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arXiv:1509.01780 · Physical review. B./Physical review. B

Striped quantum Hall state in a half-filled Landau level

Xin Wan, Kun Yang

Published May 23, 2016• 45 citations
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Abstract

The nature of the fractional quantum Hall state at Landau level filling factor 5/2 remains elusive despite intensive experimental and theoretical work. While the leading theoretical candidates are Moore-Read Pfaffian (Pf) and its particle-hole conjugate anti-Pfaffian (APf), neither received unambiguous experimental support. We show that a state that is intermediate between them, made of alternating stripes of Pf and APf in the bulk, is a viable candidate. Such a state is shown to be incompressible and thus a charge insulator in the bulk, but a heat conductor due to the presence of gapless neutral bulk modes. We argue that the properties of such a state are consistent with existing numerical and experimental work, and discuss possible experimental probes of its presence.

DOI
10.1103/physrevb.93.201303

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The nature of the fractional quantum Hall state at Landau level filling factor 5/2 remains elusive despite intensive experimental and theoretical work. While the leading theoretical candidates are Moore-Read Pfaffian (Pf) and its particle-hole conjugate anti-Pfaffian (APf), neither received unambiguous experimental support. We show that a state that is intermediate between them, made of alternating stripes of Pf and APf in the bulk, is a viable candidate. Such a state is shown to be incompressible and thus a charge insulator in the bulk, but a heat conductor due to the presence of gapless neutral bulk modes. We argue that the properties of such a state are consistent with existing numerical and experimental work, and discuss possible experimental probes of its presence.
We examine the quantum phase diagram of the fractional quantum Hall effect (FQHE) in the lowest two Landau levels in half-filled bilayer structures as a function of tunneling strength and layer separation, i.e., we revisit the lowest Landau-level filling factor 1/2 bilayer problem and make predictions involving bilayers in the half-filled second Landau level (i.e., filling factor 5/2). Using numerical exact diagonalization we investigate the important question of whether this system supports a FQHE described by the non-Abelian Moore-Read Pfaffian state in the strong-tunneling regime. In the lowest Landau level, we find that although, in principle, increasing (decreasing) tunneling strength (layer separation) could lead to a transition from the Abelian two-component Halperin 331 to non-Abelian one-component Moore-Read Pfaffian state, the FQHE excitation gap is relatively small in the lowest Landau level Pfaffian regime, and we establish that all so far observed FQHE states in half-filled lowest Landau level bilayers are most likely described by the Abelian Halperin 331 state. In the second Landau level we make the prediction that bilayer structures would manifest two distinct branches of incompressible FQHE corresponding to the Abelian 331 state (at moderate to low tunneling and large layer separation) and the non-Abelian Moore-Read Pfaffian state (at large tunneling and small layer separation). The observation of these two FQHE branches and the possible quantum phase transition between them will be compelling evidence supporting the existence of the non-Abelian Moore-Read Pfaffian state in the second Landau level. We discuss our results in the context of existing experiments and theoretical works.
The Moore-Read Pfaffian and anti-Pfaffian states have been under scrupulous review as candidates which describe the fractional quantum Hall effect at filling factor 5/2. Quantum states in the universality class of the Moore-Read Pfaffian/anti-Pfaffian have non-trivial intrinsic topological order and support low-energy non-Abelian excitations that have applications in fault-tolerant topological quantum computing schemes. Both states are exact ground states of three-body Hamiltonians that explicitly break particle-hole symmetry. We study the topological order of a competing ansatz state &PSgr;2 that is the exact ground state of a two-body Hamiltonian that preserves particle-hole symmetry. In particular, we calculate the bipartite entanglement entropy and spectra in the lowest Landau level in the spherical geometry for &PSgr; 2. We perform such calculations for a finite number of electrons up to 14. We then extrapolate to the thermodynamic limit the topological entanglement entropy γ as a measure of the topological order of the ansatz and compare to the known value of the Moore-Read Pfaffian/anti-Pfaffian state. We also study the orbital entanglement spectra for &PSgr;2 and compare with the Moore-Read Pfaffian and two-body Coulomb ground states. We show that our extrapolation of γ lies within the uncertainty of the known value of γ for the Moore-Read Pfaffian state, and that the orbital entanglement spectra of &PSgr;2 assumes a similar structure to that of the two-body Coulomb interaction.
The fractional quantum Hall effect (FQHE) observed at half filling of the second Landau level is believed to be caused by a pairing of composite fermions captured by the Moore-Read Pfaffian wave function. The generating Hamiltonian for the Moore-Read Pfaffian is a purely three-body model that breaks particle-hole symmetry and lacks other properties, such as dominate two-body repulsive interactions, expected from a physical model of the FQHE. We use exact diagonalization to study the low-energy states of a more physical two-body generator model derived from the three-body model. We find that the two-body model exhibits the essential features expected from the Moore-Read Pfaffian: pairing, non-Abelian anyon excitations, and a neutral fermion mode. The model also satisfies constraints expected for a physical model of the FQHE at half-filling because it is short range, spatially decaying, particle-hole symmetric, and supports a roton mode with a robust spectral gap in the thermodynamic limit. Hence, this two-body model offers a bridge between artificial three-body generator models for paired states and the physical Coulomb interaction and can be used to further explore properties of non-Abelian physics in the FQHE.
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.
In this work we propose a parton state as a candidate state to describe the fractional quantum Hall effect in the half-filled second Landau level. The wave function for this parton state is ${\mathcal{P}}_{\mathrm{LLL}}{\mathrm{\ensuremath{\Phi}}}_{1}^{3}{[{\mathrm{\ensuremath{\Phi}}}_{2}^{*}]}^{2}\ensuremath{\sim}{\mathrm{\ensuremath{\Psi}}}_{2/3}^{2}/{\mathrm{\ensuremath{\Phi}}}_{1}$ and in the spherical geometry it occurs at the same flux as the anti-Pfaffian state. This state has a good overlap with the anti-Pfaffian state and with the ground state obtained by exact diagonalization, using the second Landau level Coulomb interaction pseudopotentials for an ordinary semiconductor such as GaAs. By calculating the entanglement spectrum we show that this state lies in the same phase as the anti-Pfaffian state. A major advantage of this parton state is that its wave function can be evaluated for large systems, which makes it amenable to variational calculations. In the Appendix of this work, we have numerically assessed the validity of another candidate state at filling factor $\ensuremath{\nu}=\frac{5}{2}$, namely, the particle-hole-symmetric Pfaffian (PH-Pfaffian) state. We find that the proposed candidate wave function for the PH-Pfaffian state is particle-hole symmetric to a high degree but it does not appear to arise as the ground state of any simple Hamiltonian with two-body interactions.