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arXiv:1601.00403 · Physical review. B./Physical review. B

Competing states for the fractional quantum Hall effect in the 1/3-filled second Landau level

Jae‐Seung Jeong, Hantao Lu, Ki Hoon Lee, Kenji Hashimoto, Suk Bum Chung, Kwon Park

Published Sep 25, 2017• 61 citations
arXivHTMLPDF

Abstract

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 ${Z}_{4}$ 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 ${Z}_{4}$ 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 ${Z}_{4}$ 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 ${Z}_{4}$ 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 ${V}_{1}^{(1)}$ 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 ${Z}_{4}$ 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 ${Z}_{4}$ parafermion state.

DOI
10.1103/physrevb.96.125148

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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 \ensuremathν=7/3\ensuremath{\nu}=7/3\ensuremathν=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{Z}_{4}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 ddd as a variational parameter. Under the proper understanding of the ground-state degeneracy in the torus geometry, the Z4{Z}_{4}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{Z}_{4}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{Z}_{4}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 \ensuremathδV1(1)\ensuremath{\delta}{V}_{1}^{(1)}\ensuremathδV1(1)​, the variation of the first-moment Haldane pseudopotential V1(1){V}_{1}^{(1)}V1(1)​ in the SLL with respect to the pure Coulomb interaction. It is shown that, while extremely accurate at sufficiently large positive \ensuremathδV1(1)\ensuremath{\delta}{V}_{1}^{(1)}\ensuremathδV1(1)​, the Laughlin state loses its overlap with the exact 7/3 ground state significantly at \ensuremathδV1(1)\ensuremath≃0\ensuremath{\delta}{V}_{1}^{(1)}\ensuremath{\simeq}0\ensuremathδV1(1)​\ensuremath≃0. At slightly negative \ensuremathδV1(1)\ensuremath{\delta}{V}_{1}^{(1)}\ensuremathδV1(1)​, it is shown that the PH-conjugated Z4{Z}_{4}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{Z}_{4}Z4​ parafermion state.
There are several possible theoretically allowed non-Abelian fractional quantum Hall (FQH) states that could potentially be realized in one- and two-component FQH systems at total filling fraction $\ensuremath{\nu}=n+2/3$, for integer nnn. Some of these states even possess quasiparticles with non-Abelian statistics that are powerful enough for universal topological quantum computation, and are thus of particular interest. Here we initiate a systematic numerical study, using both exact diagonalization and variational Monte Carlo, to investigate the phase diagram of FQH systems at total filling fraction $\ensuremath{\nu}=n+2/3$, including in particular the possibility of the non-Abelian Z4{Z}_{4}Z4​ parafermion state. In $\ensuremath{\nu}=2/3$ bilayers we determine the phase diagram as a function of interlayer tunneling and repulsion, finding only three competing Abelian states, without the Z4{Z}_{4}Z4​ state. On the other hand, in single-component systems at $\ensuremath{\nu}=8/3$, we find that the Z4{Z}_{4}Z4​ parafermion state has significantly higher overlap with the exact ground state than the Laughlin state, together with a larger gap, suggesting that the experimentally observed $\ensuremath{\nu}=8/3$ state may be non-Abelian. Our results from the two complementary numerical techniques agree well with each other qualitatively.
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.
Abstract Fractional quantum Hall states have been observed at filling factor ν = 3/4 in GaAs hole system and bilayer graphene. In theoretical bootstrap analysis, it was revealed that non-Abelian topological orders with Ising anyons can be realized at ν = 3/4, which exhibit 12 fold ground state degeneracy on the torus. The properties of ν = 3/4 states can be analyzed using two complementary approaches. In the first one, they are treated as particle–hole conjugate of ν = 1/4 Moore–Read types states. In the second one, they are mapped to composite fermions with reverse flux attachment at effective filling factor 3/2, whose integral part realizes an integer quantum Hall state and the fractional part realizes ν = 1/2 Moore–Read type states. For bilayer graphene with appropriate Landau level mixing, numerical calculations have found 12 quasi-degenerate ground states on the torus at ν = 3/4. Chiral graviton spectral functions of these states have one low energy peak with negative chirality and one high energy peak with positive chirality. This points to a specific member of the Moore–Read type states and agrees with the deduction based on daughter states.
One of the most dominant candidates for the paired quantum Hall (QH) state at filling factor $\ensuremath{\nu}=5/2$ 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, ddd, roughly equal to the magnetic length, lB{l}_{\mathrm{B}}lB​. Specifically, by using exact diagonalization in the torus geometry, we show that the exact 5/25/25/2 state at a given Haldane pseudopotential variation is intimately connected with the antisymmetrized bilayer ground state at a corresponding d/lBd/{l}_{\mathrm{B}}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 $d\ensuremath{\gg}{l}_{\mathrm{B}}$, 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 $d\ensuremath{\gg}{l}_{\mathrm{B}}$ 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{p}_{x}px​- or py{p}_{y}py​-wave pairing instead of ${p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}$-wave pairing in the MR Pfaffian/anti-Pfaffian state.
We study the fractional quantum Hall effect at filling fractions 7/3 and 5/2 in the presence of the spin-orbit interaction, using the exact diagonalization method and the density matrix renormalization group (DMRG) method in a spherical geometry. Trial wave functions at these fillings are the Laughlin state and the Moore-Reed-Pfaffian state. The ground state excitation energy gaps and pair-correlation functions at fractional filling factor 7/3 and 5/2 in the second Landau level are calculated. We find that the spin-orbit interaction stabilizes the fractional quantum Hall states.
Topological quantum states with non-Abelian Fibonacci anyonic excitations are widely sought after for the exotic fundamental physics they would exhibit, and for universal quantum computing applications. The fractional quantum Hall (FQH) state at a filling factor of ν=12/5 is a promising candidate; however, its precise nature is still under debate and no consensus has been achieved so far. Here, we investigate the nature of the FQH ν=13/5 state and its particle-hole conjugate state at 12/5 with the Coulomb interaction, and we address the issue of possible competing states. Based on a large-scale density-matrix renormalization group calculation in spherical geometry, we present evidence that the essential physics of the Coulomb ground state (GS) at ν=13/5 and 12/5 is captured by the k=3 parafermion Read-Rezayi state (RR_{3}), including a robust excitation gap and the topological fingerprint from the entanglement spectrum and topological entanglement entropy. Furthermore, by considering the infinite-cylinder geometry (topologically equivalent to torus geometry), we expose the non-Abelian GS sector corresponding to a Fibonacci anyonic quasiparticle, which serves as a signature of the RR_{3} state at 13/5 and 12/5 filling numbers.