nLab quantum Hall effect

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Contents

Context

Topological physics

Contents

Idea

At sufficiently low tempetrature, quantum effects change the nature of the classical Hall effect, in two ways:

  1. in the integral quantum Hall effect, quantization of the energy into Landau levels of electrons that are circulating in a transverse magnetic field while confined to a plane, causes the Hall effectHall resistance?, which classically increases linearly with increasing external magnetic field, to instead intermittently form constant “plateaus” as these get “filled” by electron states,

  2. in the fractional quantum Hall effect, strong external magnetic field causes these Landau levels to be filled only partially and the strongly Coulomb-coupled electrons to form bound states “with” magnetic flux quanta that may exhibit effective fractional charge and, apparently, “fractional statistics” (anyonic braiding behaviour).

Integer quantum Hall effect

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Fractional quantum Hall effect

A neat informal account is given by Störmer 1999:

“In the FQHE, the electrons assume an even more favorable state [than in the IQHE], unforeseen by theory, by conducting an elaborate, mutual, quantum-mechanical dance. Many-particle effects are extraordinarily challenging to address theoretically. […] on occasion many-particle interactions become the essence of a physical effect. Superconductivity and superfluidity are of such intricate origin. To account for their occurrence one had to devise novel, sophisticated theoretical means. The emergence of the FQHE requires such a new kind of thinking. […]

It was an important conceptual step to realize that an impinging magnetic field BB could be viewed as creating tiny whirlpools, so-called vortices, in this lake of charge—one for each flux quantum ϕ 0=h/e\phi_0 = h/e of the magnetic field. […] Casting electron-electron correlation in terms of vortex attachment facilitates the comprehension of this intricate many-particle behavior. Regarding the vortices as little whirlpools ultimately remains a crutch for visualizing something that has no classical analog. […]

At magnetic fields higher than the i=1i = 1 IQHE, the stronger magnetic field provides more flux quanta and hence there are more vortices than there are electrons. The Pauli principle is readily satisfied by placing one vortex onto each electron [Fig. 14(a)]—but there are more vortices available. The electron system can considerably reduce its electrostatic Coulomb energy by placing more vortices onto each electron [Fig. 14(b)]. More vortices on an electron generate a bigger surrounding whirlpool, pushing further away all fellow electrons, thereby reducing the repulsive energy. […]

Vortices are the expression of flux quanta in the 2D electron system, and each vortex can be thought of as having been created by a flux quantum. Conceptually, it is advantageous to represent the vortices simply by their “generators”, the flux quanta themselves. Then the placement of vortices onto electrons becomes equivalent to the attachment of magnetic flux quanta to the carriers. Electrons plus flux quanta can be viewed as new entities, which have come to be called composite particles, CPs.

As these objects move through the liquid, the flux quanta act as an invisible shield against other electrons. Replacing the system of highly interacting electrons by a system of electrons with such a “guard ring” – compliments of the magnetic field – removes most of the electron-electron interaction from the problem and leads to composite particles which are almost void of mutual interactions. It is a minor miracle that such a transformation from a very complex many-particle problem of well-known objects (electrons in a magnetic field) to a much simpler single-particle problem of rather complex objects (electrons plus flux quanta) exists and that it was discovered.

CPs act differently from bare electrons. All of the external magnetic field has been incorporated into the particles via flux quantum attachment to the electrons. Therefore, from the perspective of CPs, the magnetic field has disappeared and they no longer are subject to it. They inhabit an apparently field-free 2D plane. Yet more importantly, the attached flux quanta change the character of the particles from fermions to bosons and back to fermions. […]

As the magnetic field deviates from exactly ν=1/3\nu = 1/3 filling to higher fields, more vortices are being created (Fig. 16). They are not attached to any electrons, since this would disturb the symmetry of the condensed state. The amount of charge deficit in any of these vortices amounts to exactly 1/3 of an electronic charge. These quasiholes (whirlpool in the electron lake) are effectively positive charges as compared to the negatively charged electrons. An analogous argument can be made for magnetic fields slightly below ν=1/3\nu = 1/3 and the creation of quasielectrons of negative charge e/3e/3. Quasiparticles can move freely through the 2D plane and transport electrical current. They are the famous 1/3 charged particles of the FQHE that have been observed by various experimental means […].

