# Has the Lie group E8 really been detected experimentally?

A few months ago there were several math talks about how the Lie group E8 had been detected in some physics experiment. I recently looked up the original paper where this was announced,

"Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry", Science 327 (5962): 177–180, doi:10.1126/science.1180085

and was less than convinced by it. The evidence for the detection of E8 appears to be that they found a couple of peaks in some experiment, at points whose ratio is close to the golden ratio, which is apparently a prediction of some paper that I have not yet tracked down. The peaks are quite fuzzy and all one can really say is that their ratio is somewhere around 1.6. This seems to me to be a rather weak reason for claiming detection of a 248 dimensional Lie group; I would guess that a significant percentage of all experimental physics papers have a pair of peaks looking somewhat like this.

Does anyone know enough about the physics to comment on whether the claim is plausible? Or has anyone heard anything more about this from a reliable source? (Most of what I found with google consisted of uninformed blogs and journalists quoting each other.)

Update added later: I had a look at the paper mentioned by Willie Wong below, where Zamoldchikov predicts the expected masses. In fact he predicts there should be 8 peaks, and while the experimental results are consistent with the first 2 peaks, there are no signs of any of the other peaks. My feeling is that the interpretation of the experimental results as confirmation of an E8 symmetry is somewhat overenthusiastic.

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Why is this getting voted down? This seems like a legitimate question in mathematical physics, albeit with an empirical bent. Not to mention, one being asked by an (I assume) Fields medalist. – Daniel Litt Jul 17 '10 at 21:40
It seems to me that "being a Fields medalist" doesn't carry any information one way or the other about the quality of the question, which should be the primary reason that people choose their votes. (That said, it seems like a completely reasonable question to me.) – JBL Jul 17 '10 at 22:11
And, at the risk of giving truth to a classical estereotype about mathematicians (nicely captured in the joke about sheep who have one black side) "having user name «bocherds»" does not carry any information one way or the other about the quality of "being a Fields medalist" :) If that is not clear, well, I happen to be a Nigerian prince!... – Mariano Suárez-Alvarez Jul 17 '10 at 22:19
My views in the philosophy of mathematics and philosophy of physics make this question nonsense. – Alexander Woo Jul 17 '10 at 23:28
But Wadim, surely the goal of an answer on MO is to answer the question posed by the original poster, rather than to guess on what "everybody who votes" wants? The latter seems neither desirable nor practical. – Yemon Choi Jul 18 '10 at 11:31

This is a great question, but I don't think a reasonable answer can be given in this short space. So I wrote an expository note jointly with a colleague who was trained as a physicist. You can read it by following the link above -- comments are welcome.

But here are a couple of highlights:

1. It's not true that they measured this one number, and so claimed to have detected E8.* There is a bit more data than that. And there is a lot more history! Back around 1990, there was a series of theoretical "deductions" (in the weak sense of physics) investigating what the appropriate theoretical model should be for the situation in the magnet experiment. This led to a unique candidate for a model, one built out of E8. I would say that the experiment corroborated the series of deductions, with the sensational bonus that the deductions led to E8.
2. Which E8 appears in the theoretical model? The obvious answer is that it is the compact real E8 and not just the root system or root lattice. For example, even though the masses of the 8 particles are given as entries in an eigenvector for the Cartan matrix (which makes it sound like it's just the root system), the proof of this statement is a calculation within the compact Lie algebra.

One can argue about both of these points, of course. But these seem to be what the physicists claim and what they use in their papers.

To address Wadim's concerns about fringe science: Whether or not you find the E8 angle interesting or plausible, it seems that the experiment is interesting for entirely different reasons. The experimenters themselves claim that their main achievement is realizing this 1-dimensional quantum Ising model in the laboratory in a situation where the external field can be tuned to be above, below, or at the critical point. The Physics Today article on the subject paraphrases Subir Sachdev:

only recently could researchers reach the high fields and low temperatures needed to access the critical point and have high enough instrumental resolution to resolve the masses of at least some of the quasiparticles they excited. (Temperatures have to be low enough to suppress any impact of thermal fluctuations.)

• Footnote: To be precise: Also, Coldea, the author of this particular article in Science, uses the more-cautious "detected evidence of E8 symmetry" as opposed to the stronger "detected E8".
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The arxiv note (linked above) is easy enough to read. But if you prefer to see a talk, I'm giving a talk about it at the AMS National Meeting, Saturday, January 8, 2011, at 2pm in Napoleon D3 on the 3rd floor of the Sheraton. – Skip Jan 4 '11 at 20:26

It should be emphasized that this is not the $E_8$ of heterotic string theory or the $E_8$ gauge group in various grand unified theories. It comes out of something much more down-to-earth, namely solid state physics. The $E_8$ in this story is an unexpected symmetry of the two-dimensional Ising model in a magnetic field that was discovered by Zamolodchikov in 1989.

