nLab Laughlin wavefunction

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Idea

In solid state quantum physics, a Laughlin wavefunction is a certain Ansatz for an n-particle wavefunction which is meant to capture at least aspects of ground states with anyonic properties, such as of fractional quantum Hall systems.

The issue is that the strongly-coupled electron dynamics, that is thought to be responsible for the fractional quantum Hall effect, cannot be solved – not even approximately — by existing theory (cf. the problem of non-perturbative quantum field theory). To make up for this, the Laughlin wavefunction is an educated guess – jargon: trial wavefunction – as to what the ground states of these systems should approximately look like to some approximation — a guess that turns out to agree accurately with experiment. As such, the Laughlin wavefunction has come to be regarded as the standard effective theory of fractional quantum Hall systems (but there is criticism of this consensus, cf. Mikhailov 2024).

Basic Laughlin wavefunction on the plane

The basic Laughlin wavefunction [Laughlin 1983 (6-7), cf. Laughlin 1999 (21)], for NN spin-less (in practice really: spin-polarized by a strong transversal magnetic field) fermions (electrons, constrained to move) in a plane, at odd “Landau level filling fraction”

q∈2ℕ+1, q \,\in\, 2\mathbb{N} + 1 \,,

is the complex-valued function on the configuration space of N N ordered points in the plane — to be thought of as the complex plane with canonical holomorphic coordinate function zz —, given (up to normalization, which we may and do disregard throughout) by

(1)Ψ La(z 1,⋯,z N)≔∏ i<j(z i−z j) qexp(−14ℓ B 2∑ i|z i| 2). \Psi_{La}\big( z_1, \cdots, z_N \big) \;\coloneqq\; \textstyle{\prod_{i \lt j}} (z_i - z_j)^q \; \exp\big( - \tfrac{1}{4 \ell_B^2} \textstyle{\sum_i} {\vert z_i\vert}^2 \big) \,.

Here the absolute value of and hence the probability density encoded by

  • the second factor drops quickly where the particles are farther away from their center of mass (here: the origin in the complex plane) than the magnetic length

    ℓ B≔ℏc|e|B \ell_B \,\coloneqq\, \sqrt{ \frac { \hbar c } { {\vert e \vert} B } }

    (where BB denotes the given external magnetic field strength while the other constants are physical units: ℏ\hbar is Planck's constant, cc is the speed of light, and ee is the electric charge of an electron),

  • while that of the first factor tends to zero wherever any pair of particles approaches each other.

Hence the particles in this quantum state are most likely to be found all close to the origin but still spread out not to be too close to each other: The image is that of a little droplet, and one speaks also of quantum fluid droplets in this context.

Moreover, if or since the first factor (the Jastrow polynomial) is actually an odd power (the qqth power) of the Vandermonde determinant

vd(z •)≔∏i<j(z i−z j), vd(z_\bullet) \;\coloneqq\; \textstyle{\underset{i \lt j}{\prod}} (z_i - z_j) \,,

which itself is skew-symmetric under permutation of the particle positions (see there), so is the Laughlin state, as it must be for a many-fermion wavefunction, by the Pauli exclusion principle.

Due to these basic properties, the Laughlin wavefunction is a plausible Ansatz for any localized bound state of fermions. But on top of this, the power of qq to which the Vandermonde determinant is taken makes these particles support “qq-fractional” quasi-particle/hole excitations, see below.

Basic Moore-Read wavefunction on the plane

At even filling fraction

q∈2ℕ, q \,\in\, 2 \mathbb{N} \,,

the Laughlin Ansatz needs modification, since here the even power of the Vandermonde determinant would be symmetric under particle exchange and hence not describe the intended Fermions.

