Bilayer mapping of the paired quantum Hall state: Instability toward anisotropic pairing

Abstract

One of the most dominant candidates for the paired quantum Hall (QH) state at filling factor $ν=5/2$ is the Moore-Read (MR) Pfaffian state. A salient problem, however, is that it does not occur exactly at the Coulomb interaction, but rather at a modified interaction, which favors particle-hole symmetry breaking. In an effort to find a better state, in this work, we investigate the possible connection between the paired QH state and the antisymmetrized bilayer ground state, which is inspired by the intriguing identity that the MR Pfaffian state is entirely equivalent to the antisymmetrized projection of the bilayer QH state called the Halperin (331) state, which is valid at interlayer distance, $d$, roughly equal to the magnetic length, $l_{\rm B}$. Specifically, by using exact diagonalization in the torus geometry, we show that the exact $5/2$ state at a given Haldane pseudopotential variation is intimately connected with the antisymmetrized bilayer ground state at a corresponding $d/l_{\rm B}$ via one-to-one mapping, which we call the bilayer mapping. One of the most important discoveries in this work is that the paired QH state occurring at the Coulomb interaction is mapped onto the antisymmetrized bilayer ground state at $d \gg l_{\rm B}$, which is equivalent to the antisymmetrized product state of two composite fermion seas at quarter filling, not the MR Pfaffian state. While maintaining high overlap with the paired QH state, the antisymmetrized bilayer ground state at $d \gg l_{\rm B}$ exhibits an abrupt change under the influence of small anisotropy. This suggests that the paired QH state occurring at the Coulomb interaction might be susceptible to anisotropic instability, opening up the possibility of anisotropic $p_x$ or $p_y$-wave pairing instead of $p_x \pm i p_y$-wave pairing in the MR Pfaffian/anti-Pfaffian state.