nLab Pfaffian

Idea

The Pfaffian of a skew-symmetric matrix is a square root of its determinant.

Definition

Let $A = (A_{i,j})$ be a skew-symmetric $(2n \times 2n)$-matrix with entries in some field (or ring) $k$.

Definition

The Pfaffian $pf(A) \in k$ is the element

(1)$$pf(A) \;\coloneqq\; \frac{1}{2^n \, n!} \textstyle{\sum_{\sigma \in Sym_{2n}}} sgn(\sigma) \textstyle{\prod_{i = 1}^n} A_{\sigma(2i -1), \sigma(2i)} \,,$$

where

• $\sigma$ runs over all permutations of $2n$ elements;

• $sgn(\sigma)$ is the signature of a permutation.

Expressed equivalently in terms of the Levi-Civita symbol $\epsilon$ and using the Einstein summation convention the Pfaffian is

(2)$$pf(A) \;\coloneqq\; \frac{1}{2^n n!} A_{i_1 j_1} A_{i_2 j_2} \cdots A_{i_n j_n} \epsilon^{ i_1 j_1 i_2 j_2 \cdots i_n j_n } \,.$$

Properties

Relation to the determinant

Proposition

(Pfaffian is square root of determinant)

Let $A = (A_{i,j})$ be a skew-symmetric $(2n \times 2n)$-matrix with entries in some field (or ring) $k$.

Then the Pfaffian of $A$ (1) is a square root of the determinant of $A$:

(3)$$\big( pf(A) \big)^2 \;=\; det(A) \,.$$

Proofs are spelled out for instance by Haber 2015, Sections 2 and 3.

In terms of Berezinian integrals

Proposition

Let $\Lambda_{2n}$ be the Grassmann algebra on $2n$ generators $\{\theta_i\}$, which we think of as a vector $\vec \theta$

Then the Pfaffian $pf(A)$ is the Berezinian integral

$$pf(A) \;=\; \textstyle{\int} \, \left( \textstyle{\prod_{i=1}^{2n}} \mathrm{d}\theta_i \right) \, \exp\big( \langle \vec \theta, A \cdot \vec \theta \rangle \big) \,.$$

Remark

Compare this to the Berezinian integral-representation of the determinant, which is instead:

$$det(A) \;\propto\; \textstyle{\int} \left( \textstyle{\prod}_{i = 1}^{2n} \mathrm{d}\theta_i \right) \left( \textstyle{\prod}_{i = 1}^{2n} \mathrm{d}\psi_i \right) \, \exp\big( \langle \vec \theta, A \cdot \vec \psi \rangle \big) \,.$$

Applications

Pfaffian state of fractional quantum Hall systems

Pfaffians appear in the expression of certain multiparticle wave functions generalizing Laughlin wavefunctions modeling ground states of fractional quantum Hall systems.

Prominent here is the pfaffian state [Moore & Read 1991 (5.1)] of an even number $N$

(In practice, $N$ 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.)

of spinless (or rather: spin-polarized) electrons, which is the (“wave”-)function on the configuration space of $N$ points in the (complex) plane

(4)$$\Psi_{Pf}(z_1,\ldots,z_{N}) \;=\; vd(z_\bullet) ^q \, pf\left( \frac{1}{z_{\bullet_1} - z_{\bullet_2}} \right) \, exp\Big( -\frac{1}{4} \textstyle{\sum}_i {|z_i|}^2 \Big) \mathrlap{\,,}$$

where

• $vd(z_\bullet) \;\coloneqq\; \prod_{i \lt j} (z_i - z_j)$ is a Vandermonde determinant,

• $pf\left( \frac{1}{z_{\bullet_1} - z_{\bullet_2}} \right)$ is the Pfaffian of the inverse complex distances between the (electron) positions,

• $q = 1/\nu \,\in\, \mathbb{N}$ is the “filling fraction”.

Historically, (4) was guessed as a deformation of the Laughlin wavefunction, which is the same expression without the Pfaffian factor. But with the above Berezinian integration method both expressions actually unify in the super-Laughlin wavefunction:

(5)$$\Psi_{super}\big( (z_1, \theta_1), \cdots (z_N, \theta_N) \big) \;\coloneqq\; \textstyle{\prod_{i \lt j}} \, \big( z_i - z_j - \theta_i \theta_j \big)^q \, exp\Big( -\frac{1}{4} \textstyle{\sum_i} {|z_i|}^2 \Big) \,,$$

of which the ordinary Laughlin wavefunction is the lowest component (the coefficient of $\theta_i = 0$ for all $i$), while the Pfaffian state is the top component (the coefficient of $\theta_1 \cdot \theta_2 \cdots \theta_N$) – since, by Prop. :

