### John Baez

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Ten kinds of matter

A cool discovery: substances can be divided into 10 kinds.

The basic idea is pretty simple.  Some substances have time-reversal symmetry: they would look the same, even on the atomic level, if you made a movie of them and ran it backwards.  Some don't - these are more rare, like certain superconductors made of yttrium barium copper oxide!   The substances that do have time reversal symmetry have a symmetry operator T that can square to 1 or to -1: please take my word for this, it's a quantum thing.  So, we get 3 choices, which are listed in the chart under T as 1, -1, or 0 (no time reversal symmetry).

Similarly, some substances have charge conjugation symmetry, meaning a symmetry where we switch particles and holes: places where a particle is missing.  The 'particles' here can be rather abstract things, like phonons - little vibrations of sound in a substance, which act like particles - or spinons - little wiggles in the spin of electrons.  Basically any sort of wave can, thanks to quantum mechanics, also act like a particle.  And sometimes we can switch particles and holes, and a substance will act the same way!

The substances that do have charge conjugation symmetry have a symmetry operator C that can square to 1 or to -1.  So again we get 3 choices, listed in the chart under C as 1, -1, or 0 (no charge conjugation symmetry).

So far we have 3 × 3 = 9 kinds of matter.  What is the tenth kind?

Some kinds of matter don't have time reversal or charge conjugation symmetry, but they're symmetrical under the combination of time reversal and charge conjugation!  You switch particles and holes and run the movie backwards, and things look the same!

This chart shows a 1 under the S when your matter has this combined symmetry, and 0 when it doesn't.  So, 0 0 1 is the tenth kind of matter (the second row in the chart).

This stuff was first discovered around 1997 by Altland and Zirnbauer.  But it's just the beginning of an amazing story.  Since then people have found substances called topological insulators that act like insulators in their interior but conduct electricity on their surface.   We can make 3-dimensional topological insulators, but also 2-dimensional ones (that is, thin films) and even 1-dimensional ones (wires).  And we can theorize about higher-dimensional ones, though this is mainly a mathematical game.

So we can ask which of the 10 kinds of substance can arise as topological insulators in various dimensions. And the answer is: in any particular dimension, only 5 kinds can show up. This chart shows how it works for dimensions 1 through 8.  The kinds that can't show up are labelled 0.

(There's more information in this chart, which I'm too lazy to explain now.)

If you look at the chart, you'll see it has some nice patterns.  And it repeats after dimension 8.  In other words, dimension 9 works just like dimension 1, and so on.

There is a huge amount of cool math lurking here, and you can see some more in my blog article:

http://golem.ph.utexas.edu/category/2014/07/the_tenfold_way.html

This math is called the ten-fold way.

The chart here comes from the paper that showed only 5 kinds of topological insulator are possible in each dimension:

• Shinsei Ryu, Andreas P Schnyder, Akira Furusaki, and Andreas W. W. Ludwig,  Topological insulators and superconductors: tenfold way and dimensional hierarchy, New J. Phys. 12 (2010) 065010, http://arxiv.org/abs/0912.2157.

#spnetwork arXiv:0912.2157 #must_read #condensed_matter #topology #physics  ﻿
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That sounds so fascinating my first thought was: bull - there's no way things are that simple/organized. You've never steered me wrong before though so..

That's pretty freaking cool :)﻿

- I'm glad you're starting to give me the benefit of the doubt.  In my blog article I refer to some papers, so you can check I'm not just making this up.  :-)﻿

This is awesome. Just shared it with my MIT topology Dr. ﻿

compliments of the division of one by zero :-)﻿

How does this square with the CPT theorem? Does it mean that for most of these kinds of matter quantum field theory cannot be an adequate description? ﻿

I work exactly on this kind of stuff, trying to understand the geometry of this quantum phases of matter. Thanks for the share!!﻿

- The CPT theorem applies to quantum systems without a preferred rest frame, where Lorentz transformations are symmetries.  But the results here are about condensed matter physics.  A chunk of solid matter picks out a preferred rest frame, so the CPT theorem doesn't apply.  (I could expand on this, but I hope you get what I mean.)﻿

Hmm, in what sense can a YBCO superconductor be non-time-reversal-symmetric?  It's all "just" atoms and electrons, that are governed by time-reversal-symmetric laws of physics at the low level, right?  Unless you're talking about increasing entropy, but that's not exactly a rare phenomenon. :)﻿

