Energetics of Pfaffian–anti-Pfaffian domains

In several recent works it has been proposed that, due to disorder, the experimentally observed $\nu =5/2$ quantum Hall state could be microscopically composed of domains of Pfaffian order along with domains of anti-Pfaffian order. We numerically examine the energetics required for forming such domains and conclude that for the parameters appropriate for recent experiments, such domains would not occur.

Figure 1

Example of extrapolation of DMRG results in the two cutoffs $N_s$ and $\chi$. (a) Extrapolation in the size of the interface region $N_s$. (b) The $N_s\to \infty$ values are extrapolated in the bond dimension $\chi$. The data is for the Pfaffian wave function at $L=26\ell$.

Figure 2

DMRG results after extrapolation in the cutoffs. $\mathrm{\Delta }\equiv E_{+e/4}-E_{-e/4}-\frac{1}{2}E\left(1\right)$, for both Pfaffian and anti-Pfaffian, plotted vs the inverse cylinder circumference $1/L$ (in units of $\ell$). We find ${\mathrm{\Delta }}^{\text{Pf}}<0$ (i.e., the Pfaffian has lower-energy quasielectrons) and the PH relation ${\mathrm{\Delta }}^{\mathrm{Pf}}+{\mathrm{\Delta }}^{\mathrm{APf}}=0$ is obeyed quite accurately for $L\ge 23\ell$. Inset: $1/L$ extrapolation of the data (outer lines indicate the uncertainty of the fit).

Figure 3

Quasielectron energy difference between Pfaffian and anti-Pfaffian as a function of system size. The numerical method is variational and exact diagonalization (see text). The upper points are for even numbers of electrons and the lower points are for odd. The fact that the two linear extrapolations do not intersect at $1/N=0$ suggests that we cannot completely trust the extrapolation. Here, $N=11,12,13,14,15,16,17$ on a hexagonal torus.