Landau Level Mixing Research Articles

Fractional Quantum Hall States with Non-Abelian Statistics

Using exact numerical diagonalization we have studied correlated many-electron ground states in a partially filled second Landau level. We consider filling fractions ? = 1/2 and 2/5, for which incompressible quantum liquids with non-Abelian anion statistics have been proposed. Our calculations include finite layer width, Landau level mixing and arbitrary deformation of the interaction pseudopotential. Computed energies, gaps, and correlation functions support the non-Abelian ground states at both ? = 1/2 (“Pfaffian”) and ? = 2/5 (“parafermion” state).

Electron quantum dynamics in closed and open potentials at high magnetic fields: Quantization and lifetime effects unified by semicoherent states

We have developed a Green's function formalism based on the use of an overcomplete semicoherent basis of vortex states specially devoted to the study of the Hamiltonian quantum dynamics of electrons at high magnetic fields and in an arbitrary potential landscape smooth on the scale of the magnetic length. This formalism is used here to derive the exact Green's function for an arbitrary quadratic potential in the special limit where Landau level mixing becomes negligible. This solution remarkably embraces under a unified form the cases of confining and unconfining quadratic potentials. This property results from the fact that the overcomplete vortex representation provides a more general type of spectral decomposition of the Hamiltonian operator than usually considered. Whereas confining potentials are naturally characterized by quantization effects, lifetime effects emerge instead in the case of saddle-point potentials. Our derivation proves that the appearance of lifetimes has for origin the instability of the dynamics due to quantum tunneling at saddle points of the potential landscape. In fact, the overcompleteness of the vortex representation reveals an intrinsic microscopic irreversibility of the states synonymous with a spontaneous breaking of the time symmetry exhibited by the Hamiltonian dynamics.

Softening of the tunneling gap in modulation-doped GaAs/AlGaAs asymmetric coupled double quantum wells in magnetic fields

Magnetophotoluminescence emissions were measured from modulation doped GaAs/AlGaAs asymmetric double quantum wells, wherein a thin barrier (25 Å) was sandwiched between a single quantum well (SQW) and a single heterojuction (SHJ). In the SQW, Landau level mixing is observed at the quantum Hall states. At ν<2, the lowest Landau level transition undergoes an exciton transition. For the SHJ region, the free carrier transitions become excitonic at the crossing point of the GaAs free exciton and the tunneling band gap shows a marked softening. An exciton-exciton interaction is shown to be responsible for the behavior of the subband energy levels in magnetic fields.

Transition from Abelian to non-Abelian quantum liquids in the second Landau level

The search for non-Abelian quantum Hall states in the second Landau level is narrowed to the range of filling factors 7/3<\nu_e<8/3. In this range, the analysis of energy spectra and correlation functions, calculated including finite width and Landau level mixing, supports the prominent non-Abelian candidates at \nu_e=5/2 (Moore--Read) and 12/5 (Read--Rezayi). Outside of it, the four-flux noninteracting composite fermion model is validated. The borderline \nu_e=7/3 state is adiabatically connected to the Laughlin liquid, but its short-range correlations and charge excitations are different.

TRANSPORT PROPERTIES OF TUNNEL-COUPLED ONE-DIMENSIONAL CHANNELS FROM AN ELECTRON BILAYER SYSTEM IN PERPENDICULAR MAGNETIC FIELD

In a tunnel coupled GaAs / AlGaAs electron bilayer system (BLS), we compare transport measurements of 1D quantum Hall (QH) edge channels with lithographically defined quantum point contacts (QPCs). The electron densities in both systems can be varied by a top-gate. In a perpendicular magnetic field, Landau level mixing is observed in the QH regime, and indications of crossings of spin split QPC subbands are detected.

