nLab Laughlin wavefunction

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Contents

Idea

In solid stae 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 to be confirmed to great accuracy by experiment.

Basic Laughlin wavefunction on the plane

The basic Laughlin wavefunction, for NN spin-less (in practice really: spin-polarized by a strong magnetic field) fermions on the plane, at odd “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 disregard throughout) by

(1)Ψ La(z 1,⋯,z N)≔∏ i<j(z i−z j) qexp(−∑ 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( - \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 far away from the origin, 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.

We are the suppressing physical units which determine the actual average radius of this droplet. Maybe to be filled in later…

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

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

it is skew-symmetric under permutation of the particle positions, 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 mm to which the Vandermonde determinant is taken makes these particles carry “fractional charge”, namely 1/m1/m times the unit charge. (…)

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 is the result of choosing this factor to be the Pfaffian pf(−)pf(-) of the matrix of inverse distances between pairs of particles – 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(−∑ 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( - \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.


(…)


References

General

The original article:

Review:

See also:

Characterization as a braid representation:

A “hierarchy” of Laughlin-like states:

Interacting generalization

(…)

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 8, 2025 at 19:12:28. See the history of this page for a list of all contributions to it.

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