Fractional Quantum Hall Liquids Research Articles

Universality of the Edge-Tunneling Exponent of Fractional Quantum Hall Liquids

In a microscopic model of fractional quantum Hall liquids with electron-electron interactions and confinement, we calculate the edge Green's function via exact diagonalization. Our results for nu=1/3 and 2/3 suggest that, in the presence of Coulomb interaction, "external" parameters such as the sharpness of the edge and the strength of the edge confining potential, which can lead to edge reconstruction, may cause deviations from universality in the edge-tunneling I-V exponent. In particular, we do not find any direct dependence of this exponent on the range of the interaction potential as suggested by recent calculations in contradiction to the topological nature of the edge.

Tilt-Induced Localization and Delocalization in the Second Landau Level

We have investigated the behavior of electronic phases of the second Landau level under tilted magnetic fields. The fractional quantum Hall liquids at nu=2+1/5 and 2+4/5 and the solid phases at nu=2.30, 2.44, 2.57, and 2.70 are quickly destroyed with tilt. This behavior can be interpreted as a tilt driven localization of the 2+1/5 and 2+4/5 fractional quantum Hall liquids and a delocalization through the melting of solid phases in the top Landau level, respectively. The evolution towards the classical Hall gas of the solid phases is suggestive of antiferromagnetic ordering.

Temperature Dependence of the Tunneling Amplitude between Quantum Hall Edges

Recent experiments have studied the tunneling current between the edges of a fractional quantum Hall liquid as a function of temperature and voltage. The results of the experiment are puzzling because at "high" temperature (600-900 mK) the behavior of the tunneling conductance is consistent with the theory of tunneling between chiral Luttinger liquids, but at low temperature it strongly deviates from that prediction dropping to zero with decreasing temperature. In this Letter we suggest a possible explanation of this behavior in terms of the strong temperature dependence of the tunneling amplitude.

Topological Numbers and Edge State of Hierarchical State in Rapidly Rotating Ultracold Atoms

The effective theory for the hierarchical fractionalquantum Hall (FQH) effectis proposed. We also derive the topological numbers K matrix and t vector and the general edge excitation from the effective theory. One canfind that the two issues in rapidly rotating ultracold atoms are similar tothose in electron FQH liquid.

Competition between a Fractional Quantum Hall Liquid and Bubble and Wigner Crystal Phases in the Third Landau Level

Magnetotransport measurements were performed in an ultrahigh mobility GaAs/AlGaAs quantum well of density approximately 3.0 x 10(11) cm(-2). The temperature dependence of the magnetoresistance Rxx was studied in detail in the vicinity of nu=9/2. In particular, we discovered new minima in Rxx at a filling factor nu approximately 41/5 and 44/5, but only at intermediate temperatures 80 approximately less than T approximately less than 120 mK. We interpret these as evidence for a fractional quantum Hall liquid forming in the N=2 Landau level and competing with bubble and Wigner crystal phases favored at lower temperatures. Our data suggest that a magnetically driven insulator-insulator quantum phase transition occurs between the bubble and Wigner crystal phases at T=0.

Strong-coupling branching between edges of fractional quantum Hall liquids

We have developed a theory of quasiparticle backscattering in a system of point contacts formed between single-mode edges of several fractional quantum Hall liquids (FQHLs) with different filling factors ${\ensuremath{\nu}}_{j}$ and one common single-mode edge ${\ensuremath{\nu}}_{0}$ of another FQHL. In the strong-tunneling limit, the model of quasiparticle backscattering is obtained by the duality transformation of the electron tunneling model. The new physics introduced by the multipoint-contact geometry of the system is coherent splitting of backscattered quasiparticles at the point contacts in the course of propagation along the common edge ${\ensuremath{\nu}}_{0}$. The ``branching ratios'' characterizing the splitting determine the charge and exchange statistics of the edge quasiparticles that can be different from those of Laughlin's quasiparticles in the bulk of FQHLs. Accounting for the edge statistics is essential for the system of more than one point contact and requires the proper description of the flux attachment to tunneling electrons.