Experimental studies of scaling behavior of a quantum Hall system with a tunable Landau level mixing

Temperature dependence of the longitudinal and Hall resistances is studied in the regime of localization-delocalization transition. We carry out measurements of scaling exponent $\ensuremath{\kappa}$ in the Landau level mixing region at several filling factors. The localization exponent $\ensuremath{\gamma}$ is extracted using an approach based on the variable-range hopping theory. The values of $\ensuremath{\gamma}$ and $\ensuremath{\kappa}$ are found to be universal and independent of the filling factor in our sample. We can conclude that although Landau level mixing can change the degeneracy of a quantum Hall state, the value of the scaling exponent remains the same for a given sample that contains a fixed disorder profile.

Phase diagram for quantum Hall states in graphene

We investigate integer and half-integer filling states (uniform and unidimensional stripe states, respectively) for graphene using the Hartree-Fock approximation. For fixed filling factor, the ratio between the scales of the Coulomb interaction and Landau level spacing $g=({e}^{2}/ϵ\ensuremath{\ell})/(\ensuremath{\hbar}{v}_{F}/\ensuremath{\ell})$, with $\ensuremath{\ell}$ as the magnetic length, is a field-independent constant. However, when $B$ decreases, the number of filled negative Landau levels increases, which surprisingly turns out to decrease the amount of Landau level mixing. The resulting states at fixed filling factor $\ensuremath{\nu}$ (for $\ensuremath{\nu}$ not too big) have very little Landau level mixing even at arbitrarily weak magnetic fields. Thus in the density-field phase diagram, many different phases may persist down to the origin, in contrast to the more standard two-dimensional electron gas, in which the origin is surrounded by Wigner crystal states. We demonstrate that the stripe amplitudes scale roughly as $B$ so that the density waves ``evaporate'' continuously as $B\ensuremath{\rightarrow}0$. Tight-binding calculations give the same scaling for stripe amplitude and demonstrate that the effect is not an artifact of the cut-off procedure used in the continuum calculations.

Quantum Hall Effect of Massless Dirac Fermions in a Vanishing Magnetic Field

The effect of strong long-range disorder on the quantization of the Hall conductivity sigma{xy} in graphene is studied numerically. It is shown that increasing Landau-level mixing progressively destroys all plateaus in sigma{xy} except the plateaus at sigma{xy}=-/+e{2}/2h (per valley and per spin). The critical state at the Dirac point is robust to strong disorder and belongs to the universality class of the conventional plateau transitions in the integer quantum Hall effect. We propose that the breaking of time-reversal symmetry by ripples in graphene can realize this quantum critical point in a vanishing magnetic field.

Influence of Landau-level mixing on Wigner crystallization in graphene

Graphene, with its massless linearly dispersing carriers, in the quantum Hall regime provides an instructive comparison to conventional two-dimensional systems in which carriers have a nonzero band mass and quadratic dispersion. We investigate the influence of Landau-level mixing in graphene on Wigner crystal states in the $n\text{th}$ Landau level obtained using single-Landau-level approximation. We show that the Landau-level mixing does not qualitatively change the phase diagram as a function of partial filling factor $\ensuremath{\nu}$ in the $n\text{th}$ level. We find that the inter-Landau-level mixing, which is quantified by relative occupations of the two Landau levels, ${\ensuremath{\rho}}_{n+1}/{\ensuremath{\rho}}_{n}$, oscillates around 2% and, in general, remains small $(l4%)$ irrespective of the Landau-level index $n$. Our results show that the single-Landau-level approximation is applicable in high Landau levels, even though the energy gap between the adjacent Landau levels vanishes.

Effects of interaction-induced second Landau level mixing in theν=1quantum Hall effect

The work by Mandal and Jain [Solid State Commun. 118, 503 (2001)] suggests that interaction-induced mixing with the second composite fermion Landau level can lead to renormalization of the electron correlation function exponent in the fractional quantum Hall effect. In the work reported here, a similar mixing with the second electronic Landau level is studied in the $\ensuremath{\nu}=1$ integer case. The ground state is calculated by use of the Hartree\char21{}Fock algorithm, and the electron density and electron correlation function on the edge are calculated. It is shown that the interaction gives rise to oscillations in the density profile. In particular, a short range interaction gives a profile qualitatively similar to the results reported by Mandal and Jain. On the other hand, no renormalization of the correlation function exponent is found.