ℂl₃ as opposed to Cl₃? Really? Otherwise fascinating stuff!
[Update: ℂl₃ is a product of my fantasy, ℂl₁ would have been a better example to nag about notation - see my comment below. Thank you John, for making me notice.]﻿

I expected just that answer. What I don't know is why the Lorentz group is so critical to CPT. Or, to put it differently, is this 10-fold way the generalization of CPT to Galilean symmetry?﻿

ZOZO makes another appearance.﻿

I love that the classification is exactly the classification of the classical symmetric spaces. Of course Hamiltonian symmetries with involutions is about symmetric spaces! But now with rigor!
It's too bad that the exceptional symmetric spaces don't get to play too, since they have fixed rank and thus don't really have thermodynamic limits. I suppose a few-particle analogue might have some exceptional behavior, but then some of the niceness coming from thermodynamic considerations might be lost. Oh well.﻿

On the other hand, it is known that CPT is an exact symmetry, assuming only Lorentz invariance and a hermitian Hamiltonian, so it would be useful to discuss just  what this more elaborate classification actually highlights.﻿

This is a nice result and a nice explanation too!  Maybe we can make quantum walks with charge conjugation symmetry, in addition to the time-asymetric 'chiral walks' we considered before.  Maybe and will want to think about this too. ﻿

- dunno if you read my reply to .  In condensed matter models we don't have Lorentz invariance, so I don't think the CPT theorem applies here. ﻿

wrote: "What I don't know is why the Lorentz group is so critical to CPT."

I've never understood this as well as I want, but part of the point is this: the parity/time-reversal operator PT lives in the connected component of the complexified Lorentz group (or more precisely, its double cover).  PT is the map

(t,x,y,z) |-> -(t,x,y,z)

which we may think of as being a 180 degree rotation in the xy plane followed by a 180 degree rotation in the complexified tz plane. (One needs to complexify precisely to get away with the second rotation.) Thus we can figure out how a field transforms under PT assuming we know how it transforms under the complexified Lorentz group.  This is fundamental to the spin-statistics theorem, and also to the CPT theorem.

Besides Lorentz invariance, we need the Hamiltonian to be bounded below to be able to define fields on a chunk of complexified Minkowski spacetime.  "Wick rotation" and all that jazz.

"Or, to put it differently, is this 10-fold way the generalization of CPT to Galilean symmetry?"

I haven't seen anyone come out and say that.  It could be true - though it actually seems that the 10-fold way follows from very minimal assumptions, not even Galilean symmetry required.﻿

wrote: " ℂl₃ as opposed to Cl₃? Really?"

Where did I mention ℂl₃?   I didn't intend to ever mention that one.﻿

wrote: "Hmm, in what sense can a YBCO superconductor be non-time-reversal-symmetric?  It's all "just" atoms and electrons, that are governed by time-reversal-symmetric laws of physics at the low level, right?"

The interesting thing about YBCO superconductors is that their ground state "spontaneously breaks" time reversal symmetry.  In other words: the fundamental laws are time reversal symmetric, but when this material settles into a lowest-energy state, it has to pick one, and there is no choice that has time reversal symmetry.

A more familiar example would be how a crystal of sugar can spontaneously break the symmetry between left and right, by "choosing" to be made of dextrose rather than levulose.

A couple of caveats:

1) I'm not a condensed matter physicist, and it seems to still be controversial whether YCBO superconductors actually do spontaneously break time reversal symmetry.  But everyone agrees it's theoretically possible - if not for this substance, then another.  Maybe some other example would be less controversial.

2) The fundamental laws of physics are not  symmetric under time reversal, only under time reversal together with switching left and right and switching particles and antiparticles!  However, it's only the weak force that violates time reversal symmetry, and that's not relevant to YCBO superconductors or any other form of condensed matter that I know.  So I didn't want to resort to this nitpick in answering your question.﻿

No, you didn't mention ℂl₃ at all! Sorry about me filling in nonesense there, only to nag about the notation. Thanks for prodding me.

