# 5/2 Enigma: ‘Odd’ fraction with even denominator

The filling factor $\nu=5/2$ state bears the most enigmatic fraction in the family of fractional quantum Hall effect (FQHE). This is the one of peculiar states that cannot be described by Laughlin’s ansatz or the hierarchical constructions by Haldane and Haperin. The reason is the even denominator in $\nu$.

The off-diagonal component of the conductance tensor has the following expression

$\sigma_{xy}=\nu \frac{e^2}{hc}$.

Inverse of  $\sigma_{xy}$ gives the Hall resistance $R_H$.

Willett, Eisenstein, English, and the trio (Tsui, Stormer, and Gossard) who  first discovered the $\nu=1/3$ FQHE, found this ‘odd’ fraction having the even denominator in 1987. Later in 1991, Moore and Read, by using conformal field theory (CFT), proposed  a new possible wavefunction

$\Psi_{\rm{Pf}}=\rm{Pf}\big(\frac{1}{z_i-z_j}\big)\prod_{i

where the Pf stands for the Pfaffian, square-root of the determinant of an anti-symmetric matrix or in other words anti-symmetrized sum over pairs:

${\rm{Pf}}\big(\frac{1}{z_i-z_j}\big)=\mathcal A \big(\frac{1}{z_1-z_2}\frac{1}{z_3-z_4}\cdots\big)$.

This Pfaffian wavefunction has a close connection to the p+ip superconductors. Therefore understanding either of these two different phases may help us to understand the other one.