For over a decade, theorists have been exploring a surprising mathematical relationship between the equations that describe two seemingly different situations–one involving curved spacetime, the other involving systems of many interacting particles. In the 3 June Physical Review Letters a team uses this connection to mathematically reproduce the operation of a standard superconducting device known as a Josephson junction starting with equations for curved spacetime. Although the results have not yet revealed any surprises about superconductors, they increase confidence in conclusions theorists might draw about other condensed matter systems.
Sometimes conditions on the boundary surface of a region of space have a powerful influence on what happens inside. Cosmologists call this the “holographic principle,” by analogy with the way a 2D hologram embodies a 3D reality. In 1997 theorist Juan Maldacena, now at the Institute for Advanced Study in Princeton, New Jersey, suggested a close mathematical relationship, or “duality,” between two very different theoretical constructs. One is string theory in a particular curved spacetime. The other is a type of quantum field theory describing strongly interacting particles in ordinary spacetime, which can be thought of as the boundary of the curved spacetime and so has one fewer dimensions.
Maldacena’s conjecture was that you could always transform the physical quantities of a curved-spacetime string theory into those of a lower-dimensional field theory on the boundary. So you could get solutions for one set of equations by transforming to the other set. Although the idea that this holds in all cases is still unproven, there is “overwhelming evidence that it’s true,” says Gary Horowitz of the University of California at Santa Barbara.
At first, theorists viewed the duality as a way to learn about string theory, without the complications of curved space. But recently they have turned it around to learn about strongly interacting particles by using the language of curved space and general relativity. “General relativity was thought of as a theory of gravity for over eighty years,” Horowitz says. “In the last ten years or so we’ve learned, through this duality, that we can use the same equations to describe non-gravitational physics as well.”
For example, in 2008 Horowitz and two collaborators used the mathematical tools of relativity to devise an analog for a two-dimensional superconductor . This superconducting sheet formed one boundary “at infinity” of a block of three-dimensional space that the team imagined had the right kind of curved spacetime. They also allowed electromagnetic fields and one additional, generic field to exist in the region. To get temperature into the system in a “relativity-friendly” way, the team added an infinite, planar black hole at the boundary opposite the superconductor, because a black hole has a well-defined temperature.
“We started with the minimal ingredients that we knew we needed to get something that would look like a superconductor,” Horowitz says. The corresponding equations showed that the fields could be zero when the black-hole temperature was high. As the team lowered the temperature, the black hole spontaneously became unstable, causing the generic field to rise above zero. Using the standard “dictionary” that translates this curved-space field into properties of the boundary, the team calculated an infinite electrical conductivity, as in a real-world superconductor.
Now Horowitz and two other collaborators have extended the work to model a device called a Josephson junction, where current flows between two hunks of superconductor across a narrow zone of normal material. As you increase the quantum phase difference between the superconductors, the current varies as a sine wave–a feature that gives a superconducting quantum interference device (SQUID) its exquisite sensitivity to magnetic fields.
Along one stripe of the sheet, the team forced the material to remain a normal conductor below the critical temperature. They found that the current flowing through this normal region depended sinusoidally on the relative phases of the two superconductors, just as in a real-life Josephson junction. Horowitz says he hoped the method would work this way, “but it certainly wasn’t obvious.” The results are “yet another confirmation that the duality is correct,” he says.
Although there have been thousands of papers on the duality, relatively few have examined such spatial variation, says Subir Sachdev of Harvard University in Cambridge, Massachusetts, because it makes the calculations much more complicated. But it remains challenging to construct holographic versions of superconductivity that match specific materials, he says. “I don’t think we’re close to really figuring that out.”
- S. A. Hartnoll, C. P. Herzog, and G. T. Horowitz, “Building a Holographic Superconductor,” Phys. Rev. Lett. 101, 031601 (2008).