Possible realization of a chiral $p$-wave paired state in a two-component system

There is much interest in the realization of systems with $p$-wave pairing in one dimension or chiral $p$-wave pairing in two dimensions, because these are believed to support Majorana modes at the ends or inside vortices. We consider a two-component system of composite fermions and provide theoretical evidence that, under appropriate conditions, the screened interaction between the minority composite fermions is such as to produce an almost exact realization of a $p$-wave paired state described by the so-called anti-Pfaffian wave function. This state is predicted to occur at filling $\nu =3/8$ or 13/8 in GaAs when the Zeeman energy is sufficiently small, and at $\nu =\pm 3/8$ or $\pm 13/8$ in single layer graphene when either the Zeeman or the valley splitting is sufficiently small.

Figure 1

The low energy spectra obtained by CF diagonalization at both “Pf shift” (right panels) and “APf shift” (left panels) at partially spin polarized $\nu =3/8$, for systems with various values of $N$ and $2Q$ shown on the panels. The statistical error, estimated from Metropolis Monte Carlo calculation is less than the diameter of the circle. The energy per electron, $E$, includes the interaction with the background; it is quoted in the units of $e^2/\epsilon l$ where $l=\sqrt{\hslash c/e{B}_{\perp }}$ is the magnetic length, $\epsilon$ is dielectric constant of the host material, and $B_{\perp }$ is the component of the magnetic field along perpendicular to the plane of the system.

Figure 2

Left panel: The ground state energy per particle, $E_{\mathrm{gs}}$, for fully and partially polarized 3/8 states at the APf shifts (circles and squares, respectively). A linear extrapolation in $1/N$, where $N$ is the number of particles, produces the ground state energies (open symbols) in the thermodynamic limit. Right panel: Dependence of the critical value of $\kappa \equiv E_{\mathrm{Z}}/(e^2/\epsilon l)$, at which a transition from the partially polarized to fully polarized state takes place (see text), on electron density for several quantum well widths indicated on the figure. Here $E_{\mathrm{Z}}$ is the Zeeman splitting, $\epsilon$ is the dielectric function of the host material, and $l$ is the magnetic length.