# What are some predictions from string theory that say some crystalline materials “will end up in one of many lowest-energy ground states?”

I am referring to this recent "news feature" by Zeeya Merali from Nature magazine www.nature.com/uidfinder/10.1038/478302a. Here is the specific quote:

"To make matters worse, some of the testable predictions from string theory look a tad bizarre from the condensed-matter viewpoint. For example, the calculations suggest that when some crystalline materials are cooled towards absolute zero, they will end up in one of many lowest-energy ground states. But that violates the third law of thermodynamics, which insists that these materials should have just one ground state."

What crystalline materials are predicted to have this degeneracy? How was string theory used to arrive at this prediction, and can it be arrived at without string theory?

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um... what? It seems like a fine question for this site to me. –  David Z Oct 27 '11 at 18:02
@Moshe I do not think this question is "about popular science books and magazines." The question is about highly degenerate, crystalline ground states, and an intersection of string theory and condensed matter physics. The question references what one might refer to as a "popular science...magazine," however that depends on how we define popular. –  jwco Mar 26 '12 at 3:06
I'm rather with jwco, here Nature is a prestige venue for publishing major results in all fields of science. The first KamLAND paper on reactor neutrino oscillations was published there (alas, I'm not on that one). –  dmckee Aug 25 '13 at 1:47
The third law of thermodynamics is more of a guideline than an actual law anyway. Lots of things can have multiple ground states, and the third law just says that for those things that do have a single ground state, we can define the entropy as zero in that case. So while breaking the second law would be a death sentence for any theory, breaking the third isn't really that bad. (Breaking the first law is pretty bad too, but general relativity seems to have gotten away with it.) –  Nathaniel Aug 25 '13 at 2:26

This is a bit old question I had missed so my answer will be short, just some references.

The stringy predictions are, of course, extracted from the AdS/CFT correspondence. It's my belief that a reader should first be familiar with the existence of the AdS/CFT correspondence due to Maldacena because the third law is just a very tiny story inside the AdS/CFT.

AdS/CFT says that physics of 4-dimensional theories of various kinds - not only gauge theories - is equivalent to (quantum) gravitational i.e. stringy physics in a higher-dimensional spacetime, five-dimensional anti de Sitter space. Various analyses of fluids and plasmas etc. without gravity may be typically translated to analyses of black holes in a higher-dimensional spacetime. I can't review the AdS/CFT here; typical reviews have hundreds of pages.

Now, the $T=0$ limit of the boundary matter - "crystal" - are typically mapped to the extremal limit of a black hole. The third law, i.e. $S\to 0$, holds because this extremal limit is simultaneously singular. The area goes to $A=0$ in the limit of low temperatures which translate to some limit of the black hole parameters.

Nevertheless, there have been arguments that with some particular bulk theories, the extremal limit may be a non-singular black hole with a nonzero area, and therefore nonzero entropy. See e.g.

http://arxiv.org/abs/0911.4518

Let me just mention that papers just a few months older were claiming that they had various proofs that the third law had to hold, e.g.

At any rate, for these exotic enough theories, one can't really say how the hypothetical material violating the third law looks like. The AdS/CFT surely doesn't construct molecules out of the usual atoms. It just says that if the description of the dual theory has a particular form, we may expect violations of the third law.

I think that all these people believe that when things are computed accurately enough to discuss the real material, the third law will always hold. In limited contexts, it can be proved, in broader contexts, it seems to have a loophole. At any rate, the discussion of the third law may be performed from a completely different angle if we borrow AdS/CFT.

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Hi Lubos: the Strominger-Vafa system already has nonzero horizon area at zero temperature, but I assumed that such degeneracies were due to highly supersymmetric nature of these states. If you have a crystal with degeneracies in the ground state, you don't have SUSY (I guess), so how do you get ground state entropy? I also thought there are other cases where you have a glassy ground state where you could have a highly degenerate ground state in principle, but I can't think of a quantum one off the top of my head. –  Ron Maimon Apr 6 '12 at 6:49
Dear Ron, SUSY is by no means necessary for a nonzero horizon area of extremal black holes. Ordinary extremal Kerr and Reissner-Nordstrom black holes in $d=4$ (without SUSY) have $T=0$ but a nonzero area as well, don't they? What was needed for just some extra charge(s) and/or angular momentum. SUSY is helpful to cancel corrections and make many calculations more doable (Strominger-Vafa brane was the minimal, highest-SUSY one with a nonzero horizon area) but it doesn't "qualitatively" change the physical behavior of the systems. –  Luboš Motl Apr 6 '12 at 7:39
@LubošMotl: yes, of course, stupid, stupid (r=M for RN). What the heck? No third law. This must be what Kaanen did, found condensed matter analogs of the glassy BH states. This is interesting. +1,nice answer. –  Ron Maimon Apr 6 '12 at 12:44