterion to me. This topological number cannot be experimentally observed in a very direct way (to my knowledge) and has to be inferred from the edge states, band structure, etc.
The reason I started thinking about this is because I did not find the definition of a charge density wave in this widely-cited paper by Johannes and Mazin appropriate. It states:
[A charge density wave is a] Peierls-like instabilit[y] that occur[s] due to a divergency in the real part of the electronic susceptibility, so that the electronic subsystem would be unstable per se, even if the ions were clamped at their high symmetry positions.
This definition bothers me in particular because it defines a charge density wave by its cause (i.e. Peierls-like instability due to a divergence in the real part of the electronic susceptibility). The main qualm I have is that one should not define a state of matter by its origin or cause. This is like trying define a superconductor by the mechanism that causes its existence (i.e. phonon-mediated electron-electron interaction for superconductors, which would exclude unconventional superconductors from its definition). This is obviously problematic. Therefore, shouldn’t one define a charge density wave by its experimentally measured properties?
So I come back to the original question: how does one define a state of matter?