When a conductor carrying electrical current is placed in a magnetic field, the flowing electrons experience a force perpendicular to both the magnetic field and the current. The production of voltage, or potential difference, by this (Lorentz) force is known as the Hall effect (Fig. 1). It is exploited in several electronic devices such as flow sensors, joysticks, car ignitions and even spacecraft propulsion.

At very low temperatures, the classical Hall effect breaks up and displays some striking quantum characteristics. In fact, this so-called integer quantum Hall effect reveals quantizations that are so precise they have been used to accurately determine some important numbers in quantum mechanics, including the fine-structure constant that characterizes the strength of electromagnetic forces.

Now, Akira Furusaki at the RIKEN Advanced Science Institute in Wako and co-workers in Japan and the USA have shown that the quantum Hall effect is strongly affected by boundaries at the edge of a material1. Their findings could alter the underlying quantum theories of condensed matter.

Quantum quirkiness in the Hall effect

During the classical Hall effect, the electrons moving under the influence of the Lorentz force experience resistance to their flow. This ‘Hall resistance’ increases linearly with the strength of the magnetic field (Fig. 2a).

The quantum version of the Hall effect was discovered in 1980 when researchers measured the properties of electrons confined to just two dimensions at very low temperatures near absolute zero. Here, the Hall resistance looks very different—it jumps up in quantized steps as the magnetic field strength increases, producing a series of plateaus (Fig. 2b).

The size of each step is determined by two fundamental constants, the electron charge e and Planck’s constant h, regardless of the material being studied. This quantum Hall effect is so precisely quantized that it is now used as a standard of resistance measurement.

Many theories have been proposed to explain exactly how electrons move between the resistance plateaus in the quantum Hall effect.

At such low temperatures the electrons experience a phenomenon called Anderson localization—they are scattered so much that they cannot propagate over any distance, and effectively stay in one place. This means that their wavefunctions are very narrow, and the sample effectively acts as an insulator against the Hall effect.

The main theoretical challenge is to work out what happens to delocalize the wavefunctions, allowing the system to jump between adjacent plateaus.

Finding fractals

Previous research completed by Furusaki and co-workers has helped them to explain the Anderson localization. In one study2, they examined the electron wavefunctions in materials that can undergo transitions from metallic (electrically conducting) to insulating behavior.

They found that the electron distribution obeys so-called multifractal statistics, meaning that they follow similar patterns on both small and large scales. However, the electrons at the sample’s boundary edges showed quite different distributions from those in the bulk of the sample.

The researchers realized that the boundary differences will influence the quantum Hall effect. Previous calculations have missed this subtlety by using bulk physical quantities that are valid only in the center of a sample.

“We showed that boundary multifractal properties are different from bulk multifractal properties,” explains Furusaki. “It was a very natural next step for us to study boundary multifractality in the integer quantum Hall effect.”

Examining the edge

The quantum Hall effect depends on impurities or defects in the sample, which can be thought of as hills that the electrons must climb over or skirt around. At the sample edges, there are limits to the directions the electrons can travel to overcome these obstacles, so their dynamics are different. However, it turns out that these restrictions at the edges are crucial in producing the quantum Hall effect.

“In a sample showing the quantum Hall effect, electron wavefunctions in the bulk are all localized and cannot carry electric current. Instead, there are ‘edge wavefunctions’ extended along the edge of the sample which can carry current,” explains Furusaki.

“When the electrons undergo a transition between two successive resistance plateaus, the number of edge states changes by 1. At the transition point the wave functions, both at bulk and at boundary, are neither extended nor localized; they are called critical.”

Furusaki and colleagues recalculated these critical wavefunctions for electrons undergoing a transition between plateaus near the edge of a sample. They found that the transitions do not follow the same multifractal statistics that have been assumed in previous studies.

Edging towards a new future

Any new theories for the quantum Hall effect will have to take these constraints into account. Furusaki looks forward to unraveling the final details of this remarkable example of quantum physics in action.

“Some recent experiments are using scanning tunneling microscopy to observe electrons in a quantum Hall sample, but they are still at a primitive stage and resolution is not high,” he says. “I suspect that the edge states in the quantum Hall effect should indirectly affect the electron distribution at boundaries, but it will take more work to get a good understanding of it.”

In the future, the quantum Hall effect could become important in the world of electronics. In a different study3, Furusaki and his collaborators have already explained an unusual quantum Hall effect caused by the relativistic nature of electrons in graphene, which could eventually replace silicon in integrated circuits.

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1. Obuse, H., Subramaniam, A.R., Furusaki, A., Gruzberg, I.A. & Ludwig, A.W.W. Boundary multifractality at the integer quantum Hall plateau transition: Implications for the critical theory. Physical Review Letters 101, 116802 (2008).

2. Obuse, H., Subramaniam, A. R., Furusaki, A., Gruzberg, I. A. & Ludwig, A. W. W. Multifractality and conformal invariance at 2D metal-insulator transition in the spin-orbit symmetry class. Physical Review Letters 98, 156802 (2007).

3. Nomura, K., Ryu, S., Koshino, M., Mudry, C. & Furusaki, A. Quantum Hall effect of massless Dirac fermions in a vanishing magnetic field. Physical Review Letters 100, 246806 (2008).

The corresponding author for this highlight is based at the RIKEN Condensed Matter Theory Laboratory

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Akira Furusaki

Akira Furusaki was born in Saitama, Japan, in 1966. He graduated from Faculty of Science, the University of Tokyo in 1988 and obtained his PhD in physics in 1993 from the same university. He became a research associate at Department of Applied Physics, the University of Tokyo in 1991, and worked as a postdoctoral associate for two years at Department of Physics, Massachusetts Institute of Technology in USA, before being appointed as an associate professor at Yukawa Institute for Theoretical Physics, Kyoto University in 1996. Since October 2002 he has been a chief scientist of Condensed Matter Theory Laboratory at RIKEN. His research focuses on developing theories of quantum electronic transport, superconductivity and magnetism in solids. He received Nishinomiya Yukawa Commemoration Prize in Theoretical Physics (2004).