The Ising model was devised as a simple mathematical model that, it was hoped, would exhibit a ferromagnetic critical point. This turned out to be the case, as Onsager showed in 1944. The model is connected to a lot of beautiful mathematics, including Kac-Moody algebras, the Yang-Baxter equation, q-series, Painlevé equations, and conformal field theory. Zamolodchikov's amazing discovery was that the conformal field theory that describes the model at its critical point can remain integrable when one perturbs away from the critical point in certain directions. One of these perturbations corresponds physically to turning on an external magnetic field. It is in this context that the $E_8$ symmetry emerges.

Since the Ising model is a toy model - real ferromagnets are messier (and 3-dimensional!) - it doesn't really need experimental test. The model is important more for the physical and mathematical insight it gives, than for any quantitative information it might yield. Nevertheless, there is a tradition of trying to find real physical systems that embody the simple microscopic picture of the Ising model. (Such systems have to behave effectively as if they are one or two dimensional.) The recent work lies within this tradition, and they have managed to verify one of the consequences of Zamolodchikov's work, namely that the mass ratio of the two lightest quasiparticles is $\phi$. I have no reason to doubt that what Coldea et al. measured in their experiment is a genuine manifestation of the $E_8$ symmetry of the model.

One consequence of the $E_8$ symmetry is that there are eight species of particles, with definite mass ratios. If I had to guess, I would say that it's going to be very difficult to observe the peaks corresponding to the remaining particles. The third particle has a mass very close to twice that of the lightest particle, which means that that peak will be buried under lots of nearby two-particle states. Furthermore, the particles with masses higher than two will be unstable. (The field theory studied by Zamolodchikov is integrable and therefore does not have unstable particles, but any attempt to realize the model experimentally will surely destroy the integrability and therefore the stability of the higher-mass particles.)

Note: The one-dimensional quantum spin chain model in the Science article is described by a Hamiltonian that has the same eigenvectors as the transfer matrices of the classical two-dimensional square lattice Ising model solved by Onsager. So for the purposes of this discussion they are equivalent.

Addendum: (This an answer to the comment of Victor Prostak that was too large to fit in the comment box.) It is widely believed (but not a theorem as far as I know) that there are only two integrable perturbations: Onsager's thermal perturbation, which has a very simple symmetry, and Zamolodchikov's magnetic perturbation with the $E_8$ symmetry. The model is not expected to be integrable if one does both perturbations simultaneously, but the question is physically just as interesting, and was studied in B.M. McCoy and T.T. Wu, Two-dimensional Ising field theory in a magnetic field: Breakup of the cut in the two-point function, Phys. Rev. D 18, 1259–1267 (1978). I am not aware of any computation of the mass ratios in the general case. I wouldn't expect any exceptional symmetries to show up.

On the other hand, exceptional groups do show up in related conformal field theories. A perturbation of the $\mathcal{M}(4,5)$ minimal model with central charge $c=7/10$ has an $E_7$ symmetry, and a perturbation of the $\mathcal{M}(6,7)$ minimal model with central charge $c=6/7$ has an $E_6$ symmetry. These models describe different kinds of critical points than the $c=1/2$ model does, and so are experimentally distinguishable. In addition, the mass ratios of the quasiparticles should be different.

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+1. A very good answer. – José Figueroa-O'Farrill Jul 18 '10 at 6:04
Can you, please, comment on whether it's only $E_8$ that shows up in Zamolodchikov's integrable perturbation, or could it be another simple Lie group? I know how Ising model is related to $c=1/2$ CFT (described holomorphically by BPZ), but the appearance of exceptional Lie groups in various models of this type always struck me as a bit numerological. If other symmetries are indeed possible, how would one experimentally differentiate between the possible groups? – Victor Protsak Jul 18 '10 at 9:26
This gives a good explanation of why the existence of an E8 symmetry implies that you get two (or rather 8) peaks with a certain ratio. However my question was about the opposite implication: does the existence of these two peaks imply an E8 symmetry? As Victor Protsak pointed out, there could be lots of other much simpler explanations for these peaks. Experimental detection of E8 symmetry is an extraordinary claim, so I would like to see some extraordinary evidence for it before believing it. – Richard Borcherds Jul 18 '10 at 17:12
I have to chime in with a rather pedestrian comment: experiments can support, but not prove a theoretical claim. (On the other hand experiments can certainly disprove a theory.) How convinced one is of a theory based on its supporting evidence is, as far as I can tell, usually a very subjective issue. – Willie Wong Jul 18 '10 at 18:23
Will I had a question about your answer, but it was too involved to put in a comment so I asked it as a regular question: mathoverflow.net/questions/32432 – Noah Snyder Jul 19 '10 at 1:08

I think the answer is no they didn't detect the E_8 Lie group, though they did detect a symmetry related to the E_8 lattice. (Or rather, they collected some evidence which is consistent with the theoretical prediction that the system have a symmetry related to the E_8 lattice.)

