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arXiv:1603.03754 · Physical Review Letters

Stabilization of the Particle-Hole Pfaffian Order by Landau-Level Mixing and Impurities That Break Particle-Hole Symmetry

Philip Zucker, Dima E. Feldman

Published Aug 22, 2016• 109 citations
arXivHTMLPDF

Abstract

Numerical results suggest that the quantum Hall effect at ν=5/2 is described by the Pfaffian or anti-Pfaffian state in the absence of disorder and Landau-level mixing. Those states are incompatible with the observed transport properties of GaAs heterostructures, where disorder and Landau-level mixing are strong. We show that the recent proposal of a particle-hole (PH)-Pfaffian topological order by Son is consistent with all experiments. The absence of particle-hole symmetry at ν=5/2 is not an obstacle to the existence of the PH-Pfaffian order since the order is robust to symmetry breaking.

DOI
10.1103/physrevlett.117.096802

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Numerical results suggest that the quantum Hall effect at ν=5/2 is described by the Pfaffian or anti-Pfaffian state in the absence of disorder and Landau-level mixing. Those states are incompatible with the observed transport properties of GaAs heterostructures, where disorder and Landau-level mixing are strong. We show that the recent proposal of a particle-hole (PH)-Pfaffian topological order by Son is consistent with all experiments. The absence of particle-hole symmetry at ν=5/2 is not an obstacle to the existence of the PH-Pfaffian order since the order is robust to symmetry breaking.
A successful probing of the neutral Majorana mode in recent thermal Hall conductivity measurements opines in favor of the particle-hole symmetric Pfaffian (PH-Pf) topological order, contrasting the theoretical predictions of Pfaffian or anti-Pfaffian phases. Here we report a reentrant anomalous quantized phase that is found to be gapped in the thermodynamic limit, distinct from the conventional Pfaffian, anti-Pfaffian, or PH-Pf phases, at an intermediate strength of Landau level mixing. Our proposed wave function consistent with the PH-Pf shift in spherical geometry rightly captures the topological order of this phase, as its overlap with the exact ground state is very high and it reproduces low-lying entanglement spectra. A unique topological order, irrespective of the flux shifts, found for this phase, possibly corroborates the experimentally found topological order.
A Compressed Particle-Hole Symmetric Pfaffian State for ν=5/2ν= 5/2ν=5/2 Quantum Hall Effect
A recent thermal Hall conductance experiment [Banerjee et al., Nature {\bf559}, 205 (2018)] for ν=5/2ν= 5/2ν=5/2 fractional quantum Hall system appears to rule out both the Pfaffian and anti-Pfaffian and be in favor of the PH-Pfaffian topological order, while the existing numerical results without disorder have shown otherwise. In this paper we offer a possible resolution by proposing a new state, termed compressed PH-Pfaffian state by "compressing" the PH-Pfaffian state with two flux quanta removed to create two abelian Laughlin type quasiparticles of the maximum avoidance from one another (or of the maximum number of zeros). The compressed PH-Pfaffian state is not particle-hole symmetric but possesses the PH-Pfaffian topological order. In spherical geometry, the compressed PH-Pfaffian state has the same magnetic flux number Nϕ=2N−3N_ϕ= 2N-3Nϕ​=2N−3 as the Pfaffian state, allowing a direct numerical comparison between the two states. Results of exact diagonalization of finite disorder-free systems in the second Landau level show that, by increasing the short range component of the Coulomb interaction, the ground state undergoes a phase transition from the Pfaffian state to the compressed PH-Pfaffian state before further entering into a gapless state. The low energy gapped excited states result from the breakup of the abelian Laughlin type quasiparticle into two non-abelian quasiparticles.
Inter-Landau-level transitions break particle hole symmetry and will choose either the Pfaffian or the anti-Pfaffian state as the absolute ground state at 5/2 filling of the fractional quantum Hall effect. An approach based on truncating the Hilbert space has favored the anti-Pfaffian. A second approach based on an effective Hamiltonian produced the Pfaffian. In this Letter, perturbation theory is applied to finite sizes without bias to any specific pseudopotential component. This method also singles out the anti-Pfaffian. A critical piece of the effective Hamiltonian, which was absent in previous studies, reverts the ground state at 5/2 to the anti-Pfaffian.
Inter-Landau-level transitions break particle hole symmetry and will choose either the Pfaffian or the anti-Pfaffian state as the absolute ground state at 5/2 filling of the fractional quantum Hall effect. An approach based on truncating the Hilbert space has favored the anti-Pfaffian. A second approach based on an effective Hamiltonian produced the Pfaffian. In this Letter, perturbation theory is applied to finite sizes without bias to any specific pseudopotential component. This method also singles out the anti-Pfaffian. A critical piece of the effective Hamiltonian, which was absent in previous studies, reverts the ground state at 5/2 to the anti-Pfaffian.
Particle-hole (PH) Pfaffian topological phases may exist in a uniform system due to strong Landau level (LL) mixing according to theoretical predictions based on the Son-Dirac composite fermion theory. Numerical investigations in the presence of large LL mixing are limited due to numerical complexities, when taking into account at least one more LL. Because of this, we apply the same field theoretical approach to the quantum Hall bilayer at total filling factor equal to one, for which many numerical studies exist. The most advanced in Zhu et al. [Zheng Zhu, Liang Fu, D. N. Sheng, Phys. Rev. Lett. 119, 177601 (2017)] predicts an intermediate phase (for intermediate distances between layers) with an even-odd effect. According to our approach, the intermediate phase represents a mixed negative-flux ppp-wave pairing i.e., coexisting intra (PH Pfaffian in each layer) and inter (a la PH Pfaffian) pairing correlations. This again underlines a necessity for strong entanglement with additional degrees of freedom, i.e., at least one more (additional) LL in the search for a stable PH Pfaffian phase in a single layer. Based on the analogy with the bilayer physics, we propose a PH Pfaffian wave function that resides in two LLs.