Possible observation of phase coexistence of the nu=1/3 fractional quantum hall liquid and a solid.

We have measured the magnetoresistance of a very low density and extremely high quality two-dimensional hole system. With increasing magnetic field applied perpendicularly to the sample we observe the sequence of insulating, nu=1/3 fractional quantum Hall liquid, and insulating phases. In both of the insulating phases in the vicinity of the nu=1/3 filling the magnetoresistance has an unexpected oscillatory behavior with the magnetic field. These oscillations are not of the Shubnikov-de Haas type and cannot be explained by spin effects. They are most likely the consequence of the formation of a new electronic phase which is intermediate between the correlated Hall liquid and a disorder pinned solid.

Possible persistence of fractional quantum Hall effect down to ultralow fillings

A recent theoretical study indicating that the fractional quantum Hall liquid is the ground state at $\ensuremath{\nu}=1/9$ is inconsistent with an excitonic instability of the fractional quantum Hall liquid found earlier at the same filling factor. This paper shows that, when the calculation is improved perturbatively, by allowing mixing between composite fermion Landau levels, the instability disappears. In fact, no instability occurs in our theory for filling factors as low as $\ensuremath{\nu}=1/31,$ suggesting that the fractional quantum Hall effect may be robust down to much smaller filling factors than presently believed.

Edge reconstruction in the fractional quantum Hall regime

The interplay of electron-electron interaction and confining potential can lead to the reconstruction of fractional quantum Hall edges. We have performed exact diagonalization studies on microscopic models of fractional quantum Hall liquids, in finite size systems with disk geometry, and found numerical evidence of edge reconstruction under rather general conditions. In the present work we have taken into account effects like layer thickness and Landau level mixing, which are found to be of quantitative importance in edge physics. Due to edge reconstruction, additional nonchiral edge modes arise for both incompressible and compressible states. These additional modes couple to electromagnetic fields and thus can be detected in microwave conductivity measurements. They are also expected to affect the exponent of electron Green's function, which has been measured in tunneling experiments. We have studied in this work the electric dipole spectral function that is directly related to the microwave conductivity measurement. Our results are consistent with the enhanced microwave conductivity observed in experiments performed on samples with an array of antidots at low temperatures, and its suppression at higher temperatures. We also discuss the effects of the edge reconstruction on the single electron spectral function at the edge.

Quantum coherent equilibration in multipoint electron tunneling into a fractional quantum Hall edge

We have found a solution to a model of tunneling between a multichannel Fermi liquid reservoir and an edge of the principal fractional quantum Hall liquid (FQHL) in the strong-coupling limit. The model explicitly takes into account quantum coherence of electrons in different reservoir channels, the fact that makes it important to determine their mutual exchange statistics. Our solution shows that the statistics is in general hyperfermionic, with the phase branches of the statistical factor $\ensuremath{-}1$ taking different values $\ensuremath{\pi}\ifmmode\times\else\texttimes\fi{}\mathrm{odd}$ depending on the distance between the points of tunneling from the reservoir channels into the chiral edge. The choice of the statistics determines the saturated current between the edge and reservoir under nonvanishing bias voltage. If the tunneling points are well separated from one another, the outgoing edge is equilibrated with the reservoir as the number of reservoir channels is increased. The equilibration implies that the universal two-terminal conductance of the FQHL is fractionally quantized, in contrast to a one-dimensional Tomonaga-Luttinger liquid wire, where a similar tunneling model predicts unsuppressed free-electron conductance. On the fundamental level, this difference is associated with the absence of time-reversal symmetry at high energies in the FQHL case manifested in the chiral edge propagation.

Entanglement properties of some fractional quantum Hall liquids

We study the entanglement properties of some fractional quantum Hall liquids. We calculate the entanglement of the Laughlin wave function and the wave functions that are generated by the K matrix using the modified entanglement measure of indistinguishable fermions that is first proposed by Pa\ifmmode \check{s}\else \v{s}\fi{}kauskas and You [Phys. Rev. A64, 042310 (2001)].

