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Pfaffian

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Contents

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×2n)-matrix with entries in some field (or ring) k.

Definition

The Pfaffian Pf(A)∈k is the element

12 nn!∑ σ∈S 2nsgn(σ)∏ i=1 nA σ(2i−1),σ(2i),\frac{1}{2^n n!} \sum_{\sigma \in S_{2n}} sgn(\sigma) \prod_{i = 1}^n A_{\sigma(2i -1), \sigma(2i)} \,,

where

Properties

In terms of Berezinian integrals

Proposition

Let Λ 2n be the Grassmann algebra on 2n generators {θ i}, which we think of as a vector θ⇀

Then the Pfaffian Pf(A) is the Berezinian integral

Pf(A)=∫exp(⟨θ⇀,A⋅θ⇀⟩)dθ 1dθ 2⋯dθ 2n.Pf(A) = \int \exp( \langle \vec \theta, A \cdot \vec \theta \rangle ) d \theta_1 d \theta_2 \cdots d \theta_{2n} \,.
Remark

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

det(A)∝∫exp(⟨θ⇀,A⋅ψ⇀⟩)dθ 1dθ 2⋯dθ 2ndψ 1dψ 2⋯dψ 2n.det(A) \propto \int \exp( \langle \vec \theta, A \cdot \vec \psi \rangle ) d \theta_1 d \theta_2 \cdots d \theta_{2n} d \psi_1 d \psi_2 \cdots d \psi_{2n} \,.

Pfaffian state

Pfaffians appear in the expression of certain multiparticle wave functions. Most notable is the pfaffian state of N spinless electrons

Ψ Pf(z 1,…,z N)=pfaff(1z k−z l)∏ i<j(z i−z j) qexp(−14∑∣z∣ 2)\Psi_{Pf}(z_1,\ldots,z_N) = pfaff\left(\frac{1}{z_k-z_l}\right)\prod_{i\lt j}(z_i-z_j)^q exp(-\frac{1}{4}\sum |z|^2)

where pfaff(M kl) denotes the Pfaffian of the matrix whose labels are k,l and q=1/ν is the filling fraction, which is an even integer. For Pfaffian state see

  • Gregory Moore, N. Read, Nonabelions in the fractional quantum hall effect, Nucl. Phys. 360B(1991)362 pdf

Revised on October 9, 2012 18:49:26 by Zoran Škoda (161.53.130.104)
Edit | Back in time (3 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: determinant, Pfaffian line bundle, matrix, Gauss-Bonnet theorem, fermionic path integral, Connections, Curvature, and Cohomology, hafnian