Breaking of Particle-Hole Symmetry by Landau Level Mixing in the $\nu =5/2$ Quantized Hall State

We perform numerical studies to determine if the fractional quantum Hall state observed at a filling factor of $\nu =5/2$ is the Moore-Read wave function or its particle-hole conjugate, the so-called anti-Pfaffian. Using a truncated Hilbert space approach we find that, for realistic interactions, including Landau-level mixing, the ground state remains fully polarized and the anti-Pfaffian is strongly favored.

Figure 1

The overlaps of the projected ground state with the Moore-Read Pfaffian and the anti-Pfaffian for ${\mathcal{H}}_{r,15}$, $(N_{\varphi },N)=(20,50)$. Varying $V_1$ slightly, the projected ground state obtains a large overlap with the APf, while the overlap with the Pf remains relatively small. A substantial part of the Pf overlap appears to be caused by the nonorthogonality of Pf with APf. Further increasing $V_1$, the system crosses over to a composite fermion Fermi-liquid [4], which in this case has the same crystal momentum. The top curve (also in Fig. 2 for the ZB case) is the projection of the ground state to the two-dimensional subspace spanned by the Pf and APf.

Figure 2

Same as Fig. 1 but for a square periodic unit cell. ZC and ZB refer to the zone corner and boundary of the Brillouin zone. Note that for this geometry the overlap of the Pfaffian with the anti-Pfaffian at the ZC is extremely small. A first order transition to the composite fermion state occurs for $\delta V_1>0.04$. For negative $\delta V_1$ we expect a stripe-type order [4]. Here, as well as in Fig. 1, the completely symmetry reduced Hilbert space dimension within ${\mathcal{H}}_{p,10}$ is 263.