A further educated guess suggests to multiply, at even filling fraction, the Laughlin wavefunction by some other factor which is skew-symmetric in the particule positions. The Moore-Read wavefunction [Moore & Read 1991 (5.1)] is the result of choosing this factor to be the Pfaffian pf(−)pf(-) of the matrix of inverse distances between pairs of particles

(For the Pfaffian to be defined in this situation, we must consider an even number NN of electrons. But in the practice of the fractional quantum Hall effect, NN is a humongous “macroscopic” number on the scale of the Avogadro constant, and hence may be assumed to be even without any conceivable restriction of practical generality.)

– whence often known as the Pfaffian state:

(2)Ψ MR(z 1,⋯,z N)≔pf(1z • 1−z • 2)∏ i<j(z i−z j) qexp(−14ℓ B 2∑ i|z i| 2). \Psi_{MR}\big( z_1, \cdots, z_N \big) \;\coloneqq\; pf \left( \tfrac { 1 } { z_{\bullet_1} - z_{\bullet_2} } \right) \textstyle{\prod_{i \lt j}} (z_i - z_j)^q \, \exp\big( - \tfrac{1}{4 \ell_B^2} \textstyle{\sum_i} {\vert z_i\vert}^2 \big) \,.

That the Pfaffian of the (inverse) distance matrix is indeed skew-symmetric is readily seen in its Berezinian integral-formulation (see there). That formulation also reveals a kind of supersymmetry between the Laughlin and the Moore-Read wavefunctions:

Supersymmetry between Laughlin- and Moore&Read-wavefunctions

Consider the supermanifold (“Cartesian superspace”) version ℂ 1|1\mathbb{C}^{1 \vert 1} of the complex plane, with canonical holomorphic super-coordinates (z,θ)(z,\theta). This carries the structure of a super Lie group, namely of the complex super translation group, whose group operation is (see there):

(z i,θ i)+(z j,θ j)=(z i+z j+θ iθ i,θ i+θ j),−(z,θ)=(−z,−θ). (z_i, \theta_i) + (z_j, \theta_j) \;=\; \big( z_i + z_j + \theta_i \theta_i ,\, \theta_i + \theta_j \big) \,, \;\;\;\; -(z, \theta) = (-z, -\theta) \,.

Via this super-translation structure, the basic Laughlin state (1) lifts to superspace as

(3)Ψ sLa((z 1,θ 1),⋯(z N,θ N))≔∏ i<j(z i−z j−θ iθ j) qexp(−14∑ i|z i| 2). \Psi_{sLa}\big( (z_1, \theta_1), \cdots (z_N, \theta_N) \big) \;\coloneqq\; \textstyle{\prod_{i \lt j}} \, \big( z_i - z_j - \theta_i \theta_j \big)^q \, exp\Big( -\frac{1}{4} \textstyle{\sum_i} {|z_i|}^2 \Big) \,.

This super-Laughlin wavefunction unifies the basic Laughlin state (1) with the basic Moore&Read-state (2) in the sense of superfields:

