The Holey Grail and its Dual: From String Theory to Strange Metals

Quantum Gravity: The Holey Grail.

Jonathan Granot’s colloquium slides have an amusing typo rendering quantum gravity the “holey grail of physicists.” How appropriate, I thought, and many of these holes are black. But typo aside, how holy really is this grail to physicists? After all, other areas of physics have their own holey grails: quantum computing for example, or high temperature superconductivity.

High temperature superconductors are badly understood theoretically, yet this understanding might allow us one day to create superconducting materials that save energy by avoiding resistive losses in long distance power lines. Imagine the potential! (Alternatively, read this.)

Presently the temperature at which these materials become superconducting is “high” only to the physicist who spends his days playing with liquid nitrogen: The transition temperature below which superconductivity sets in, also called the critical temperature, is in all known cases below -70°C. (The value depends on properties of the material as well as external fields.)

“Normal” superconductivity is described by the theory of Bardeen, Cooper and Schrieffer. At low temperatures, but in the not-superconducting phases, these metals are well described as Fermi liquids. But metals who display high temperature superconductivity are an entirely different story, and one that is largely unwritten.

One thing we know from experiment is that high temperature superconductors are “strange metals” whose electric resistance in the normal, non-superconducting, phase increases linearly with the temperature rather than with the square of the temperature. The latter is what one finds for a Fermi liquid with weakly coupled quasi-particles. Thus, plausibly the reason for our lacking theoretical understanding is that strange metals are strongly coupled system, which are notoriously hard to understand. “But darling,” said the string theorist, “I can explain everything.” And so he puts a black hole into an Anti-DeSitter (AdS) space and looks at the boundary.

The celebrated AdS/CFT correspondence makes it possible to deal with strongly coupled systems by mapping them to a weakly coupled gravitational system in a space-time with one more dimension. This is computationally more manageable, or at least one hopes so. So far, this correspondence, also called “duality”, between the gravity in the AdS space and the strongly coupled theory on the boundary of this space (thus one dimension less) is an unproved conjecture put forward by Juan Maldacena. However, it has been extensively tested for a few cases and many people are confident that it captures a deep truth about nature (though they might disagree on the extent to which it holds). We previously discussed this idea here and here.

For a high-temperature superconductor, one puts a planar black hole in the AdS space and decorates it with some U(1) vector fields and a scalar field, φ, and then goes on to calculate the free energy for different configurations of the scalar field. For temperatures above a critical value, the free energy is minimal if the scalar field vanishes identically. However, if the temperature drops below this critical value, configurations with a non-vanishing scalar field minimize the free energy, so the system must make a transition. In the figure below, you see the free energy, F, (with some normalization) as a function of the temperature (again with some normalization) for the case of φ = 0 (dotted line) and a case with non-vanishing φ (solid line). The latter solution doesn’t exist for all values of the temperature. But note that when it exists, its free energy is lower than that of the φ=0 solution.

[Image credits: Hartnoll, Herzog and Horowitz]

For these different configurations one can then calculate thermodynamic quantities of interest, such as the electric conductivity (AC and DC) or heat conductivity, and… compare the results with actual measurements.

As you can tell already from my brief summary, this approach to understand strange metals is, presently, far too rough to give quantitative predictions. It can however describe qualitative behavior, such as the scaling of the resistance with temperature that is so puzzling. And that it does quite well!

A bunch of smart people have been studying the strange metal duals for a couple of years now, among others Subir Sachdev, Sean Hartnoll, Hong Liu (who wrote a recent article for Physics Today on the topic), Shamit Kachru, Gary Horowitz, and a group here at Nordita around Lárus Thorlacius.

An exciting recent development is that Horowitz et al added a lattice structure by a periodic boundary condition, which is a big step towards modeling more realistic systems. Amazingly, despite the simplicity of the model, they scaling they find for the optical conductivity (the ability of photons to pass through a material) as a function of the photon’s frequency is in excellent agreement with experiment. (See “Optical Conductivity with Holographic Lattices” Gary T. Horowitz, Jorge E. Santos and David Tong, arXiv:1204.0519 [hep-th]).

One of the side-effects of commuting from Heidelberg to Stockholm is that the door sign with my name spontaneously relocates when I’m not at the institute, and I acquire new office mates in this process. Which is how I came to talk to Blaise Goutéraux, who arrived at Nordita this fall.

Blaise is among the AdS/CFT correspondents of Nordita’s “subatomic” group. (In fact, at this point I seem to be the only one in this group who doesn’t have anything to do with bulks and branes.) Blaise has taken on another challenge in this area, which is to describe the landscape of holographic quantum critical points, from which the strange metallic behavior at finite temperature is believed to originate. For this, he is working with more complicated geometries that exhibit different scaling behaviors from AdS.

What do we learn from this? The AdS/CFT correspondence is a useful tool, and if you’ve got a hammer quantum critical points might start looking like nails. But the only reason we call the bulk theory gravitational is that we first encountered a theory of this type when we wanted to describe the gravitational interaction. Leaving aside this scientific history, in the end it’s just a mathematical model to calculate observables that can be compared to experiment. And that’s all fine with me.

The big question is however whether this approach will ever be able to deliver quantitative predictions. For this, a connection would have to be made to the microscopic description of the material, a connection to the theories we already know. While this is not presently possible, one can hope that one day it will be. Then one could no longer think of the duality as merely useful computational tool with an educated guess for the geometry – the bulk theory would have to be a truly equivalent description for whatever is going on with the lattice of atoms on the boundary. But the cases for which the AdS/CFT correspondence has been well tested are very different from the ones that are being used here, and the connection to string theory, the original inspiration for the duality, has almost vanished. It wouldn’t be the first time though that physicists’ intuitions are ahead of formal proof.