Intrinsic Gap of theν=5/2Fractional Quantum Hall State

The fractional quantum Hall effect is observed at low magnetic field where the cyclotron energy is smaller than the Coulomb interaction energy. The nu=5/2 excitation gap at 2.63 T is measured to be 262+/-15 mK, similar to values obtained in samples with twice the electronic density. Examining the role of disorder on the 5/2 state, we find that a large discrepancy remains between theory and experiment for the intrinsic gap extrapolated from the infinite mobility limit. The observation of a 5/2 state in the low-field regime suggests that inclusion of nonperturbative Landau level mixing may be necessary to fully understand the energetics of half-filled fractional quantum Hall liquids.

Effect of Landau Level Mixing on Braiding Statistics

We examine the effect of Landau level mixing on the braiding statistics of quasiparticles of Abelian and non-Abelian quantum Hall states. While path dependent geometric phases can perturb the Abelian part of the statistics, we find that the non-Abelian properties remain unchanged to an accuracy that is exponentially small in the distance between quasiparticles.

Particle-Hole Symmetry and theν=52Quantum Hall State

We discuss the implications of approximate particle-hole symmetry in a half-filled Landau level in which a paired quantum Hall state forms. We note that the Pfaffian state is not particle-hole symmetric. Therefore, in the limit of vanishing Landau-level mixing, in which particle-hole transformation is an exact symmetry, the Pfaffian spontaneously breaks this symmetry. There is a particle-hole conjugate state, which we call the anti-Pfaffian, which is degenerate with the Pfaffian in this limit. We observe that strong Landau-level mixing should favor the Pfaffian, but it is an open problem which state is favored for the moderate Landau-level mixing which is present in experiments. We discuss the bulk and edge physics of the anti-Pfaffian. We analyze a simplified model in which transitions between analogs of the two states can be studied in detail. Finally, we discuss experimental implications.

Particle-Hole Symmetry and the Pfaffian State

We show that the particle-hole conjugate of the Pfaffian state-or "anti-Pfaffian" state-is in a different universality class from the Pfaffian state, with different topological order. The two states can be distinguished easily by their edge physics: their edges differ in both their thermal Hall conductance and their tunneling exponents. At the same time, the two states are exactly degenerate in energy for a nu=5/2 quantum Hall system in the idealized limit of zero Landau level mixing. Thus, both are good candidates for the observed sigma_{xy}=5/2(e;{2}/h) quantum Hall plateau.

Intrinsic Zeeman Effect in Graphene

The intrinsic Zeeman energy is precisely one half of the cyclotron energy for electrons in graphene. As a result a Landau-level mixing occurs to create the energy spectrum comprised of the 4 j -fold degenerated zero-energy level and 4-fold degenerated nonzero-energy levels in the j -layer graphene, where j =1,2,3 for monolayer, bilayer and trilayer, respectively. The degeneracy manifests itself in the quantum Hall (QH) effect. We study how the degeneracy is removed by the Coulomb interactions. With respect to the zero-energy level, an excitonic gap opens by making a BCS-type condensation of electron–hole pairs at the filling factor ν=0. It gives birth to the Ising QH ferromagnet at ν=±1 for monolayer, ν=±1,±3 for bilayer, and ν=±1,±3,±5 for trilayer graphene from the zero-energy degeneracy. With respect to the nonzero-energy level, a remarkable consequence is derived that the effective Coulomb potential depends on spins, since a single energy level contains up-spin and down-spin electrons belonging to different Landau levels. The spin-dependent Coulomb interaction leads to the valley polarization at ν=±4,±8,±12,... for monolayer, ν=±2,±6,±10,... for bilayer, and ν=±2,±4,±8,±12,... for trilayer graphene.

