nLab quantum Hall effect

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Contents

Idea

In short:

At sufficiently low temperature, 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 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).


In a tad more detail:


Quantum Hall effect. In a ∼\sim 2D sheet Σ 2\Sigma^2 of conducting material threaded by magnetic flux density BB

the energy of electron quantum states is quantized by Landau levels i∈ℕi \in \mathbb{N} as E=ℏω B(i+12), E \,=\, \hbar \omega_B \big( i + \tfrac{1}{2} \big) \,,

where each level comprises one state per magnetic flux quantum: n deg=B/Φ 0=Beh. n_{\mathrm{deg}} \,=\, B/\Phi_0 \,=\, B \tfrac{e}{h} \,.

Integer quantum Hall effect. Fermi theory of idealized free electrons hence predicts the system to be a conductor away from the energy gaps between a completely filled and the next empty Landau level, hence away from the number of electrons being integer multiples n el=νB/Φ 0n_{\mathrm{el}} = \nu B/\Phi_0, ν∈ℕ\nu \in \mathbb{N} of the number of flux quanta, where longitudinal conductivity should vanish.

This is indeed observed and is called the integer quantum Hall effect — in fact the vanishing conductivity is observed in sizeable neighbourhoods of the critical filling fractions (“Hall plateaux”, attributed to disorder effects).

Fractional quantum Hall effect. But electrons in a conductor are far from free. While there is little to no theory for strongly interacting quantum systems, experiment shows that the Fermi idealization breaks down at low enough temperature, where longitudinal conductivity decreases also in neighbourhoods of certain fractional filling factors ν∈ℚ\nu \,\in\, \mathbb{Q}, prominently so at ν=1/k\nu = 1/k for k∈2ℕ+1k \in 2\mathbb{N} + 1.

The heuristic idea is that at these filling fractions the interacting electrons form a kind of bound state with kk flux quanta each, making “composite bosons” that as such condense to produce an insulating mass gap, after all.

Anyonic quasi-particles. But this suggests that in the Hall plateau neighbourhood around such filling fraction, there are unpaired flux quanta each “bound to” one 1/k1/kth of a (missing) electron: called “quasi-particles” (“quasi-holes”). These evidently have fractional charge ±e/k\pm e/k and are expected to be anyonic with pair exchange phase e iπ/ke^{\mathrm{i} \pi/k}. There is claim that this anyonic phase has been experimentally observed.


Hall effect and Hall resistivity

The setup of any Hall effect is a plane sheet of conducting material placed in a transverse magnetic field (constant across the plane, directed perpendicular to it).

The classical Hall effect is the phenomenon that a voltage V xV_x applied along the conducting sheet in some direction – to be called the xx-direction – induces a Hall voltage V yV_y in the perpendicular direction – to be called the yy-direction – across the conducting sheet.

The cause of this effect is the Lorentz force, exerted on the electrons by the magnetic field, which is proportional in magnitude to the magnetic field and to the electron velocity but perpendicular in direction to both the magnetic field and to their direction of motion.

Due to this force, the electrons which start to follow the applied voltage gradient quickly drift to one side of the conducting sheet until their mutual electrostatic repulsion there counterbalances the Lorentz force. At this point the electrons move straight along the applied voltage gradient, with the Lorentz force now exactly compensated by the Hall voltage due to the gradient in electron concentration.

For more details on the classical Hall effect see there; here we further just need the formulas for conductivity and resistivity:

Consider in the plane ℝ 2\mathbb{R}^2, with canonical coordinates xx and yy, the

With the current running in the xx-direction

J→≔[J x 0] \vec J \;\coloneqq\; \left[ \begin{array}{c} J_{x} \\ 0 \end{array} \right]

the statement of the classical Hall effect is that

  • not just the longitudinal field E x≠0E_x \neq 0

but also

  • the Hall field E y≠0E_y \neq 0.

To say this more formally, recall that in a conductor the current J→\vec J is a linear function of the field E→\vec E with proportionality being the conductivity tensor σ\sigma, here a 2×22 \times 2 matrix, such that Ohm's law holds:

J→=σ⋅E→. \vec J \,=\, \sigma \cdot \vec E \,.

Assuming that the conducting sheet has no preferred direction, the conductivity tensor is of the form

σ=[σ xx σ xy −σ xy σ xx] \sigma \;=\; \left[ \begin{array}{cc} \sigma_{x x} & \sigma_{x y} \\ - \sigma_{x y} & \sigma_{xx} \end{array} \right]

for σ xx,σ xy∈ℝ\sigma_{x x}, \sigma_{x y} \,\in\, \mathbb{R}.

