laboratoire pierre aigrain
électronique et photonique quantiques
laboratoire pierre aigrain

Colorful approach to the fractional quantum Hall effect


The classical Hall effect in a conductor is understood in termes of a voltage drop perpendicular to the applied current in a magnetic field. In 1985, von Klitzing received the Nobel price in physics for the discovery of the quantum Hall effect: in a two dimensional electronic gas, which can be formed at the interface between two semiconductors, the ratio between the applied current and voltage is a multiple of the conductance quantum e^2/h. This property being universal is since used as a new practical standard for the electrical resistance.

The fractional quantum Hall effect occurs at stronger magnetic field and lower temperature (a few milliKelvin). The ratio between the applied current and voltage is a fraction of the conductance quantum, in contradiction with what was expected. The origin of the effect is believed to be an exotic quantum liquid with fractionally charged particle excitations. Although the effect was discovered more than twenty years ago and crowned with the Nobel price in 1998, this phenomenon is still an active field of research. Among new topics that have emerged recently, one is concerned with rotating ultracold atomic gases, where similar phases are expected to exist. One of the most promising proposals is the possible realization of exotic statistics, the so-called non-Abelian statistics, that go beyond the standard fermionic, bosonic, or even fractional statistics. It has been shown that non-Abelian statistics may be used as way to implement quantum computation while being robust by design to decoherence, which is the major bottleneck for the realization of a quantum computer.

A complete understanding of the fractional quantum Hall effect is still one of the open issues in theoretical physics. This system is the typical case where quantum phenomena and interactions are so tightly entangled that usual approximation schemes fail. Depending on the number of particles and magnetic field intensity, different models allow one to capture some essential parts of the problem. The most famous are the Laughlin liquid (Nobel price 1998), Jain’s composite fermions or the Moore-Read state. Still none of them is universal.

Nicolas Regnault from the Laboratoire Pierre Aigrain at the École Normale Supérieure, in collaboration with Mark Goerbig (LPS, Université Paris Sud) and Thierry Jolicoeur (LPTMS, Université Paris Sud), all being CNRS members, have developed a novel approach based on multi-color quantum liquids. Each color group of particles forms a Laughlin liquid. Using colors can be viewed as an artificial marking to distinguish particles whereas they are all identical. Actually, there is only one type of particles, say gray. In order to get rid of these artificial colors, the authors have proposed a procedure which renders all particles gray again. This may be viewed as taking a black-and-white photo of a colorful painting. Amazingly, the black-and-white photo reveals an internal structure of the quantum liquid - it consists of distinct droplets the number of particles of which is that of the original colors. The performed numerical calculations indicate that this structure may be a common feature of the different types of fractional quantum Hall states. This works is the first step in the construction of a bridge between the different theoretical approaches and may lead to a global picture of one of the most prominent problems in modern Physics.

Bridge Between Abelian and Non-Abelian Fractional Quantum Hall States

N. Regnault, M. O. Goerbig, and Th. Jolicoeur

Phys. Rev. Lett. 101, 066803 (2008)

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