Plateau formation in the FQHE arises, in analogy to plateau formation in the IQHE from potential fluctuations and the resulting localization of carriers. In the case of the FQHE the carriers are not electrons, but, instead, the bizarre fractionally charged quasiparticles.

[end of excerpt from Störmer 1999]


Properties

As a topological insulator

The bulk/edge behaviour in a quantum Hall effect is is that of a topological insulator. (While topological insulator materials typically show this behaviour without the need of a strong magnetic field.)

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References

General

Review:

Discussion via Newton-Cartan theory:

  • William Wolf, James Read, Nicholas Teh, Edge modes and dressing fields for the Newton-Cartan quantum Hall effect [arXiv:2111.08052]

See also:

Integral quantum Hall effect

Experiment

Original experimental detection:

Theory

While an intuitive understanding for the quantization of the Hall conductance has been given in

a theoretical derivation of the effect was obtained only much later in

with closely related results in

Review of this theory behind the quantum Hall effect:

Fractional quantum Hall effect

General

Review and survey of the FQHE:

See also:

A quick review of the description via Chern-Simons theory with further pointers is in the introduction of

  • Spencer D. Stirling, Abelian Chern-Simons theory with toral gauge group, modular tensor categories, and group categories, [arXiv:0807.2857]

Realization via AdS/CFT in condensed matter physics:

Experiment

Observation of the FQHE in GaASGaAS:

in graphene:

in oxide interfaces:

  • A. Tsukazaki et al.: Observation of the fractional quantum Hall effect in an oxide, Nature Materials 9 (2010) 889–893 [doi:10.1038/nmat2874]

in CdTeCdTe:

  • B. A. Piot, J. Kunc, M. Potemski, D. K. Maude, C. Betthausen, A. Vogl, D. Weiss, G. Karczewski, T. Wojtowicz: Fractional quantum Hall effect in CdTeCdTe, PhysRev B. 82 (2010) 081307 (R) [doi:10.1103/PhysRevB.82.081307, arXiv:1006.0908]

Phenomenological models

Phenomenological models for the fractional quantum Hall effect:

The original Laughlin wavefunction:

The Halperin multi-component model:

The Haldane-Halperin model:

The composite-fermion model (CF) which explains the FQHE as the integer quantum Hall effect not of the bare electrons but of quasi-particles which they form (for reasons not explained by the model):

Introducing abelian Chern-Simons theory to the picture:

Further discussion:

Discussion highlighting the lack of microscopic explanation of these phenomenological models:

“Though the Laughlin function very well approximates the true ground state at ν=1q\nu = \tfrac{1}{q}, the physical mechanism of related correlations and of the whole hierarchy of the FQHE remained, however, still obscure.”

“The so-called HH (Halperin–Haldane) model of consecutive generations of Laughlin states of anyonic quasiparticle excitations from the preceding Laughlin state has been abandoned early because of the rapid growth of the daughter quasiparticle size, which quickly exceeded the sample size.”

“the Halperin multicomponent theory and of the CF model advanced the understanding of correlations in FQHE, however, on a phenomenological level only. CFs were assumed to be hypothetical quasi-particles consisting of electrons and flux quanta of an auxiliary fictitious magnetic field pinned to them. The origin of this field and the manner of attachment of its flux quanta to electrons have been neither explained nor discussed.”

Quantum Hall effect via non-commutative geometry

Discussion of the integer quantum Hall effect via a Brillouin torus with noncommutative geometry and using the Connes-Chern character:

Generalization of BvESB94 to the fractional quantum Hall effect:

See also exposition in:

Discussion of the fractional quantum Hall effect via abelian but noncommutative (matrix model-)Chern-Simons theory:

Anyons in quantum Hall liquids

References on anyon-excitations (satisfying braid group statistics) in the quantum Hall effect (for more on the application to topological quantum computation see the references there):

The prediction of abelian anyon-excitations in the quantum Hall effect (i.e. satisfying braid group statistics in 1-dimensional linear representations of the braid group):

The original discussion of non-abelian anyon-excitations in the quantum Hall effect (i.e. satisfying braid group statistics in higher dimensional linear representations of the braid group, related to modular tensor categories):

Review:

Claims of experimental observation:

In string/M-theory

On geometric engineering of aspects of the quantum Hall effect on M5-brane worldvolumes via an effective noncommutative geometry induced by a constant B-field flux density:

category: physics

Last revised on September 23, 2024 at 10:34:51. See the history of this page for a list of all contributions to it.

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