I just got to the end of your nice post at the n-Category Cafe, where you clearly say ℂl₁ is one of the non-real clifford algebras on the list, ℂl₀ = ℂ the other one, and no sight of any other ℂl. I'm now wondering what a super Brauer algebra is, and how to get 8 things (7 especially symmetric) that when you put them on a thread, yield the associative real super division algebras.﻿

My understanding is that there are a number of assumptions needed to arrive at this nice table of groups, and that the groups need to be understood in some "stable sense", which makes the physical interpretation more subtle (involving K-theory and Bott periodicity).  Roughly speaking, in higher spatial dimensions, the topology of the translational symmetries is responsible for the "dimension shifts" in the groups. Unlike C and T, the parity-reversal P acts on the translational symmetries, and one needs to consider equivariance under this action, which complicates things. The particular way in which C and T sit inside the full symmetry group is very important, as emphasised by Freed-Moore http://arxiv.org/abs/1208.5055, and I have a preprint discussing these matters http://arxiv.org/abs/1406.7366.﻿

wrote: "It's too bad that the exceptional symmetric spaces don't get to play too, since they have fixed rank and thus don't really have thermodynamic limits."

That's a great point.  But I'm actually not sure they play no role.  If you look at Table A1 in the appendix of the paper I linked to, you'll see that symmetric spaces are in 1-1 correspondence with condensed matter systems in three different ways!   In one of these ways, the symmetric space G/H shows up as the 'target space of a nonlinear sigma-model'.  This is an incredibly distracting piece of jargon that means our chunk of matter can be described by a field theory with a field taking values in G/H.   I see no reason this G/H couldn't involve an exceptional Lie group G.

So, while the exceptional symmetric spaces may be 'second-class citizens' with fewer rights, I bet they will show up in some aspects of this game.  And I can't help but hope they're related to the nonassociative super division algebras.   There are too many nonassociative division algebras to classify them all in a useful way, but 'alternative' ones are nice - and at one point, years ago, Todd Trimble and I tried to classify the alternative super division algebras.  I now want to return to this... except that I have other work to do.﻿

If the excitations satisfy a wave equation,  with a rotation invariant d'Alembertian operator, the relevant Lorentz transformations are those that leave the characteristic velocity invariant-and the issue is that these are an effective description, i.e. that they break down at the scale where the microscopic degrees of freedom, whose collective excitations are these waves, can be resolved. In these circumstances an effective CPT theorem can be relevant. So that was my question: what is the effective description, where this classification becomes relevant: is this classification sensitive to some crystallographic structure?﻿

The Ten Fold Way! I love it :)

Always back to symmetry it seems. This is a fascinating idea, great post, thank you :)﻿

So, CPT only holds in 3+1 dimensions, too?﻿

- There will indeed be condensed matter systems that have an approximate 'effective Lorentz invariance', but I'd guess these are exceptional, not geneic.   There's an approximate Lorentz invariance when in a linear approximation waves in a medium obey a dispersion relation that's approximately of the form

E^2 = p^2 c^2 + m^2 c^4

for various different m's but one specific velocity c, the 'effective speed of light'.   But we can have media where different waves have different c's, and they interact enough that we have to treat them together.  We can also have anisotropic linear media where waves of different kinds move in ways governed by different "effective metrics".

(There's also no reason the dispersion relation needs to take this form at all, except to second order in p: if E has a local minimum at p = 0 and it's smooth this will be a decent approximation.)

As for the effect of crystal structure, that's the topic of the paper by Freed and Moore cited in my blog post on the n-Cafe.  It gets quite elaborate!   I don't understand it, but I can tell it's a great paper.﻿

Of course, I was just giving an example. But it does have a wider applicability, than it might have been thought. (The speed isn't the speed of light-it's the speed of the excitations.) And, indeed, the effective description shows its limitations when more than one ``species'' can interact.﻿

- I don't know the details, but there has to be a version of the CPT theorem in special relativistic theories in (almost?)  any dimension.  Same for spin-statistics.  But the spin-statistics theorem is much more elaborate in 2+1 dimensions than higher dimensions, thanks to the existence of 'anyons' - the universal cover of the rotation group in 2d space is the real line, completely different from the way it works in all higher dimensions.  So, there are many more kinds of 'spin', and also many more kinds of statistics, governed by the braid group instead of the permutation group.﻿

- thanks for pointing out that paper.  One of the authors, Ryu, was the lead author of the paper I got the chart from.