∫(∏ idθ i)∏i<j(z i−z j−θ iθ j) q =∫(∏ idθ i)∏i<j((z i−z j) qexp(−q(z i−z j) −1θ iθ j)) =∫(∏ idθ i)(∏i<j(z i−z j) q)(∏i<jexp(−q(z i−z j) −1θ iθ j)) =(∏i<j(z i−z j) q)∫(∏ idθ i)(∏i<jexp(−q(z i−z j) −1θ iθ j)) =(∏i<j(z i−z j) q)∫(∏ idθ i)exp(−q∑ i<j(z i−z j) −1θ iθ j) =(−q2) N/2vd(z •) qpf(1z • 1−z • 2). \begin{array}{l} \textstyle{\int} \left( \textstyle{\prod_i} \mathrm{d}\theta_i \right) \, \underset{i \lt j}{\prod} ( z_i - z_j - \theta_i \theta_j )^q \\ \;=\; \textstyle{\int} \left( \textstyle{\prod_i} \mathrm{d}\theta_i \right) \, \textstyle{\underset{i \lt j}{\prod}} \Big( ( z_i - z_j )^q \, \exp\big( -q\, ( z_i - z_j )^{-1} \, \theta_i \theta_j \big) \, \Big) \\ \;=\; \textstyle{\int} \left( \textstyle{\prod_i} \mathrm{d}\theta_i \right) \, \Big( \textstyle{\underset{i \lt j}{\prod}} ( z_i - z_j )^q \Big) \Big( \textstyle{\underset{i \lt j}{\prod}} \exp\big( -q\, ( z_i - z_j )^{-1} \, \theta_i \theta_j \big) \Big) \\ \;=\; \Big( \textstyle{\underset{i \lt j}{\prod}} ( z_i - z_j )^q \Big) \, \textstyle{\int} \left( \textstyle{\prod_i} \mathrm{d}\theta_i \right) \, \Big( \textstyle{\underset{i \lt j}{\prod}} \exp\big( -q \, ( z_i - z_j )^{-1} \, \theta_i \theta_j \big) \Big) \\ \;=\; \Big( \textstyle{\underset{i \lt j}{\prod}} ( z_i - z_j )^q \Big) \, \textstyle{\int} \left( \textstyle{\prod_i} \mathrm{d}\theta_i \right) \, \exp\big( -q \, \textstyle{\sum_{i \lt j}} ( z_i - z_j )^{-1} \, \theta_i \theta_j \big) \\ \;=\; \left(-\tfrac{q}{2}\right)^{N/2} \, vd\big(z_\bullet\big)^q \, pf\left( \tfrac { 1 } { z_{\bullet_1} - z_{\bullet_2} } \right) \,. \end{array}

This observation is due to Hasebe 2008, cf. Gromov, Martinec & Ryu 2020 (13).

To re-amplify how the “statistics” matches across this transformation:

Remark

The Pfaffian pf((z • 1−z • 2) −1)pf\Big( (z_{\bullet_1} - z_{\bullet_2})^{-1} \Big) changes sign when swapping any pair of variables z r↔z sz_r \leftrightarrow z_s (which is manifest in the Berezinian presentation, where it corresponds to equivalently to swapping θ r↔θ s\theta_r \leftrightarrow \theta_s).

But also the Vandermonde determinant changes sign when swapping pairs of variables (see there).

This means that:

  1. for odd filling fraction qq:

    1. the ordinary Laughlin state is skew-symmetric in its arguments — as befits the wavefunction of multiple fermions,

    2. the Pfaffian Moore-Read state is symmetric in its arguments — as befits the wavefunction of multiple bosons.

  2. for even filling fraction qq it is the other way around.

Basic Laughlin wavefunction with quasi-holes

The above Laughlin- and Moore-Read-wavefunctions are meant to model the ground states of fractional quantum Hall systems at exact Landau level-filling fraction ν=1/q\nu = 1/q where, so the idea, every electron forms a kind of bound state with exactly qq quanta of magnetic flux (see there).

But if then the magnetic field is increased by n∈ℕn \in \mathbb{N} units of flux, a net total of nn flux quanta must remain un-paired with electrons. Still imagining that also these un-paired flux quanta are localized in the quantum Hall material, reflected in little vortices in the “electron fluid”, they are imagined to correspond to “holes” (jargon “quasi-holes”) in the electron fluid: points where electrons are forced to be absent.

A now common proposal for how to modify the basic Laughlin state (1) to reflect the presence of nn such quasi-holes at positions ξ a∈ℂ\xi_a \in \mathbb{C} is to mak these be the external parameters for a wavefunction of the form [Laughlin 1983 (13)]