$$\begin{array}{l} \textstyle{\int} \left( \textstyle{\prod_i} \mathrm{d}\theta_i \right) \, \underset{i \lt j}{\prod} ( z_i - z_j - \theta_i \theta_j )^q \\ \;=\; \textstyle{\int} \left( \textstyle{\prod_i} \mathrm{d}\theta_i \right) \, \textstyle{\underset{i \lt j}{\prod}} \Big( ( z_i - z_j )^q \, \exp\big( -q\, ( z_i - z_j )^{-1} \, \theta_i \theta_j \big) \, \Big) \\ \;=\; \textstyle{\int} \left( \textstyle{\prod_i} \mathrm{d}\theta_i \right) \, \Big( \textstyle{\underset{i \lt j}{\prod}} ( z_i - z_j )^q \Big) \Big( \textstyle{\underset{i \lt j}{\prod}} \exp\big( -q\, ( z_i - z_j )^{-1} \, \theta_i \theta_j \big) \Big) \\ \;=\; \Big( \textstyle{\underset{i \lt j}{\prod}} ( z_i - z_j )^q \Big) \, \textstyle{\int} \left( \textstyle{\prod_i} \mathrm{d}\theta_i \right) \, \Big( \textstyle{\underset{i \lt j}{\prod}} \exp\big( -q \, ( z_i - z_j )^{-1} \, \theta_i \theta_j \big) \Big) \\ \;=\; \Big( \textstyle{\underset{i \lt j}{\prod}} ( z_i - z_j )^q \Big) \, \textstyle{\int} \left( \textstyle{\prod_i} \mathrm{d}\theta_i \right) \, \exp\big( -q \, \textstyle{\sum_{i \lt j}} ( z_i - z_j )^{-1} \, \theta_i \theta_j \big) \\ \;=\; \left(-\tfrac{q}{2}\right)^{N/2} \, vd\big(z_\bullet\big)^q \, pf\left( \tfrac { 1 } { z_{\bullet_1} - z_{\bullet_2} } \right) \,. \end{array}$$

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

Remark

The argument of the super-Laughlin wavefunction (5) is indeed the difference of “super-positions” as seen in the super-translation group, where (cf. there):

$$(z_i, \theta_i) - (z_j, \theta_j) \;=\; \big( z_i - z_j - \theta_i \theta_i ,\, \theta_i - \theta_j \big) \,.$$

Remark

The Pfaffian $pf\Big( (z_{\bullet_1} - z_{\bullet_2})^{-1} \Big)$ changes sign when swapping any pair of variables $z_r \leftrightarrow z_s$ (which is manifest in the Berezinian presentation, where it corresponds equivalently to instead swapping $\theta_r \leftrightarrow \theta_s$).

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

1. for odd filling fraction $q$:

   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 $q$ it is the other way around.

Related entries

• determinant,

• Pfaffian line bundle,

• hafnian

References

General

Basics:

• Howard E. Haber, Notes on antisymmetric matrices and the pfaffian, 2015 (pdf, pdf)

See also

• J.-G. Luque, J.-Y. Thibon, Pfaffian and hafnian identities in shuffle algebras, math.CO/0204026

• Claudiu Raicu, Jerzy Weyman, Local cohomology with support in ideals of symmetric minors and Pfaffians, arxiv/1509.03954

• Haber, Notes on antisymmetric matrices and the pfaffian, pdf

There is also a deformed noncommutative version of Pfaffian related to quantum linear groups:

• Naihuan Jing, Jian Zhang, Quantum Pfaffians and hyper-Pfaffians, Adv. Math. 265 (2014), 336–361, arxiv/1309.5530

Pfaffian variety is subject of 4.4 in

• Alexander Kuznetsov, Semiorthogonal decompositions in algebraic geometry, arxiv/1404.3143

Relation to $\tau$-functions is discussed in

• J. W. van de Leur, A. Yu. Orlov, Pfaffian and determinantal tau functions I, arxiv/1404.6076

Other articles:

• András C. Lőrincz, Claudiu Raicu, Uli Walther, Jerzy Weyman, Bernstein-Sato polynomials for maximal minors and sub-maximal Pfaffians, arxiv/1601.06688

• Mauro Spera, A C–algebraic approach to determinants and Pfaffians, Acta Cosmologica Fasc. XXI-2, (1995) 203–208.

• Donald E. Knuth, Overlapping Pfaffians, Electronic Journal of Combinatorics 3:2 (1996) pdf

  “A combinatorial construction proves an identity for the product of the Pfaffian of a skew-symmetric matrix by the Pfaffian of one of its submatrices. Several applications of this identity are followed by a brief history of Pfaffians.”

• Jacques Distler, Nathan Donagi, Ron Donagi: On Generalized Pfaffians [arXiv:2409.06871]

Euler forms

Discussion of Euler forms (differential form-representatives of Euler classes in de Rham cohomology) as Pfaffians of curvature forms:

• Shiing-Shen Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Annals of Mathematics Second Series 45:4 (1944) 747-752 (jstor:1969302)

• Varghese Mathai, Daniel Quillen, below (7.3) of Superconnections, Thom classes, and equivariant differential forms, Topology Volume 25, Issue 1, 1986 (10.1016/0040-9383(86)90007-8)

• Siye Wu, Section 2.2 of Mathai-Quillen formalism, pages 390-399 in Encyclopedia of Mathematical Physics 2006 (arXiv:hep-th/0505003)

• Gerard Walschap, ch. 6.3 of Metric structures in differential deometry, Graduate Texts in Mathematics, Springer 2004

• Hiro Lee Tanaka, Pfaffians and the Euler class, 2014 (pdf)

• Liviu Nicolaescu, Section 8.3.2 of Lectures on the Geometry of Manifolds, 2018 (pdf, MO comment)

category: algebraic geometry, representation theory