The corresponding resistivity tensor is the inverse matrix rho=σ −1rho = \sigma^{-1}

(1)ρ=[ρ xx ρ xy −ρ xy ρ yy]=1σ xx 2+σ xy 2[σ xx −σ xy σ xy σ xx]. \rho \;=\; \left[ \begin{array}{cc} \rho_{x x} & \rho_{x y} \\ - \rho_{x y} & \rho_{y y} \end{array} \right] \;=\; \tfrac{1}{ \sigma_{x x}^2 + \sigma_{x y}^2 } \left[ \begin{array}{cc} \sigma_{x x} & -\sigma_{x y} \\ \sigma_{x y} & \sigma_{x x} \end{array} \right] \,.

in terms of which Ohm's law reads

E→=ρ⋅J→. \vec E \,=\, \rho \cdot \vec J \,.

In this tensorial language, the classical Hall effect is the statement that for transverse magnetic field B>0B \gt 0 the non-diagonal elements of the conductivity/resistivity tensors are non-vanishing, in that we have

(2)Hall conductivityσ xy≠0andHall resistanceρ xy≠0. \text{Hall conductivity} \;\; \sigma_{x y} \neq 0 \;\;\;\;\;\; \text{and} \;\;\;\;\;\; \text{Hall resistance} \;\; \rho_{x y} \neq 0 \,.

In this case the basic matrix relation (1) is of some importance for understanding the measurement results in the integer quantum Hall effect below, since it implies the (maybe surprising-sounding) phenomenon that for non-vanishing Hall effect the longitudinal conductivity and resistivity may jointly vanish, see (4) below.

Concretely, the Hall resistivity turns out to be related to

by the formula

(3)ρ xy=Bne. \rho_{x y} \;=\; \frac{ B }{ n e } \mathrlap{\,.}

Integer quantum Hall effect

At extremely low temperature and extreme thin-ness of the conducting sheet, the above classical Hall effect exhibits modifications by quantum mechanical effects, due to the fact that the energy of electrons in a transverse magnetic field is quantized into discrete units known as Landau levels.

Since electrons are fermions, the Pauli exclusion principle demands that in their ground state the electrons fill the available Fermi sea with one electron per available state, below a given energy, the “chemical potential”. (Here, due to the strong external magnetic field, all electrons may be assumed to have their spin aligned along this field, so that the states in question concern just the remaining electron momenta.) The larger the magnetic field, the more quantum states are comprised by one Landau level.

In the case that the electrons fill exactly i∈ℕi \in \mathbb{N} Landau levels – one speaks of filling fraction ν=i\nu = i –, the next excited state, needed for the transport of charge, is separated by the energy gap to the next Landau level, and hence at an integer number of exactly filled Landau levels the Hall system behaves like an insulator with vanishing longitudinal conductivity σ xx∼0\sigma_{x x} \sim 0.

What is measured in experiments is the longitudinal resistivity, which — by (1) with (2) — also goes to zero at these points of exactly filled Landau levels:

(4)R x≔ρ xx=σ xxσ xx 2+σ xy 2⟶σ xx→00. R_x \coloneqq \rho_{x x} \;=\; \frac{ \sigma_{x x} }{ \sigma^2_{x x} + \sigma^2_{x y} } \;\; \underset{\sigma_{x x} \to 0}{\longrightarrow} \;\; 0 \,.

But the hallmark of the integer quantum Hall effect is that this vanishing of the longitudinal (conductivity and hence) resistivity is observed not just right when the magnetic field strength is at the critical value B=B iB = B_i, but in a whole neighbourhood of these values:

Remarkably, the height of these Hall plateaus is an experimental constant to high precision, and is independent of the detailed nature of the underlying material, unaffected even by punching holes into the conducting sheet.

Yet more remarkably, the explanation for the horizontal extension of these plateaux is thought to be related to impurities in the material — in an ideally pure conductor the quantum Hall effect is expected to be invisible! The idea is that, due to the impurities, the idealized picture of Landau levels applies only to some of the electrons in the sample, while others are “localized” at/by the imporitites; and as the magnetic field is varied it is only after the reservoir of localized electrons has changed energy levels that it becomes the turn of the “quantum Hall electrons”.

To compute the Hall plataux values:

The density of available states (number per surface area) available in a Landau level is

(5)d=ehB=BΦ 0 d \;=\; \frac{ e }{ h } B \;=\; \frac{B}{\Phi_0}

  • where h=2πℏh = 2\pi \hbar is Planck's constant,

  • Φ 0=he\Phi_0 = \frac{h}{e} is the unit magnetic flux quantum,

hence there us room for one electron per magnetic flux quanta.

Therefore the iith Landau level is exactly filled when

  • there are exactly ii electrons per magnetic flux quanta

hence when

  • the electron density nn is
n=i⋅d, n \,=\, i \cdot d \mathrlap{\,,}

which, according to (5), occurs theoretically right at (and in practice in a neighbourhood around) the critical magnetic field values

(6)B i≔Φ 0ni, B_i \;\coloneqq\; \Phi_0 \frac{n}{i} \mathrlap{\,,}

for which in turn, by (3), the Hall resistivity is

(7)(ρ xy) i=B ine=he 21i. (\rho_{x y})_i \;=\; \frac{ B_i }{ n e } \;=\; \frac{ h }{ e^2 } \frac{1}{i} \,.