(4)ψ La (ξ 1,⋯,ξ n)(z 1,⋯,z N) ≔ ∏ a∏ i(z i−ξ a)ψ La(z 1,⋯,z N) = ∏ a∏ i(z i−ξ a)∏ i<j(z i−z j) qexp(−14ℓ B 2∑ i|z i| 2). \begin{array}{ccl} \psi ^{(\xi_1, \cdots, \xi_n)} _{La}(z_1, \cdots, z_N) &\coloneqq& \textstyle{\prod_a} \prod_{i} (z_i - \xi_a) \, \psi_{La}(z_1, \cdots, z_N) \\ &=& \textstyle{\prod_a} \prod_{i} (z_i - \xi_a) \, \textstyle{\prod_{i \lt j}} (z_i - z_j)^q \; \exp\big( - \tfrac{1}{4 \ell_B^2} \textstyle{\sum_i} {\vert z_i\vert}^2 \big) \mathrlap{\,.} \end{array}

Clearly, the probability density of this wavefunction vanishes whenever at least one of the electron positions z iz_i coincides with one of the “electron hole” positions ξ a\xi_a, and there is non-vanishing angular momentum concentrated around these holes, as expected for electon-vortices centered at the ξ a\xi_a.

Beware that these holes at positions ξ a\xi_a are not fundamental particles like the electrons at positions z iz_i. Instead, the ξ i\xi_i are external parameters: In a less guess-work heavy discussion one would consider the Hamiltonian of the fractional quantum Hall system parameterized by hole positions, and the above wavefunction (4) would be an approximation to its ground state, also depending on these parameters.

As such, the hole positions ξ a\xi_a are “not dynamical” as the electrons are, but may, at least in principle, be moved by forces from outside the system, by changing the Hamiltonian across its parameter space. Such an external parameter-movement of a quantum system is described by the quantum adiabatic theorem, which says that, in the suitable adiabatic limit, a rotation of one parameter ξ a\xi_a around another will return the quantum state (4) to itself up to (an Aharonov-Bohm effect due to the magnetic field and) a Berry phase. This Berry phase is claimed to be [Arovas, Schrieffer & Wilczek 1984 (12)]

e 2πi/q, e^{ 2 \pi \mathrm{i} / q } \,,

which means that away from the case of the integer quantum Hall effect where q=1q = 1, the ground state wavefunction will not quite return to itself as its defect positions are rotated around each other, but only up to a phase.

Such fractional braiding-Berry phases of defects under adiabatic movement are said to identify the defects as being (abelian) anyons.

Beware that adiabatic Berry phases of defect parameters are conceptually different from “quantum statistics” of fundamental particles (even if most expositions will say otherwise, and even if the very term “anyon” suggests otherwise): For one, the wavefunction (4) is symmetric in the hole positions which, if these were positions of dynamical particles, would make them bosons.

In particular, some of the quasi-hole positions may as well coincide, ξ a=ξ b\xi_a = \xi_b. Noticing that the Laughlin wavefunction of qq coincident quasi-holes (at some position ξ a≔ξ∈ℂ\xi_a \coloneqq \xi \in \mathbb{C}) among NN electrons equals the Laughlin wavefunction of N+1N + 1 electrons with one held fixed at ξ\xi:

ψ La (ξ 1=ξ,⋯,ξ q=ξ)(z 1,⋯,z N)=ψ La(ξ,z 1,⋯,z N) \psi_{La}^{(\xi_1 = \xi, \cdots, \xi_q = \xi)}( z_1, \cdots, z_N) \;=\; \psi_{La}( \xi, z_1, \cdots, z_N )

gives another view of how the quasi-holes are qq-fractional objects as compared to the electrons.


References

General

The Laughlin wavefunction is due to

and the argument for the anyonic nature of its quasi-hole excitations is due to

Review:

See also:

A “hierarchy” of Laughlin-like states:

The pfaffian modification for even-fractional filling factor is due to

Critical discussion:

Laughlin wavefunctions as conformal blocks

Relating anyonic topologically ordered Laughlin wavefunctions to conformal blocks:

Review in the broader context of the CS-WZW correspondence:

Specifically for logarithmic CFT:

Specifically for su(2)-anyons:

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:

Last revised on January 12, 2025 at 09:30:48. See the history of this page for a list of all contributions to it.

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