We propose a state described by the second Landau level (SLL) projection of a generalized Moore-Read Pfaffian wave function with an antiholomorphic pairing component. Unlike the PH-Pfaffian state which is described by the lowest Landau level (LLL) projection of the same antiholomorphic Pfaffian wave function, we find the proposed state, which we call the SLL PH-Pfaffian, represents a gapped, incompressible phase, and provides an excellent description for the exact ground state of finite-size SLL Coulomb-interacting electrons over a range of the short-distance interaction strength. We also find the SLL PH-Pfaffian, when mapped to the LLL, has a very large overlap and the same low-energy orbital entanglement spectrum structures with the anti-Pfaffian state. We discuss possible interpretations of our results on the nature of the topological order of both states.
Recently we have shown that the anti-Pfaffian state and a state described by the second Landau level (SLL) projection of antiholomorphic Pfaffian wavefunctions have large overlap and almost identical low-energy orbital entanglement structures, suggesting they have the same topological order. In this paper, we further show that the similar entanglement structure observed in the SLL projected state is also identifiable in other Landau level projected states, indicating it is not the result of the SLL projection but rather a "fingerprint" of the PH-Pfaffian topological order of the original unprojected antiholomorphic Pfaffian wavefunctions. Consequently, we argue that the SLL projected state, and therefore the anti-Pfaffian state, is a PH-Pfaffian topological order state. We also discuss the implications on the edge physics.
The PH-Pfaffian state having 1/2 central charge is consistent with the thermal Hall conductance measurement of 5/2 fractional quantum Hall system, but lacks support from the existing numerical results. In this paper we propose a new state described by a wavefunction obtained by multiplying a Hafnian factor to a PH-Pfaffian wavefunction. We call this new state the Hafnian PH-Pfaffian state. In spherical geometry, the Hafnian PH-Pfaffian state has the same magnetic flux number as the Pfaffian state, allowing a direct numerical comparison between the two states. Results of exact diagonalization of finite systems in the second Landau level show that the overlap of the exact ground state with the Hafnian PH-Pfaffian state exceeds that with the Pfaffian state when the short range component of the Coulomb interaction increases to a certain level, lending a numerical support to the Hafnian PH-Pfaffian state. We further show the Hafnian PH-Pfaffian state is mathematically identical to the newly proposed compressed PH-Pfaffian state [arxiv: 2001.01915 (2020)] formed 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. As the result we argue that the Hafnian PH-Pfaffian state has the same central charge as the PH-Pfaffian state, and is therefore consistent with the thermal Hall conductance measurement. Finally we present numerical results on two new wavefunctions formed by increasing the relative angular momentum by two for each of the paired composite fermions of the PH-Pfaffian state.
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.
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 investigate the nature of the fractional quantum Hall state in the 1/3-filled second Landau level (SLL) at filling factor $\ensuremath{\nu}=7/3$ (and 8/3 in the presence of the particle-hole symmetry) via exact diagonalization in both torus and spherical geometries. Specifically, we compute the overlap between the exact 7/3 ground state and various competing states including (i) the Laughlin state, (ii) the fermionic Haffnian state, (iii) the antisymmetrized product state of two composite fermion seas at 1/6 filling, and (iv) the particle-hole (PH) conjugate of the
Z4 parafermion state. All these trial states are constructed according to a guiding principle called the bilayer mapping approach, where a trial state is obtained as the antisymmetrized projection of a bilayer quantum Hall state with interlayer distance
d as a variational parameter. Under the proper understanding of the ground-state degeneracy in the torus geometry, the
Z4 parafermion state can be obtained as the antisymmetrized projection of the Halperin (330) state. Specifically, while unclear at other momentum sectors, all degenerate copies of the
Z4 parafermion state can be obtained by antisymmetrizing those of the Halperin (330) state at the zero-momentum sector, where both states occur as the exact ground states of their respective model Hamiltonians with the same degeneracy. Meanwhile, in the spherical geometry, the
Z4 parafermion state is shown to be entirely equivalent to the antisymmetrized Halperin (330) state without any ground-state degeneracy issue. Similarly, it is proved in this work that the fermionic Haffnian state can be obtained as the antisymmetrized projection of the Halperin (551) state. The exact 7/3 ground state is obtained as a function of $\ensuremath{\delta}{V}_{1}^{(1)}$, the variation of the first-moment Haldane pseudopotential
V1(1) in the SLL with respect to the pure Coulomb interaction. It is shown that, while extremely accurate at sufficiently large positive $\ensuremath{\delta}{V}_{1}^{(1)}$, the Laughlin state loses its overlap with the exact 7/3 ground state significantly at $\ensuremath{\delta}{V}_{1}^{(1)}\ensuremath{\simeq}0$. At slightly negative $\ensuremath{\delta}{V}_{1}^{(1)}$, it is shown that the PH-conjugated
Z4 parafermion state has a substantial overlap with the exact 7/3 ground state, which is the highest among the above four trial states. Around the Coulomb point, the energy spectrum exhibits an intriguing change from the spectrum with the Laughlin-type magnetoroton structure to that with the specific quasidegeneracy of the ground state, which is characteristic to the PH-conjugated
Z4 parafermion state.