Here's something that paper says, which should make both of us happy:

We will also show that, once parity symmetry or parity symmetry combined with other discrete symmetries is included into our consideration, the CPT theorem plays an important role in classifying topological states of matter.  The CPT theorem holds in Lorentz invariant quantum field theories, which says, C, P, T, when combined into CPT, is always conserved, i.e., CPT = 1, schematically. For example, a Lorentz invariant CP symmetric field theory also possesses time reversal symmetry, and vice versa.

In condensed matter systems, however, such relations between these discrete symmetries (T, C, and P) do not arise since we are not to be restricted to relativistic systems; symmetries can be imposed independently. Nevertheless, some physical properties of these non-relativistic systems at long wavelength limit, such as the band topology or the electromagnetic response, can be encoded in the so-called topological field theory, which respects the Lorentz symmetry.

So, Lorentz symmetry and CPT doesn't apply to everything about condensed matter physics, but they do apply to topological degrees of freedom.﻿

That's the remarkable feature,indeed-that the``topological degrees of freedom'' can be classified this way. And, naturally, raises the question of  how to describe ``topology-changing'' transitions.﻿

What does AZ stand for?  ﻿

I would say that the groups in the above table, as well as in the paper on TIs and CPT, are K-theoretic in nature, being defined via certain classifying spaces and their homotopy groups. Without passing to the stable regime, I don't think that a "Bott periodicity" in the classification groups arises yet.

In that case, the discrete symmetries C, P, T, (via an associated Clifford algebra), as well as spatial translations, (via some sort of Thom isomorphism), all get involved in determining the relevant K-theory group. I may be misunderstanding Ryu's paper, but it seems like they assume the presence of a certain CPT symmetry, to link e.g. the ''topological'' classification in the presence of C and P, to that in the presence of T, and hence one of the groups considered in the above table. However, I'm not sure that they mean that a ''CPT symmetry'' is necessarily present in general.

Of course, one needs to be more precise about what the terms "topological" (non-commutative Brillouin zone in IQHE?), "classification" (systems or differences of systems?), "dimension" (discrete/continuous translations?) etc. mean. A great attempt in this direction is of course the paper of Freed--Moore. There is also interesting work by Bellissard, Schulz-Baldes, Kellendonk, etc., utilising K-theoretic ideas to link, e.g., bulk conductivity and edge conductivity in the IQHE. Perhaps, that might be some sort of ``topology-changing transition".﻿

The way I think roughly about CPT is that it corresponds to turning a Feynman diagram upside down. So if amplitudes are invariant under twisting Feynman diagrams this way and that, then CPT must apply as a special case. Lorentz invariance plus some other assumptions of quantum field theory actually get you the more general case.﻿

Oh Great FSM, Don't let Deepak Chopra find out this is called the Ten-Fold Way.  Nothing good can come from that....﻿

- AZ stands for Altland and Zirnbauer, the guys who discovered the 10-fold way back around 1996.  ﻿

अदीन् ऐक्लर् - M-theory actually lives in 11 dimensions, and it's quite poorly understood.  But superstring theory lives in 10 dimensions, and this appearance of the number 10 is related to the ten-fold way!

Briefly: the 10-fold way splits into 8 and 2 as shown in the chart: 2 kinds of matter that have neither T nor C symmetry, and 8 others.  Mathematically the 2 are related to the 2 dimensions of a string worldsheet (which are described using the complex numbers, a 2d number system), while the 8 are related to the 8 extra dimensions of spacetime (which are described using the octonions, an 8-dimensional number system).

I explained this in more detail on my blog article; I've never heard anyone discuss it before.  It's hard to tell how significant it is, but it's not a "mere coincidence".﻿

अदीन् ऐक्लर् - thanks!  It gives me great pleasure to share the fun.  So much pop physics focuses on flashy stuff that doesn't pan out, merely because it seems easy to explain.  A lot of more solid research is just as exciting; the journalists just don't know about it or how to explain it.﻿

Why isn't there an 11th type (0, 0, -1)?﻿

- that's a great question!!!  I was wondering about that.  It turns out that this is equivalent to (0, 0, 1).  It's hard to see why without getting into the math a bit deeper.  The space of states for a substance of type (0, 0, 1) is a complex Hilbert space equipped with a unitary operator S with S^2 = 1.  For (0, 0, 1) we have a unitary operator S with S^2 = -1.  But we can turn one kind of operator into the other by multiplying by i.  This trick does not work for T and C since those operators are antilinear.﻿