Let me try to make that more precise. When people say that there's and SU(3) symmetry to QCD which explains the eightfold way, what are they saying? They're saying that the particles can be identified with certain vectors in certain representations of the group SU(3). For example, the quarks transform like the standard rep of SU(3), the anti-quarks like the dual rep of SU(3), and the meson octet transforms like the adjoint representation of SU(3).

If you were to discover a physical system with an E_8 symmetry then the particles should correspond to vectors in a certain representation of E_8. For example, you might expect to find particles corresponding to the weight vectors of the adjoint representation of E_8. That would mean 248 different particles!

Zamolodchikov's E_8 symmetry is an entirely different sort of beast. Instead of 248 particles there are only 8 particles. These particles correspond only to the 8 simple roots of E_8. So the system is closely related to the E_8 root system, but it is not the sort of thing one usually thinks of when thinking of a system which exhibits symmetry with respect to the E_8 Lie group.

In some ways this is good. Certainly you would need a lot less evidence to suggest that a system exhibited icosahedral symmetry (which hardly seems surprising since it's a relatively small group) than you would to think a system showed symmetry with respect to the affine Lie algebra E~_8 (which is huge and complicated). Nonetheless the Grothendieck group of the icosahedral group A_5 can be identified with the affine E~_8 root system. Similarly you should not be too surprised to find that a system exhibits Zamolodchikov's type of "E_8 symmetry" which just says that the particles are the objects in a small fusion category whose Grothendieck group can be identified with the E_8 lattice.

All of this should be taken with a grain of salt. The Science paper certainly claims "Remarkably, the simplest of systems, the Ising chain, prom- ises a very complex symmetry, described mathematically by the E8 Lie group." I was unable to find a claim in the theoretical literature that the E_8 Lie group occurs, rather than just the E_8 lattice. Nonetheless either I've seriously misunderstood something, or the authors are being imprecise in their use of mathematical language. I'm not terribly confident in my ability to understand physics, but I'm also not terribly confident that physicists use mathematical language in an extremely precise fashion which agrees exactly with how mathematicians use that language.

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The general viewpoint Noah is explaining here seems quite related to the recent article in the AMS Bulletin (July 2010) by John Baez and John Huerta. As someone who knows little physics, I found their article pretty interesting and understandable (although I haven't finished reading it yet...). The article is available here: ams.org/journals/bull/2010-47-03/S0273-0979-10-01294-2/… – Dan Ramras Jul 19 '10 at 21:22
Here's another example. I don't think you'd hear people call something an SU(3) symmetry just because there are two particle types whose energy levels correspond to the Frobenius-Perron eigenvector for the Dynkin diagram A_2 (that is two particles with the same energy). (Though I could be wrong, if anyone has seen such examples please share them!) – Noah Snyder Jul 19 '10 at 21:53
You may be right that only root system, and not the algebra itself, is relevant to the Ising model - I can't say for certain at the moment. If there is a connection with the Lie group $E_8$, it would probably be a consequence of the fact that the the minimal CFT that describes the Ising critical point can be generated by a coset construction involving WZW models that are invariant under $E_8$. How this relates to what happens when the system is perturbed away from criticality is something I don't yet understand. – Will Orrick Jul 20 '10 at 2:47
Could you give me a reference for something that says that the E_8 CFT has to do with a coset construction involving E_8? For example, in Kawihashi and Longo's classification of c<1 CFTs (arxiv.org/abs/math-ph/0201015) they quote Boeckenhauer-Evans saying that the (E_8,A_30) modular invariant should come from a coset SU(2)_29 < (G2)_1 x SU(2)_1 while the (A_28,E_8) does not come from any coset construction. – Noah Snyder Jul 20 '10 at 3:13
I may not have made it clear that the Ising CFT is the $A_1$ CFT. It has two distinct coset constructions: $\hat{su}(2)_2<\hat{su}(2)_1\times\hat{su}(2)_1$ and $(\hat{E}_8)_2<(\hat{E}_8)_1\times(\hat{E}_8)_1$. See Table 3b in P. Bowcock and P. Goddard, Virasoro algebras with central charge $c<1$, Nucl. Phys. B 285[FS19] (1987) 651-670, or Section 18.4.1 of Di Francesco, Mathieu, and Senechal's book, or Section 14.2.2 of G. Mussardo's book, Statistical Field Theory, which contains the most detailed information about this topic. – Will Orrick Jul 20 '10 at 4:44