book review
## Are superconductors holograms?A new theory called AdS/CFT duality, which comes from superstring theory, is finding holograms everywhere. by T. J. Nelson |

Holographic Duality in Condensed Matter Physics

by Zaanen, Sun, Liu, and Schalm

Reviewed by T.J. Nelson

ike everybody else, scientists sometimes fall for fads. The latest fad is to call everything a hologram. But what scientists are calling a hologram is not quite the same as the hologram we see on the front of our credit cards.

It's also not the same as the holographic universe theory, where the entire universe is said to be two-dimensional information encoded on a colossal black hole. (A recent paper on holographic noise was reported in the popular press to have disproved this idea, but that is not the case.)

What it is is a new application for a theory in physics called anti-de
Sitter space / conformal field theory duality (AdS/CFT for short) that came out of
string theory. This theory keeps popping up everywhere. No longer is it, as everybody
thinks, some abstract thing that's only useful in string theory. Zaanen * et al.*
say it solves many problems for which conventional theories about superconductors are
conceptually inadequate.

That includes, these authors propose, high-critical temperature (Tc) superconductors and Fermi liquids. There's much more to superconductivity than Cooper pairs, but the current theory is not good enough, which might explain why it's been so hard to create a room temperature superconductor.

High Tc superconductors lack a Fermi surface (which I will get to in a minute), so they are examples of non-Fermi liquids, also known as strange metals. In strange metal one invariably finds Mott insulators, where electrical resistance is not caused by large band gaps, but by electron repulsions becoming dominant due to a strong lattice potential in the material. These Mott insulators, according to standard theories, destroy the Fermi liquidness. By doping a Mott insulator to add electron holes, one can produce high Tc superconductors. This has led to a huge theoretical effort to explain it which has so far been unsuccessful.

Bose-Einstein condensate (top) and Fermi gas (bottom)

The authors treat superconductors above the Tc as undergoing a quantum phase change from a Fermi liquid superconductor to a non-Fermi liquid. Cuprates, they suggest, might be forming quantum critical (or conformal) metallic phases. Their idea is that quantum phenomena account for superconductivity, and AdS/CFT theory can be used to explain it. For that we need some background on these exotic quantum materials. There's a lot to know before we can see how AdS/CFT fits in.

Quantum matter is matter that exhibits long-range quantum entanglements; that is to say, entanglements on a macroscopic scale. A familiar example is Bose-Einstein condensates, which is a gas composed of bosons cooled to nearly absolute zero. At that temperature all the particles have the same quantum state. Bosons all get along, like a herd of little dogies, and scientists head them up and move them out to observe quantum phenomena.

A boson is a particle with integral spin (0, 1, or 2). For example, photons, gluons, Cooper pairs, Z and W gauge bosons, and elements with even Z (mass numbers) such as helium-4 are all bosons. This integral spin means that more than one of them can occupy the same quantum state. It also means only bosons can form Bose-Einstein condensates, and they have weird properties. Superfluidic helium-4, in which the liquid loses all viscosity, is an example.

Thus,
despite the fact that the formal name of cattle is * Bos taurus * and the
fact that cows can all be in the same state, cows are technically not bosons
because they don't have an integral spin.

By contrast, fermions are particles that have 1/2 spins. This includes elements with odd mass numbers and particles like electrons. Fermions can't occupy the same quantum state, and so they form a type of quantum matter called a Fermi gas, which has quite different properties than a Bose condensate. Fermions can only form a B-E condensate if they pair up to form Cooper pairs.

A Fermi gas or liquid is more like a “traffic jam” of particles. Its behavior is radically different from a B-E condensate: even at absolute zero, a Fermi gas, unlike an ordinary gas, would still have a finite viscosity.

Another weird property is that transverse sound waves can propagate in a strongly interacting Fermi liquid as if in an elastic medium. This is impossible in ordinary liquids. The Fermi surface of a Fermi liquid is the order parameter of the state. As a result, a Fermi liquid behaves like a two-dimensional system—that is, its behavior is defined by its surface.

Okay, simple enough, you might say. Even my dog knows about Bose-Einstein condensates. (As well he should: he chewed over my quantum mechanics book long enough.) But how can superconductors be holograms?

Remember the analogy with black holes, about which Hawking famously said the entropy is proportional to their surface area. When you have something that can be fully described by its surface, then mathematically you can call it a hologram. And, according to the authors, mathematically that's what we have with superconductors.

The holographic part comes from the dual representation provided by the theory because the two aspects of the theory have different dimensions. One aspect has four, and the other has five:

N=4 in super-Yang-mills lives in (3+1) dimensions while the dual closed-string theory lives in the (9+1)-dimensional spacetime formed by AdS

_{5}× s^{5}. [p.138]

What this means plain English is that the extra radial dimension called for by the mismatch between the dual representations is not a spatial dimension at all, but a variable that describes the successive stages of the conformal field theory under coarse graining. That is, it is the dimension of a scale factor as one goes from the UV to the infrared. This distinguishes it from the older theory, which can't do that.

But why do we need duality at all? In the simplest terms, a duality simply means that two theories are describing the same phenomena using a different mathematical approach. There is nothing mysterious, say the authors, about duality. Fourier transforms are an example: by doing a Fourier transform, one changes one's viewpoint so that position becomes momentum and particles appear as waves. Each viewpoint describes a different aspect of the same behavior.

The theory is somewhat conjectural. The book is mainly a sales job for their new theory, but the authors are very enthusiastic about their bottom-up approach. They write:

AdS black holes can un-collapse ... this realisation that black-holes in AdS are not stable states at all has been a stunning surprise. At the same time, AdS/CFT's ability to nearly effortlessly compute near equilibrium many-body correlations in real time with a seamless transition to the hydrodynamic regime should be nearly as astonishing to a condensed matter physicist. [p.116]

As stretches go, this is a whopper. But their goal is to convince condensed matter physicists to try the new theory. Maybe they overdo it a little. Can we say these physicists have found a connection between black holes and superconductors? That is too early to say. But everybody loves black holes, even condensed matter physicists, and AdS is a difficult theory, so I suspect a little enthusiasm couldn't hurt.

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Holographic Duality in Condensed Matter Physics
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by Zaanen, Sun, Liu, and Schalm