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String theory (!) and "bad metals"

I saw a remarkable talk today by Hong Liu from MIT, about quantum gravity and what it has to say about high temperature superconductivity. Yes, you read that correctly. It was (at least for a nonexpert) a reasonably accessible look at a genuinely useful physics result to come from string theory. I doubt I can do it justice, so I'll just give the bare-bones idea. Within string theory, Maldacena (and others following) showed that there is a duality (that is, a precise mathematical correspondence) between some [quantum theories of gravity in some volume of d+1 dimensions] and some [quantum field theories w/o gravity on the d-dimensional boundary of that volume]. This sounds esoteric - what could it be good for? Well, we know what we think the classical limit of quantum gravity should be: Einstein's general relativity, and we know a decent number of solutions to the Einstein equations. The duality means that it is possible to take what could be a very painful interacting many-body quantum mechanics problem (say, the quantum field theory approach to dealing with a large number of interacting electrons), and instead of solving it directly, we could convert it into a (mathematically equivalent) general relativity problem that might be much simpler with a known solution. People have already used this approach to make predictions about the strongly-interacting quark-gluon plasma produced at RHIC, for example.

I'd known about this basic idea, but I always assumed that it would be of very limited utility in general. After all, there are a whole lot of possible hard many-body problems in solid state physics, and it seemed like we'd have to be very lucky for the duals of those problems to turn out to be easy to find or solve. Well, perhaps I was wrong. Prof. Liu showed an example (or at least the results), in which a particular general relativity solution (an extremal charged blackhole) turns out to give deep insights into a long-standing issue in the strongly-correlated electron community. Some conducting materials are said to be "bad metals". While they conduct electricity moderately well, and their conductivity improves as temperature goes down (one definition of metal), the *way* that the conductivity improves is weird. Copper, a good metal, has an electrical resistance that scales like *T*^{2} at low temperatures. This is well understood, and is a consequence of the fact that the low-energy excitations of the electrons in copper act basically like noninteracting electrons. A bad metal, in contrast, has a resistance that scales like *T*, which implies that the low energy excitations in the bad metal are very complex, rather than electron-like. Well, looking at the dual to the extremal black hole problem actually seems to explain the properties of this funny metallic state. A version of Prof. Liu's talk is online at the KITP. Wild stuff! It's amazing to me that we're so fortunate that this particular correspondence exists.

## 14 comments:

Cute. I skimmed through the talk but I'm left with a question. How important is the role of supersymmetry in the correspondence, and does that limit the validity of the "bad metal" side of the correspondence?

Ok, that was 2 questions. But they're related.

It is indeed a wild and crazy thing, and more widely useful than one might have guessed. My understanding is that the correspondence is still an unproven conjecture, but everybody thinks it's true.

Theorists like Subir Sachdev and Jan Zaanen have been applying the correspondence to high-Tc as well. Intriguingly, the correspondence connects strong-coupling theories to weak-coupling theories in curved space. Or so I'm told.

At the string theory level, the correspondence explains why you can't easily say whether the theory is in ten or eleven dimensions, because they are equally good mathematically.

I've tried at times to think of ways to turn this into a popular science story, but it's not easy. Outside of the string-theory context, this duality seems to be not "real" but a mapping between different theoretical models. That's very powerful, but not very interesting to a general reader. It's hard to imagine a popular story about conformal mapping, for example, which similarly transforms a difficult problem into an easy one but doesn't solve it. Anyone have any ideas for making this less abstract and more compelling for a general audience?

Doug, this duality was recently highlighted in the CondMat Journal Club by Varma and others: http://www.condmatjournalclub.org/?p=732

As an alternative to Quantum Theory there is a new theory that describes and explains the mysteries of physical reality. While not disrespecting the value of Quantum Mechanics as a tool to explain the role of quanta in our universe. This theory states that there is also a classical explanation for the paradoxes such as EPR and the Wave-Particle Duality. The Theory is called the Theory of Super Relativity and is located at: Super Relativity

This theory is a philosophical attempt to reconnect the physical universe to realism and deterministic concepts. It explains the mysterious.

Doug wrote:

>Copper, a good metal, has an

>electrical resistance that scales

>like T2 at low temperatures.

Not to derail the conversation here... But does anyone know of a reference where this is actually shown to be experimentally true in a simple metal? It is of course the conventional wisdom, but I don't know where it was actually shown. Anyone? I've looked extensively, but not exhaustively but found nothing.

Anon@8:51 - One of my theorist colleagues asked exactly this question. The answer was: while in general symmetries of the gravity dual are reflected by symmetries in the QFT side, in this

particularcase, supersymmetry is not important to the end result.Don, you also emphasized something i should have pointed out more strongly. The limit that we really know how to handle on the gravity side is the classical GR case, which (conveniently or annoyingly, depending on your point of view) is the classical limit of many many string theories.

Anon@1:51, thanks for the journal club link. I need to get back in the habit of looking there more often. I'm not surprised that Chandra Varma would be up on this, since he's a prime mover of "marginal Fermi liquids" like the bad metals here.

Peter, I've browsed a bit, and that's a really good point. The T^2 assumes that e-phonon scattering is completely negligible (that is, the Debye temperature is very very high). Otherwise one gets the Bloch T^5 result in the low T limit. I'll look a bit more....

I've been trying to decide whether it is worth the effort for me to learn some of these techniques. So far, I'm not compelled, but i still have an open mind to it. Even if there is no physical system where this tells us anything interesting yet, it is still worth studying because someday there might be one --- the analogy to fractional statistics / Chern-Simons theory comes to mind: many of the mathematical details were worked out before there was any physical system that it applied to.

