How do we define states of matter?

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:

  1. Zero Resistivity
  2. Meissner Effect
  3. 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:

  1. 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)
  2. 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?

Comments welcome…

3 responses to “How do we define states of matter?

  1. After a law school friend asked me about what defines a state of matter after he read this Vice article (, I’m not sure I find state of matter to be an incredibly meaningful term compared to phase. I sort of blame it on the fact that we tend to learn about “states of matter” really early in school and just reduce it to solid, liquid, gas, and if your science teacher is really on their game, plasma. And then we do intro chemistry or physics (or at least my teachers), linked states of matter to equations of state. And although I could tell you the properties that distinguish a metal from a semiconductor from a superconductor, I’m not sure I know what could be a state equation to reliably distinguish those.


  2. If you pursue a bit your idea, you can come out with a radically different way of thinking a phase. Why not defining a state of matter as its constitutive relation from Maxwell’s equations ? Then clearly you can not distinguish whether the system is liquid, gas or solid, but you can distinguish a superconductors from a topological insulator, from others … The problem with this classification is that it’s blind to neutral matter. So to refine, you ask : why not defining a state of matter as its constitutive relation from thermodynamics ? And then you recover the usual gas/liquid/solid classification. But in the spirit of the XVIII-th century thermodynamics, you are now blind to charge. So in fact you need (at least !) to know how a system respond to i) a gradient temperature, ii) an electric field and iii) a magnetic field to classify the states I can think about. To conclude, it’s clearly its responses which decide which type of phase/state you have. But that’s they are intrinsically related to the microscopic origin of the response (the constitutive relation in Maxwell construction for instance).


    • Actually, I don’t think what I was suggesting was so radical. I was just saying that we should define states of matter from their experimental properties, which we (by and large) do already. This is not to say we should not value theoretical frameworks such as the Landau symmetry-breaking paradigm, but to just realize its limitations.


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