This is hence the height of the iith Hall plateau in the integer quantum Hall effect.


These formulas, at least, generalize immediately from (positive) integers ii to (positive) rational numbers ν\nu:

In particular, for the 1st Landau level to be filled up to an integer fraction ν=1/k\nu = 1/k, there must be exactly kk magnetic flux quanta per electron.

Nothing special is expected to happen at these fractional fillings of Landau level from the above understanding based all on the energy gap seen by non-interacting electrons (only) at the Fermi surface of a filled Landau level. But electron interaction changes this picture, leading to the fractional quantum Hall effect:


Fractional quantum Hall effect

Even though the integer quantum Hall effect (above) involves many electrons (a macroscopic number on the scale of the Avogadro constant), which necessarily interact strongly via their mutual Coulomb force, for understanding the effect it turns out (as indicated above) to be sufficient to consider the energy of single electrons right at the Fermi sea surface of a filled Landau level as if they were “free” (non-interacting). That such a radical (and conceptually unjustified!) approximation works so well is surprising on a fundamental level, but is entirely common in traditional solid state physics, notably in Landau's Fermi liquid theory.

However, yet closer experimental analysis at yet smaller temperatures shows that this approximation breaks down at some point, and that the strong interaction between the electrons makes them collectively behave in exotic ways.

Concretely, experiments show that Hall plateaus appear not just at integer filling levels, but (smaller) Hall plateaus appear also at certain rational filling fractions

(8)ν∈ℚ,notably atν=1qforq∈2ℕ+1. \nu \in \mathbb{Q} \,, \;\;\; \text{notably at} \;\; \nu = \tfrac{1}{q} \;\text{for}\; q \in 2\mathbb{N} + 1 \,.

Concretely, by the same computation as for (7), the fractional Hall plateaux are at

(9)(ρ xy) ν=he 21ν. (\rho_{x y})_\nu \;=\; \frac{ h }{ e^2 } \frac{1}{\nu} \,.

This experimentally observed phenomenon is thus called the fractional quantum Hall effect.

Unfortunately, due to the general open problem of formulating and analyzing non-perturbative quantum field theory, there is essentially no first-principles understanding of what causes the fractional quantum Hall effect!

What people have come up with, instead, are:

  1. ad hoc (mental) models of how the electrons form supposedly “bound states” with magnetic flux quanta: “composite fermions”,

    suggesting that the fractional quantum Hall effect is just the integer quantum Hall effect again, now not for plain electrons but for their exotic “fractional” quasi-particle/quasi-hole bound states,

  2. some educated guesses as to the many-electron wavefunction describing the fractional Hall quantum state – Laughlin wavefunctions,

    which, while just guessed, are confirmed well by experiment and have in practice become the effective theory for FQH systems,

  3. some actual effectice quantum field theory description by variants of abelian Chern-Simons theory.

For more on Laughlin wavefunctions and on effective abelian Chern-Simons theory in the FQH context, see there.

A neat account of the commonly accepted composite fermion picture 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.)

(…)

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 abelian 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:

[p 3:] “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.”

Abelian Chern-Simons for frational quantum Hall effect

The idea of abelian Chern-Simons theory as an effective field theory exhibiting the fractional quantum Hall effect (abelian topological order) goes back to

  • Steven M. Girvin, around (10.7.15) in: Summary, Omissions and Unanswered Questions, Chapter 10 of: The Quantum Hall Effect, Graduate Texts in Contemporary Physics, Springer (1986, 1990) [doi:10.1007/978-1-4612-3350-3]

  • Steven M. Girvin, A. H. MacDonald, around (10) of: Off-diagonal long-range order, oblique confinement, and the fractional quantum Hall effect, Phys. Rev. Lett. 58 12 (1987) (1987) 1252-1255 [doi:10.1103/PhysRevLett.58.1252]

  • S. C. Zhang, T. H. Hansson S. Kivelson: Effective-Field-Theory Model for the Fractional Quantum Hall Effect, Phys. Rev. Lett. 62 (1989) 82 [doi:10.1103/PhysRevLett.62.82]

and was made more explicit in:

Early review:

Further review and exposition:

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

On edge modes:

Further developments:

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:

Supersymmetry in fractional quantum Hall systems

On hidden supersymmetry in fractional quantum Hall systems between even- and odd-level (filling-fraction) quantum states (Laughlin wavefunctions and their variants):

(for a similar phenomenon cf. also hadron supersymmetry)

The use of supergeometry in the description of fractional quantum Hall systems, and the observation that the Moore&Read state is the top super field-component of a super-Laughlin wavefunction was promoted in:

Based on this, the proposal that specifically the two collective modes of the ν=5/2\nu = 5/2 Moore&Read-state should be superpartners of each other, is due to:

further discussed in:

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 January 13, 2025 at 08:36:44. See the history of this page for a list of all contributions to it.

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