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arXiv:1512.00889 · Physical review. B./Physical review. B

Phase diagram of ν=12+12 bilayer bosons with interlayer couplings

Zhao Liu, Abolhassan Vaezi, Cécile Repellin, Nicolas Regnault

Published Feb 9, 2016• 8 citations
arXivHTMLPDF

Abstract

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 $\mathrm{SU}(2)$ symmetric, while the interlayer tunneling acts as a Zeeman field polarizing the system. Besides the Moore-Read phase, the $(220)$ Halperin state and the coupled Moore-Read state are also realized in this model. We study their stability against each other.

DOI
10.1103/physrevb.93.085115

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Journal of Applied Mathematics and Physics•Jan 1, 2018•0 citations
We present the quantitative phase diagram of the bilayer bosonic fractional quantum Hall system on the torus geometry at total filling factor \ensuremathν=1\ensuremath{\nu}=1\ensuremathν=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.
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.
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 $4\ifmmode\times\else\texttimes\fi{}4\phantom{\rule{4pt}{0ex}}K$ 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.
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.
We study phase transitions in bilayer and trilayer bosonic quantum Hall systems. In the absence of interlayer tunneling and interaction, each layer is chosen to have filling factor ν=1/2ν=1/2ν=1/2 or 111 to realize the Laughlin state or the Moore-Read state. By tuning interlayer tunneling and/or interaction, multiple phases can be generated. In the absence of interlayer interaction, three phase transitions appear when interlayer tunneling becomes sufficiently strong: (1) from two decoupled ν=1/2ν=1/2ν=1/2 Laughlin states to the Moore-Read state in bilayer systems; (2) from one ν=1/2ν=1/2ν=1/2 Laughlin state plus one ν=1ν=1ν=1 Moore-Read state to the Read-Rezayi Z3\mathbb{Z}_{3}Z3​ state in bilayer systems; (3) from three decoupled ν=1/2ν=1/2ν=1/2 Laughlin states to the Read-Rezayi Z3\mathbb{Z}_{3}Z3​ state in trilayer systems. Numerical calculations suggest that these transitions are third-order ones. We propose non-Abelian Chern-Simons-Higgs theory to describe them. If both interlayer tunneling and interaction are present, one-component or multi-component composite fermion liquids and Jain states can be realized. This leads to intricate phase diagrams that host multiple phase transitions and possibly exotic critical points.
We study a quantum Hall bilayer system of bosons at total filling factor $\ensuremath{\nu}=1$, and study the phase that results from short-ranged pair tunneling combined with short-ranged interlayer interactions. We introduce two exactly solvable model Hamiltonians which both yield the coupled Moore-Read state [Phys. Rev. Lett. 108, 256809 (2012)] as a ground state, when projected onto fixed particle numbers in each layer. One of these Hamiltonians describes a gapped topological phase, while the other is gapless. However, on introduction of a pair-tunneling term, the second system becomes gapped and develops the same topological order as the gapped Hamiltonian. Supported by the exact solution of the full zero-energy quasihole spectrum and a conformal field-theory approach, we develop an intuitive picture of this system as two coupled composite fermion superconductors. In this language, pair tunneling provides a Josephson coupling of the superconducting phases of the two layers, and gaps out the Goldstone mode associated with particle transport between the layers. In particular, this implies that quasiparticles are confined between the layers. In the bulk, the resulting phase has the topological order of the Halperin 220 phase with U(1)${}_{2}\ifmmode\times\else\texttimes\fi{}\mathrm{U}$(1)2{}_{2}2​ topological order, but it is realized in the symmetric/antisymmetric basis of the layer index. Consequently, the edge spectrum at a fixed particle number reveals an unexpected U(1)${}_{4}\ifmmode\times\else\texttimes\fi{}\mathrm{U}$(1) structure.
The phase diagrams and phase transitions of a typical bilayer fractional quantum Hall (QH) system with filling factor ν = 2/3 at the layer balanced point are investigated theoretically by finite size exact-diagonalization calculations and an exactly solvable model. We find some basic features essentially different from the bilayer integer QH systems at ν = 2, reflecting the special characteristics of the fractional QH systems. The degeneracy of the ground states occurs depending on the difference between intralayer and interlayer Coulomb energies, when interlayer tunneling energy (ΔSAS) gets close to zero. The continuous transitions of the finite size systems between the spin-polarized and spin-unpolarized phases are determined by the competition between the Zeeman energy (ΔZ) and the electron Coulomb energy, and are almost not affected by ΔSAS.