Quantum Hall effect gets new dimensions
Oct 30, 2001
There is a long history of connections between condensed matter and particle physics. The Higgs mechanism that is thought to explain the masses of elementary particles, for instance, has its origins in studies of superconductivity. Now two condensed matter theorists have proposed a four-dimensional version of a widely studied two-dimensional effect, the quantum Hall effect, that could tell us more about the basic properties of space. The work by Shou-Cheng Zhang and Jianping Hu of Stanford University in California and Tsinghua University in China might even represent a small step towards one of the ultimate goals in theoretical physics - a quantum theory of gravity (S-C Zhang and J Hu 2001 Science 294 823).
‘The higher-dimensional generalization of the quantum Hall effect has been sought for a long time, but no one has succeeded before,’ Zhang told PhysicsWeb. ‘The mathematical structure could be very relevant to string theory, but the model is far, far from a realistic model of the universe.’
Most condensed matter systems can be explained by ignoring the interactions or correlations between electrons and calculating the properties of charged excitations in the ‘sea’ of electrons in the system. However, there is a growing number of strongly correlated systems - such as high-temperature superconductors - in which electron interactions are important and the conventional approach breaks down. Most of these systems develop long range order in their ground state. However, the quantum Hall effect and the ‘Luttinger liquid’ are the only known examples of quantum disordered ground states.
The quantum Hall effect is observed when the resistance of a two-dimensional gas of electrons is measured in a magnetic field. In 1980 Klaus von Klitzing discovered that the resistance of the gas is quantized when the magnetic field is high and the temperature is very low. This integer quantum Hall effect could be explained without electron correlations, but this conventional approach failed when the fractional quantum Hall effect was discovered in 1982. This effect was subsequently explained by Robert Laughlin in terms of electron correlations leading to fractionally charged excitations. The Luttinger liquid can also be understood in terms of fractionally charged excitations.
When Zhang and Hu extended the theory of the quantum Hall effect to four dimensions, they found that the equations that described excitations at the boundary of the system were similar to Maxwell’s equations of classical electromagnetism, and also to the linear version of Einstein’s General Theory of Relativity. They also found that the excitations could be used to model relativistic particles without mass, such as the photon and the graviton, and also other particles without analogues in high-energy physics. The results suggest that it might be possible to think of special and general relativity as theories that emerge from quantum mechanics, rather than as completely different theories.
‘Although this work is still very limited,’ write Zhang and Hu, ‘we hope that this framework will stimulate investigations on the deep connection between condensed matter and elementary particle physics.’
About the author
Peter Rodgers is Editor of Physics World