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, lB. 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/lB 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 ≫ lB, 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 ≫ lB 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 px or py-wave pairing instead of px ± ipy-wave pairing in the MR Pfaffian/anti-Pfaffian state.

Observation of an anomalous density-dependent energy gap of the ν=5/2 fractional quantum Hall state in the low-density regime

“…As discussed in the second paragraph of this Article, there are several topologically distinct candidate ground states for the ν = 5/2 FQHS: the Pfaffian 52 , anti-Pfaffian 53,54 , the (3,3,1) Abelian state 55 , a variational wavefunction based on an antisymmetrized bilayer state 56 , the particle-hole symmetric Pfaffian 57,58 , a stripe-like alternation of the Pfaffian and anti-Pfaffian 59 , and other exotic states 60,61 . One thus may consider a potentially large number of topological phase transitions between the pairwise distinct candidate ground states.…”

“…Alternative candidate ground states for a FQHS at this filling factor are the anti-Pfaffian 53,54 , the (3,3,1) Abelian state 55 , a variational wavefunction based on an antisymmetrized bilayer state 56 , the particle-hole symmetric Pfaffian 57,58 , a stripe-like alternation of the Pfaffian and anti-Pfaffian 59 , and other exotic states 60,61 . An ongoing intense experimental effort is not yet able to unambiguously discriminate between these gapped candidate states [33][34][35][36][37][38][39] .…”

Fractional quantum Hall bilayers at half filling: Tunneling-driven non-Abelian phase

“…Numerical simulations [33][34][35][36] have confirmed that the Abelian (331) Halperin state dominates at vanishing interlayer tunneling (t ⊥ = 0) 14 . Remarkably, through uncovering the underlying pairing nature of the (331) Halperin state 24,25 , it has been suggested that the tunneling effect may drive the system into a non-Abelian phase [16][17][18][26][27][28][29][30][31] , which motivates intensive efforts [36][37][38][39][40] to establish its existence. However, previous numerical studies, primarily utilizing exact diagonalization on small system sizes, are still too limited to reach a consensus.…”

Pairing instabilities of Dirac composite fermions

“…By using the Dirac description of the dipole nature of CFs, we can identify the paired quantum Hall state of Ref. [21] with PH Pfaffian, and its closeness to anisotropy as a sign of the relevance of anisotropic solutions discussed in Sec. II.…”

“…(20) or (21) likely present when considering pairing instabilities in the half-filled LL, consistent with the exact diagonalization results of Refs. [21,22]. Finally, we point out that the Dirac based microscopic wave functions of pairing instabilities have not been proposed and tested yet.…”