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arXiv:1907.08364 · Physical review. B./Physical review. B

Interlayer correlated fractional quantum Hall state in the ν=4/5 bilayer system

Hao Wang, Alexander Seidel, Kun Yang, Fu‐Chun Zhang

Published Dec 12, 2019• 2 citations
arXivHTMLPDF

Abstract

We perform exact diagonalization studies for fractional quantum Hall states at filling factor $4/5$ in a bilayer system, on a torus with various aspect ratios and angles. We find that in the absence of tunneling, two weakly coupled $2/5$ layers undergo a phase transition into an interlayer-correlated regime, which is also Abelian with the fivefold degeneracy on the torus. In the limit of zero layer separation, this phase becomes a singlet in the pseudospin variable describing the layer degree of freedom. By studying the Chern-number matrix, we show that the $K$ matrix describing the interlayer-correlated regime requires a matrix dimension larger than two and this regime is in particular not described by a Halperin state. A detailed analysis of possible $4\ifmmode\times\else\texttimes\fi{}4\phantom{\rule{4pt}{0ex}}K$ matrices having the requisite symmetries and quantum numbers shows that there is only one equivalence class of such matrices. A model wave function representing this universality class is constructed. The role of separate particle number conservation in both layers is discussed, and it is argued that this additional symmetry allows for the further distinction of two different symmetry-protected Abelian phases in the interlayer correlated regime. Interlayer tunneling breaks this symmetry and can drive the system into a single-layer regime when strong enough. A qualitative phase diagram in the tunneling-layer separation parameter space is proposed based on our numerical results.

DOI
10.1103/physrevb.100.245122

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We perform exact diagonalization studies for fractional quantum Hall states at filling factor 4/54/54/5 in a bilayer system, on a torus with various aspect ratios and angles. We find that in the absence of tunneling, two weakly coupled 2/52/52/5 layers undergo a phase transition into an interlayer-correlated regime, which is also Abelian with the fivefold degeneracy on the torus. In the limit of zero layer separation, this phase becomes a singlet in the pseudospin variable describing the layer degree of freedom. By studying the Chern-number matrix, we show that the KKK matrix describing the interlayer-correlated regime requires a matrix dimension larger than two and this regime is in particular not described by a Halperin state. A detailed analysis of possible 4\ifmmode×\else\texttimes\fi4K4\ifmmode\times\else\texttimes\fi{}4\phantom{\rule{4pt}{0ex}}K4\ifmmode×\else\texttimes\fi4K matrices having the requisite symmetries and quantum numbers shows that there is only one equivalence class of such matrices. A model wave function representing this universality class is constructed. The role of separate particle number conservation in both layers is discussed, and it is argued that this additional symmetry allows for the further distinction of two different symmetry-protected Abelian phases in the interlayer correlated regime. Interlayer tunneling breaks this symmetry and can drive the system into a single-layer regime when strong enough. A qualitative phase diagram in the tunneling-layer separation parameter space is proposed based on our numerical results.
We present the quantitative phase diagram of the bilayer bosonic fractional quantum Hall system on the torus geometry at total filling factor $\ensuremath{\nu}=1$ in the lowest Landau level. We consider short-range interactions within and between the two layers, as well as the interlayer tunneling. In the fully polarized regime, we provide an updated detailed numerical analysis to establish the presence of the Moore-Read phase of both even and odd numbers of particles. In the actual bilayer situation, we find that both interlayer interactions and tunneling can provide the physical mechanism necessary for the low-energy physics to be driven by the fully polarized regime, thus leading to the emergence of the Moore-Read phase. Interlayer interactions favor a ferromagnetic phase when the system is SU(2)\mathrm{SU}(2)SU(2) symmetric, while the interlayer tunneling acts as a Zeeman field polarizing the system. Besides the Moore-Read phase, the (220)(220)(220) Halperin state and the coupled Moore-Read state are also realized in this model. We study their stability against each other.
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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.
Identifying and understanding interacting systems that can host non-Abelian topological phases with fractionalized quasiparticles have attracted intense attentions in the past twenty years. Theoretically, it is possible to realize a rich variety of such states by coupling two Abelian fractional quantum Hall (FQH) states together through gapping out part of the low energy degrees of freedom. So far, there are some indications, but no robust example has been established in bilayer systems for realizing the non-Abelian state in the past. Here, we present a phase diagram of a double-layer bosonic FQH system based on the exact diagonalization and density-matrix renormalization group (DMRG) calculations, which demonstrate a potential regime with the emergence of the non-Abelian bosonic Moore-Read state. We start from the Abelian phase with fourfold topological degeneracies on torus geometry when the two layers are weakly coupled. With the increase of interlayer tunneling, we find an intermediate regime with a threefold groundstate degeneracy and a finite fractional drag Hall conductance. We find the different topological sectors in consistent with Moore-Read state by inserting different fluxes in adiabatic DMRG study. We also extract the modular S−\mathcal{S}-S− matrix, which supports the emergence of the non-Abelian Ising anyon quasiparticle in this system.
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