To add to John's answer, Wigner's Theorem is hiding behind the choices of +1 and -1. For T,C, they need only be anti-unitary projective involutions, so they square to some phase. If you consider T(T^2)=(T^2)T, it follows that this phase must be a real number, so it's either +1 or -1.﻿

No CPT theorem, OK; but why isn't P in this chart at all?﻿

Hi Toby, my own take is that the above tenfold way is simply a list of possible combinations of antiunitary symmetries T and C. The list of groups appearing on the right, suitably interpreted and assuming no other symmetries, tell you the ways in which Hamiltonians compatible with T,C can be realised. If there are other unitary symmetry constraints, such as P, the tenfold classification still makes sense. However, the groups classifying the Hamiltonians compatible with T,C, and P have to be modified to take those into account.﻿

Because this is solid state physics and the spatial symmetries of the crystal are dealt with separately? (and have been classified using group theory for about 80 years)﻿

Indeed, Wigner had already thought about time reversal, symmetry and group theory a long time ago, but his ideas are also relevant to relativistic systems, e.g. in his classification of elementary particles. What's quite new here is taking charge-conjugation seriously as a symmetry, at least on par with T. Surprisingly, lots of rich mathematical structures appear.

One could very well consider the full Poincare group, augmented with C, and look for its ``gapped'' representations. An example of such a representation would be solutions (both positive and negative frequency) to the massive Dirac equation. This is possibly a pleasant alternative to having C arise as an "accidental" symmetry of the Dirac equation solutions when regarded as a representation of the ordinary Poincare group without C.

Just as the Wigner-Dyson threefold way applies generally, the tenfold way should also make sense outside of a solid-state context.﻿

wrote: "Because this is solid state physics and the spatial symmetries of the crystal are dealt with separately? (and have been classified using group theory for about 80 years)"

It's really worthwhile looking at Freed and Moore's paper "Twisted equivariant matter", which combines the ten-fold way with a study of crystal symmetries.  They've discovered a lot of new stuff:

http://arxiv.org/abs/1208.5055

Just read the table of contents if you like!  I'm glad these guys aren't just working on string theory: the humbler but more practical world of condensed matter holds a lot of mathematical fun too!﻿

Back to basics: I'm confused about "topological insulators that act like insulators in their interior but conduct electricity on their surface".  I think I remember from freshman physics that all charges on a conductor are on the surface, and E is zero in the interior.  So, how does this differ from a substance that is an insulator inside but conductive on the surface?  Maybe I remember my physics wrong - it has been a long time.﻿

- Hi!   Indeed, these materials are insulators on the inside and conductive on the surface... but if you cut a chunk of this stuff in half, the newly revealed surface is now a conductor!﻿

This has been puzzling me for a long time as well. A remark: the newer versions of the tenfold way have extra structure: they are ordered (through Clifford algebras, or the super-Brauer group). Unfortunately, I'm unable to see a similar structure in Dyson's old tenfold way. The grading does seem crucial for Brauer-multiplication to make sense.﻿

(third time I've tried to comment: Google+ seems to be malfunctioning)
I guess my question is this: all the charges on "topological insulators" and on solid conductors lie on the conductive surfaces, and if you cut them the new surfaces are also conductive; so, what is the difference between the properties of topological insulators and solid conductors?  I'm missing something.﻿

It's not true that all the charges on conductors lie on the surface.  All the excess charge lies on the surface, but if you have current flowing down a copper wire, electrons are flowing through the middle of the wire, not just the surface.  But on a topological insulator, there's no current except on the surface!﻿

John, thanks.  How did I get the idea that current traveled on the outside of a wire?!  If we live long enough I'm sure you will eventually correct all my misconceptions about physics.﻿

And you may teach me statistics.﻿

:  It's confusing, because the statement mixes things that usually don't go together: static electricty and conducting materials.  So it's easier to remember a wrong version (moving charges only move on the conductor's outside, or excess charges only appears on the outside of a charged object) rather than the correct version (excess charges only appear on the outside of a charged object made of conductive material).﻿