CM In the literature: contested

Which topological order describes the $5/2$ quantum Hall state

In plain words

Several candidate wavefunctions (named Pfaffian, anti-Pfaffian and particle-hole Pfaffian) predict different edge structures for the $5/2$ state, and experiments and computer simulations currently point to different answers.

Precise statement

For $\nu = 5/2$ in high-mobility GaAs quantum wells, determine the topological order. The thermal Hall conductance $\kappa_{xy} = c \kappa_0$, $\kappa_0 = \pi^2 k_B^2 T/(3h)$, distinguishes Pfaffian ($c = 7/2$), PH-Pfaffian ($c = 5/2$) and anti-Pfaffian ($c = 3/2$); the measured value is $5/2$ (Banerjee et al., 2018), while numerics with Landau-level mixing favour anti-Pfaffian or Pfaffian. An answer is an identification consistent with thermal Hall, upstream neutral modes, tunnelling exponents and numerics.

What would settle it

Thermal Hall measurements with controlled edge equilibration, combined with numerics including realistic Landau-level mixing and disorder.

Status in the literature

Unverified note

Experiments favour PH-Pfaffian (thermal Hall 2018; neutral-mode noise, Dutta et al. 2022; time-domain braiding noise consistent with PH-Pfaffian, arXiv:2608.12897, 2026), while numerics favour anti-Pfaffian or Pfaffian; proposed reconciliations (incomplete edge equilibration, domains, disorder-stabilized PH-Pfaffian) are debated as of 2026.