(500) Phases of Matter

Oh galleons, remember when you were in primary school and learned about the three states of matter? Matter could be a solid, a liquid, or a gas. Those were its only phases.

And then you got a bit older, and your science textbooks introduced you to plasma, a special, ionized state of matter that doesn’t occur terribly often on Earth, but is actually the most common state of matter out in spaceland.

So, four states of matter. Solid, liquid, gas, plasma. And that’s it. Those have been the sturdy pillars of our understanding of condensed matter physics for ages.

Turns out… our “four phases of matter” belief is off by, oh, about 496 phases or so.

To be fair, to say that there are 500 or so phases of matter isn’t actually saying we have solid, liquid, gas, plasma PLUS 496 unnamed states of matter insanity. Instead, it’s a matter of reclassifying phase states. And to understand this reclassification, we first have to look at where the four original matter states came from.

The phases of matter have traditionally been classified by their symmetry in a method known as the Landau paradigm. Liquids are extremely symmetric, for example, while solids are much less so (which isn’t to say they aren’t symmetric- solids possess their own symmetry, but it isn’t the universal symmetry of liquids). Landau’s paradigm allows scientists to not only arrange the phases of matter on a chart, but to understand the behaviors of known phases.

And then…

We found a handful of phases Landau’s paradigm couldn’t describe.

Well, shit.

It’s all the fault of that pesky quantum mechanics, which always seems to wiggle in and throw a wrench into long-held scientific beliefs. When investigating quantum systems, some condensed matter researchers found numerous ground states that existed with the same symmetry.

Well, shit.

It became apparent that a new classification system was needed for states of matter. And thus, topological order was born. Topological order doesn’t describe phases of matter by their symmetry, but rather by patterns of entanglement.

And this new method of phase classification was powerful, but… well, it wasn’t perfect. Turns out, there were a few phases that still didn’t fit into the new classification system. These were short-range entangled phases that didn’t not break the symmetry (and therefore couldn’t be classified by topological order, which requires breaking through the symmetry to study the underlying patterns of quantum effects), aptly named symmetry-protected topological phases.


And so, some condensed matter researchers have recently come up with a new system to classify the phases of matter, one that should be able to classify all phases, including the symmetry-protected ones.

The new classification system (currently nameless) uses group cohomology theory and group super-cohomology theory. These two algebraic theories study groups by using functors, which are ways of mapping from one category to another by mapping objects to objects and morphisms to morphisms in such a manner that the composition of morphisms and the identities are preserved- which seems to make sense for phase description, no? This new classification system has allowed the researchers to describe all states of matter (including the symmetry-protected phases) in any number of dimensions and for any symmetries.

This new system, while not necessary for children understanding how liquid water transforms into ice, will help us understand quantum phases of matter, allowing us to design specialized matter states for use in quantum computers and superconductors.

Fascinating, to be sure, but don’t worry about having to run out and memorize 500 new states of matter- good ol’ solid, liquid, gas, and plasma are good enough for those of us who don’t work in the fields of theoretical, experimental, and condensed matter physics.

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