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Modern Physics Letters B

HARTREE–FOCK–BOGOLIUBOV STUDIES OF BILAYER ν = 2 QUANTUM HALL SYSTEMS

YUICHI SHIMODA, Tatsuya Nakajima, Anju Sawada

Published May 10, 2005• 3 citations
Publisher

Abstract

Excitation spectra in bilayer quantum Hall systems at total Landau-level filling ν = 2 are studied by the Hartree–Fock–Bogoliubov (HFB) approximation. The systems have the spin degrees of freedom in addition to the layer degrees of freedom described in terms of pseudospin. On the excitation spectrum from the spin-singlet and pseudospin-polarized ground state, this approximation fully preserves the spin rotational symmetry and thus can give both spin-triplet and spin-singlet excitations systematically. It is also found that the ground state is pseudospin squeezed because of pseudospin correlations and that the HFB approximation can describe ground-state properties better than other approximations.

DOI
10.1142/s0217984905008517

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Excitation spectra in bilayer quantum Hall systems at total Landau-level filling ν = 2 are studied by the Hartree–Fock–Bogoliubov (HFB) approximation. The systems have the spin degrees of freedom in addition to the layer degrees of freedom described in terms of pseudospin. On the excitation spectrum from the spin-singlet and pseudospin-polarized ground state, this approximation fully preserves the spin rotational symmetry and thus can give both spin-triplet and spin-singlet excitations systematically. It is also found that the ground state is pseudospin squeezed because of pseudospin correlations and that the HFB approximation can describe ground-state properties better than other approximations.
Excitation spectrum of ν=2\nu=2ν=2 Bilayer Quantum Hall Systems
Excitation spectra in bilayer quantum Hall systems at total Landau-level filling ν=2\nu=2ν=2 are studied by the Hartree-Fock-Bogoliubov approximation. The systems have the spin degrees of freedom in addition to the layer degrees of freedom described in terms of pseudospin. On the excitation spectra from spin-unpolarized and pseudospin-polarized ground state, this approximation fully preserves the spin rotational symmetry and thus can give not only spin-triplet but also spin-singlet excitations systematically. It is also found that the ground-state properties are well described by this approximation.
Quantum Hall systems have a one-body energy spectrum consisting of dispersion-less Landau levels. Electron-electron interactions thus dominate in partially filled Landau levels, which exhibit a myriad of strongly correlated phases such as quantum hall ferromagnets and fractional quantum Hall phases. We study two examples of these phenomena. In the first project, we explore the ground state of a system with an interface between two semi-infinite regions with fillings ν= 4 and ν= 3 respectively. The width of the interface can be controlled by varying the background potential, which provides an additional tuning parameter. For a certain range of interaction strengths, the ν= 4 bulk is unpolarized whereas the ν= 3 bulk is fully polarized. In the parameter space spanned by the interaction strength and width of the interface, we find two phases at the interface. Phase A has spin as a good quantum number, and the long-wavelength spin edge excitations are gapped. In phase B, spin rotation symmetry is spontaneously broken at the mean-field level. Using symmetry arguments we find the effective theory near the interface of phase B. This effective theory is known to have gapless long-wavelength spin excitations. In the second project, we study the ground state of a tunnel-decoupled double-layer graphene system when both layers are undoped. We find a simple Hamiltonian in the continuum limit from symmetries of the system. Using the Hartree-Fock approximation we find a state with inter-layer coherence with broken layer U(1) symmetry. This phase becomes magnetized in presence of a non-zero Zeeman field. A first-order phase transition can be driven from the ferromagnetic phase to the magnetized inter-layer coherent phase by increasing the Zeeman field. We predict the number of gapless modes in the bulk.
We present an exact diagonalization study of bilayer quantum Hall systems at filling factor nu = 2 in the spherical geometry. We find the high-Zeeman-coupling phase boundary of the broken symmetry canted antiferromagnet is given exactly by previous Hartree-Fock mean-field theories, but that the state's stability at weak Zeeman coupling has been qualitatively overestimated. In the absence of interlayer tunneling, degeneracies occur between total spin multiplets due to the Hamiltonian's invariance under independent spin rotations in top and bottom two-dimensional electron layers.
We study the correlation energy, the effective anisotropy parameter, and quantum fluctuations of the pseudospin magnetization in bilayer quantum Hall systems at total filling factor \ensuremath{\nu}=1 by means of exact diagonalizations of the Hamiltonian in the spherical geometry. We compare exact-diagonalization results for the ground-state energy with finite-size Hartree-Fock values. In the ordered ground-state phase at small layer separations the Hartree-Fock data compare reasonably with the exact results. Above the critical layer separation, however, the Hartree-Fock findings still predict an increase in the ground-state energy, while the exact ground-state energy is in this regime independent of the layer separation indicating the decoupling of layers and the loss of spontaneous phase coherence between them. We also find accurate values for the pseudospin anisotropy constant, whose dependence of the layer separation provides another very clear indication for the strong interlayer correlations in the ordered phase and shows an inflection point at the phase boundary. Finally, we discuss the possibility of interlayer correlations in biased systems even above the phase boundary for the balanced case. Certain features of our data for the pseudospin anisotropy constant as well as for quantum fluctuations of the pseudospin magnetization are not inconsistent with the occurrence of this effect. However, it appears to be rather weak at least in the limit of vanishing tunneling amplitude.
We present details of earlier studies [Zheng et al., Phys. Rev. Lett. 78, 310 (1997) and Das Sarma et al., 79, 917 (1997)] and additional results on double-layer quantum Hall systems at a total filling $\ensuremath{\nu}=2{\ensuremath{\nu}}_{1}$, where a single layer at filling ${\ensuremath{\nu}}_{1}$ forms a ferromagnetic, fully spin-polarized, gapped incompressible quantum Hall state. For the case ${\ensuremath{\nu}}_{1}=1$, a detailed Hartree-Fock analysis is carried out on a realistic, microscopic Hamiltonian. Apart from the state continuously connected to the ground state of two well-separated layers, we find two double-layer quantum Hall phases: one with a finite interlayer antiferromagnetic spin ordering in the plane orthogonal to the applied field (the ``canted'' state), and the other a spin singlet. The quantum transitions between the various quantum Hall states are continuous, and are signaled by the softening of collective intersubband spin-density excitations. For the case of general ${\ensuremath{\nu}}_{1}$, closely related results are obtained by a semi-phenomenological continuum quantum field theory description of the low-lying spin excitations using a nonlinear $\ensuremath{\sigma}$ model. Because of its broken symmetry, the canted phase supports a linearly dispersing Goldstone mode and has a finite-temperature Kosterlitz-Thouless transition. We present results on the form of the phase diagram, the magnitude of the canted order parameter, the collective excitation dispersions, the specific heat, the form of the dynamic light-scattering spectrum at finite temperature, and the Kosterlitz-Thouless critical temperature. Our findings are consistent with recent experimental results.