Exact-diagonalization studies of trion energy spectra in high magnetic fields

Binding energies of negative and positive trions in doped GaAs quantum wells in high magnetic fields are studied by exact numerical diagonalization in spherical geometry. Compared to earlier calculations, finite width of the quantum well and its asymmetry caused by one-sided doping are both fully taken into account by using self-consistent subband wave functions in the integration of Coulomb matrix elements, and by inclusion of higher subbands along with several Landau levels in the Hilbert space. Detailed analysis of the accuracy and convergence of the exact diagonalization scheme is presented, including dependence on Landau level and subband mixing, sensitivity to the (not well known) single-particle spectrum in the valence band, and the estimate of finite-size errors. The main results are the exciton dispersion and trion binding energy spectrum calculated as a function of the magnetic field, quantum well width, electron concentration, and the presence of an ionized impurity. As a complementary approach, a combination of the exact diagonalization in the quantum well plane and the variational calculation in the normal direction is used as well.

Landau level mixing in theν=5∕2fractional quantum Hall state

The nu=5/2 fractional quantum Hall state is studied numerically, directly including the effects of electron scattering between neighboring Landau levels. Significant reduction of the excitation gap caused by the LL mixing explains the discrepancy between earlier calculations and experiments. On the other hand, LL mixing also considerably reduces overlaps with the Moore--Read wavefunction, raising a question of the actual realization of nonabelian quasiparticles in present experiments.

Transport gap in aν=1∕3quantum Hall system: A probe for skyrmions

Transport measurements of the activation gap at fractional filling factor $1∕3$ are compared to results of exact diagonalization, allowing identification of a small anti-skyrmion as the lowest excitation in the low-field regime. In agreement with theory, a crossover to spinless excitations at higher electron densities is observed. Two samples of different quality are investigated. A detailed description of the theoretical calculation of activation gaps is presented and features which should be taken into account are summarized: finite thickness, Landau level mixing, and comparison between different sizes of the model system and---whenever possible---also between different geometries (torus and sphere). Within the chosen model of disorder (entailing a single fit parameter) we obtain a good agreement between calculated energies and experimental results.

Physical Picture behind the Oscillating Sign of Drag in High Landau Levels

We consider the oscillating sign of the drag resistivity and its anomalous temperature dependence discovered experimentally in a bilayer system in the regime of the integer quantum Hall effect. We attribute the oscillating sign to the effect of disorder on the relation between an adiabatic momentum transfer to an electron and the displacement of its position. While in the absence of any Landau level mixing a momentum transfer implies a displacement of (with being the magnetic length), Landau level mixing induced by short range disorder adds a potentially large displacement that depends on the electron's energy, with the sign being odd with respect to the distance of that energy from the center of the Landau level. We show how the oscillating sign of drag disappears when the disorder is smooth and when the electronic states are localized.

Tunneling spectroscopy of Landau band gaps at a quantum Hall line junction with adjustable Fermi level

Two laterally adjacent quantum Hall systems separated by an extended barrier of a thickness on the order of the magnetic length possess a complex Landau band structure in the vicinity of the line junction. The energy dispersion is obtained from an exact quantum-mechanical calculation of the single electron eigenstates for the coupled system by representing the wave functions as a superposition of parabolic cylinder functions. For orbit centers approaching the barrier, the separation of two subsequent Landau levels is reduced from the cyclotron energy to gaps which are much smaller. The position of the anticrossings increases on the scale of the cyclotron energy as the magnetic field is raised. In order to experimentally investigate a particular gap at different field strengths but under constant filling factor, a GaAs/AlGaAs heterostructure with a 52 Angstrom thick tunneling barrier and a gate electrode for inducing the two-dimensional electron systems was fabricated by the cleaved edge overgrowth method. The shift of the gaps is observed as a displacement of the conductance peaks on the scale of the filling factor. Besides this effect, which is explained within the picture of Landau level mixing for an ideal barrier, we report on signatures of quantum interferences at imperfections of the barrier which act as tunneling centers. The main features of the recent experiment of Yang, Kang et al. are reproduced and discussed for different gate voltages. Quasiperiodic oscillations, similar to the Aharonov Bohm effect at the quenched peak, are revealed for low magnetic fields before the onset of the regular conductance peaks.