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arXiv:2106.03475 · Physical review. B./Physical review. B

Particle-hole Pfaffian intracorrelations and intercorrelations in the quantum Hall bilayer

Milica V. Milovanović, Stevan Djurdjević

Published Dec 10, 2021• 3 citations
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Abstract

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 $p$-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.

DOI
10.1103/physrevb.104.245303

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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.
We discuss monolayer and bilayer quantum Hall systems in which each layer is a half-filled Landau level (LL) system. In the mean field approximation of the Son's formalism, there is a common pairing structure that underlines the possibilities for paired ground states in both systems. We argue that the particle-hole (PH) Pfaffian state in the (particle-hole symmetric) half-filled LL of a monolayer, and analogous state in the PH symmetric bilayer (in which each layer is half-filled LL) can be considered as critical states, i.e., states that cannot describe a phase under PH symmetry. We point out that the inclusion of a PH symmetry breaking (like LL mixing) may stabilize the PH Pfaffian in a monolayer. In the bilayer case, in numerical experiments on a sphere, by choosing the PH symmetric shift, we can stabilize the interlayer correlated (111) excitonic state or critical state, for any distance between the layers, but in general, with no bias, the evolution of the bilayer includes other phases.
In this review, the physics of Pfaffian paired states, in the context of fractional quantum Hall effect, is discussed using field-theoretical approaches. The Pfaffian states are prime examples of topological ([Formula: see text]-wave) Cooper pairing and are characterized by non-Abelian statistics of their quasiparticles. Here we focus on conditions for their realization and competition among them at half-integer filling factors. Using the Dirac composite fermion description, in the presence of a mass term, we study the influence of Landau level mixing in selecting a particular Pfaffian state. While Pfaffian and anti-Pfaffian are selected when Landau level mixing is not strong, and can be taken into account perturbatively, the particle–hole (PH) Pfaffian state requires non-perturbative inclusion of at least two Landau levels. Our findings, for small Landau level mixing, are in accordance with numerical investigations in the literature, and call for a non-perturbative approach in the search for PH Pfaffian correlations. We demonstrated that a method based on the Chern–Simons field-theoretical approach can be used to generate characteristic interaction pseudo-potentials for Pfaffian paired states.
We study Cooper pairing in the Dirac composite fermion (CF) system. The presence of the mass term in the Dirac CF description (which may simulate Landau level mixing), i.e., breaking of particle-hole (PH) symmetry in this system, is a necessary condition for the existence of a PH Pfaffian-like topological state. In the scope of the random-phase approximation (RPA) and hydrodynamic approach, we find some signatures of pairing at finite frequencies. Motivated by this insight, we extend our analysis to the case of a different but still Dirac quasiparticle (CF) representation, appropriate in the presence of a mass term, and discuss the likelihood of PH Pfaffian and Pfaffian pairings in general. On the basis of gauge field effects, we find for a small Dirac mass, an anti-Pfaffian or Pfaffian instability depending on the sign of mass, while for large mass (Landau level mixing), irrespective of its sign, we find a PH Pfaffian-like instability.
We numerically assess model wave functions for the recently proposed particle-hole-symmetric Pfaffian (``PH-Pfaffian'') topological order, a phase consistent with the recently reported thermal Hall conductance [M. Banerjee et al., Nature 559, 205 (2018)] at the ever enigmatic $\ensuremath{\nu}=5/2$ quantum Hall plateau. We find that the most natural Moore-Read-inspired trial state for the PH-Pfaffian, when projected into the lowest Landau level, exhibits a remarkable numerical similarity on accessible system sizes with the corresponding (compressible) composite Fermi liquid. Consequently, this PH-Pfaffian trial state performs reasonably well energetically in the half-filled lowest Landau level, but is likely not a good starting point for understanding the $\ensuremath{\nu}=5/2$ ground state. Our results suggest that the PH-Pfaffian model wave function either encodes anomalously weak ppp-wave pairing of composite fermions or fails to represent a gapped, incompressible phase altogether.
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.