Holographic Electron Systems
Gravity describes the motion of the planets and other celestial objects in addition to the evolution of the Universe itself. Because of its weakness, however, it can often be ignored in problems where one of the other three forces is involved.
But recent developments in string theory suggest that gravity could provide unexpected insight into dynamics involving these "stronger forces." The reason is not because the effects of gravity are significant. Rather, it relies on an amazing equivalence between theories of gravity and the theories describing other interactions—for example, the interactions between electrons moving in a metal.
This equivalence is known as Maldacena duality or the AdS/CFT correspondence. It states that a non-gravitational, strongly interacting field theory may be described by a weakly interacting gravitational theory. An example of such a field theory is Quantum Chromodynamics (QCD)—the theory that describes the strong force. This equivalence between gravitational and non-gravitational theories is not yet proved, and it is not at all obvious; however, it has passed every check since its proposal.
The proposal begins with gravity in a curved background space called anti-de Sitter space (AdS). In the language of general relativity, this space has maximal symmetry consistent with a negative cosmological constant, and, hence, negative scalar curvature. In AdS, light can travel to spatial infinity in a finite time. Thus, the boundary at infinity must be treated as part of the space. Juan Maldacena of the Institute for Advanced Study had the idea that the gravitational theory in AdS would be completely equivalent to a field theory living on the boundary. This idea makes a weird connection between a gravity theory, say, in five dimensions to a field theory like QCD in four dimensions. But it is not completely crazy. The entropy of a gravitational theory in (d+1) dimensions and the that of an ordinary field theory in d dimensions scale in the same way with the size of the system.
A fundamental quantity in the field theory is a so-called two-point correlation function of two fields or operators. The two-point function gives precise information about how two operators placed at different points in space-time talk to one another. Within the correspondence, there should be an equivalent quantity in the gravitational theory. The Maldacena conjecture says that, for each operator in the field theory, there is a corresponding gravitational particle living in AdS. The two-point correlation function is computed from the scattering of two particles in the gravity theory.
Now, why would one want to work with a complicated gravitational theory when one can do calculations in the field theory? The answer is that we have few methods to calculate in field theories when their interactions become strong. However, Maldacena duality relates a strongly coupled theory to a weakly coupled one. Thus it saves the day by allowing calculations in a strongly coupled field theory to be done in the weakly coupled gravitational dual!
Working here at SLAC with Shamit Kachru and with Oliver DeWolfe of the University of Colorado at Boulder, I studied an application of these ideas to theories related to QCD. It had been suggested that these theories possessed a metastable state that could play a role in supersymmetric models of particle physics. We constructed this state as a gravity dual and were able to work out many of its properties.
This suggests that gravity could be important in studying strongly correlated electron systems. If we could find a gravitational dual to such a system, we could study properties of the electronic state. There are still many mysteries about electronic systems in two dimensions, especially those related to high temperature superconductors, electronic systems in which superconductivity persists to high temperatures and for which standard BCS theory is inadequate. A great deal is known about these systems. However, many ideas remain untested because of the difficulty of dealing with strongly coupled systems. If we could find a gravitational dual to such a system, we could study its properties from this new approach.
With Shamit Kachru and Xiao Liu of the Perimeter Institute, I am studying possible gravitational duals for systems related to the high temperature superconductors. We have computed two-point correlation functions in the gravitational theory and find agreement with expectations from the field theory dual. We are in the midst of understanding how other properties of these electronic systems are reflected in the gravitational theory. There is much yet to be discovered.
Michael Mulligan, SLAC Today, May 29, 2008