One should be warned however, that much of the AdS/CFT correspondence remains at the level of conjecture unless you are talking about very particular model systems in d=6 dimensions with N=4 supersymmetry (or whatever the particular solvable case is).

Quantum field theory algebraic topology has come a long way in the last 5 years, and now several of the concepts of "String theory(!) and "bad metals"" have new dimensions. That all depends on the correlation function solution for the atomic topological equation's set of virtual force photons. The images of strings develop in chapter 2 of The Crystalon Door.

Recent advancements in quantum science have produced the picoyoctometric, 3D, interactive video atomic model imaging function, in terms of chronons and spacons for exact, quantized, relativistic animation. This format returns clear numerical data for a full spectrum of variables. The atom's RQT (relative quantum topological) data point imaging function is built by combination of the relativistic Einstein-Lorenz transform functions for time, mass, and energy with the workon quantized electromagnetic wave equations for frequency and wavelength.

The atom labeled psi (Z) pulsates at the frequency {Nhu=e/h} by cycles of {e=m(c^2)} transformation of nuclear surface mass to forcons with joule values, followed by nuclear force absorption. This radiation process is limited only by spacetime boundaries of {Gravity-Time}, where gravity is the force binding space to psi, forming the GT integral atomic wavefunction. The expression is defined as the series expansion differential of nuclear output rates with quantum symmetry numbers assigned along the progression to give topology to the solutions.

Next, the correlation function for the manifold of internal heat capacity energy particle 3D functions is extracted by rearranging the total internal momentum function to the photon gain rule and integrating it for GT limits. This produces a series of 26 topological waveparticle functions of the five classes; {+Positron, Workon, Thermon, -Electromagneton, Magnemedon}, each the 3D data image of a type of energy intermedon of the 5/2 kT J internal energy cloud, accounting for all of them.

Those 26 energy data values intersect the sizes of the fundamental physical constants: h, h-bar, delta, nuclear magneton, beta magneton, k (series). They quantize atomic dynamics by acting as fulcrum particles. The result is the picoyoctometric, 3D, interactive video atomic model data point imaging function, responsive to keyboard input of virtual photon gain events by relativistic, quantized shifts of electron, force, and energy field states and positions.

Images of the h-bar magnetic energy waveparticle of ~175 picoyoctometers are available online at http://www.symmecon.com with the complete RQT atomic modeling manual titled The Crystalon Door, copyright TXu1-266-788. TCD conforms to the unopposed motion of disclosure in U.S. District (NM) Court of 04/02/2001 titled The Solution to the Equation of Schrodinger.

Actually after some more poking around I found some of these. See them and references therein...

http://prola.aps.org/abstract/PRB/v19/i8/p3873_1

http://prola.aps.org/showrefs/PRB/v23/i6/p2718_1

The take away message is that you have to work really really hard to see T^2 in a simple metal. In the experimental paper above, they felt that their dilution fridge temperatures were essentially to seeing T^2, because it was only that low that they could get a good measure of the residual resistivity.

Materials like high-Tc's have alot of unusual properties, but not having a T^2 rho is not one of them. That is commonplace.

"Within string theory, Maldacena (and others following) showed that there is a duality (that is, a precise mathematical correspondence) between some [quantum theories of gravity in some volume of d+1 dimensions] and some [quantum field theories w/o gravity on the d-dimensional boundary of that volume]."

If your description is fair to the speaker, then this would basically a generalization of the divergence theorem made interesting by the context. No wonder the chatter about the Net the last few months regarding this was skeptical about the string theorists' claim that it was a real world application. The consensus was that it's interesting math, which can have broader application than just within the string theory community, much in the same way that wave equations and their solutions show up in everywhere from particle in a box to geophysics to optics.

But I'm wondering. The whole point of quantum gravity is that quantum mechanics and general relativity are incompatible because of the contrast between discrete and continuous behavior. Only in limits of statistical behavior could they behave similarly. How did Liu show that the two systems can reasonably be expected to approach those limits, physically (as opposed to saying here's what happens if these systems were to approach the limit we need to make our description applicable)?

And what do you lose in taking the limit, what's the trade off? For example, you must sacrifice the ability to describe a lot of really important physics at the particle level in order to build MHD formalisms that are computationally tractable.

Anon@12:41 - I'm no mathematician, but there is definitely more to this than a generalization of Green's/Stokes'/the divergence theorem. The point is that this lets you relate certain strong-coupling theories to other weak-coupling (and therefore tractable) theories.

I also think that the GR vs QM/QFT problem is more than an issue of discrete vs. continuous. QM/QFT doesn't really care about absolute energy scales. That's why renormalizing away the vacuum energy (which in a handwavy way should contain zero-point modes from all quantum fields) is viewed as ok w/in QM/QFT. GR, on the other hand, does care about absolute energy scales, and says that absolute zero energy density should correspond to flat Minkowski spacetime. One problem (but not the only one) in quantum gravity is, how do you reconcile these issues?

a T^2 resistivity at low temperatures is what one gets in a Fermi liquid metal with strong electron-electron scattering and is seen in a wide range of strongly correlated metals.

For references see this recent Nature Physics article

http://www.nature.com/doifinder/10.1038/nphys1249

The talk of Liu appears to be about non-Fermi liquids not "bad metals". Some do not make the distinction but there is an important distinction.

Non-Fermi liquids don't have well-defined quasiparticles as the temperature approaches zero.

Bad metals can mostly be understood as the absence of quasi-particles above some temperature. See for example:

http://condensedconcepts.blogspot.com/search/label/bad%20metals

Hi Ross. My point/question was about "simple metals".... not strongly correlated ones.

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