Fermionizing a small gas of ultracold bosons

We study the physics of a rapidly-rotating gas of ultracold atomic bosons, with an internal degree of freedom. We show that in the limit of rapid rotation of the trap the problem exactly maps onto that of non-interacting fermions with spin in the lowest Landau level. The spectrum of the real bosonic system is identical to the one of the effective fermions, with the same eigenvalues and the same density of states. When the ratio of the number of atoms to the spin degeneracy is an integer number, the ground state for the effective fermions is an integer quantum Hall state. The corresponding bosonic state is a fractional quantum Hall liquid whose filling factor ranges in the sequence nu=1/2, 2/3, 3/4, ..., as the spin degeneracy increases. Anyons with 1/2,1/3,1/4, ... statistics can be created by inserting lasers with the appropriate polarizations. A special situation appears when the spin degeneracy equals the number of atoms in the gas. The ground state is then the product of a completely antisymmetric spin state and a nu=1 Laughlin state. In this case the system exhibits fermionic excitations with fermionic statistics though the real components are bosonic atoms.

Two-phonon scattering of magnetorotons in fractional quantum Hall liquids

We study the phonon-assisted process of dissociation of a magnetoroton, in a fractional quantum Hall liquid, into an unbound pair of quasiparticles. Whilst the dissociation is forbidden to first order in the electron-phonon interaction, it can occur as a two-phonon process. Depending on the value of final separation between the quasiparticles, the dissociation is either a single event involving absorption of one phonon and emission of another phonon of similar energy, or a two-phonon diffusion of a quasiexciton in momentum space. The dependence of the magnetoroton dissociation time on the filling factor of the incompressible liquid is found.

Klein factors in multiple fractional quantum Hall edge tunneling

A fractional quantum Hall liquid with multiple edges is considered. The computation of transport quantities such as current, noise, and noise cross correlations in such multiple edge samples requires the implementation of so-called Klein factors, which insure the correct quasiparticle exchange properties. The commutation relations of these factors are obtained by closing the system into a single edge. The nonequilibrium Green's-function formalism associated with such factors is derived for a simple Laughlin fraction of the Hall effect. It is shown explicitly how Klein factors enter the calculation of the noise cross correlations, as well as the correction to the Poisson limit for the noise.

Light scattering by magnetorotons of collective excitations in the fractional quantum Hall regime

Magnetorotons in the dispersion curves of collective gap excitation modes of fractional quantum Hall liquids are measured by resonant inelastic light scattering. Two deep magnetoroton minima are observed at ν= 2 5 , while a single deep minimum is resolved at ν= 1 3 . The results support Chern–Simons and composite fermion calculations that predict multiple roton minima for states with ν> 1 3 .

Two-terminal conductance of a fractional quantum Hall edge

We have found solution to a model of tunneling between a multi-channel Fermi liquid reservoir and an edge of the principal fractional quantum Hall liquid (FQHL) in the strong coupling limit. The solution explains how the absence of the time-reversal symmetry at high energies due to chiral edge propagation makes the universal two-terminal conductance of the FQHL fractionally quantized and different from that of a 1D Tomonaga-Luttinger liquid wire, where a similar model but preserving the time-reversal symmetry predicts unsuppressed free-electron conductance.

Magnetoroton scattering by phonons in fractional quantum Hall liquids

Motivated by recent phonon spectroscopy experiments in the fractional quantum Hall regime we consider processes in which thermally excited magnetoroton excitations are scattered by low energy phonons. We show that such scattering processes can never give rise to dissociation of magnetorotons into unbound charged quasiparticles as had been proposed previously. In addition we show that scattering of magnetorotons to longer wavelengths by phonon absorption is possible because of the shape of the magnetoroton dispersion curve and it is shown that there is a characteristic cross-over temperature above which the rate of energy transfer to the electron gas changes from an exponential (activated) to a power law dependence on the effective phonon temperature.

Partition Noise and Statistics in the Fractional Quantum Hall Effect

A microscopic theory of current partition in fractional quantum Hall liquids, described by chiral Luttinger liquids, is developed to compute the noise correlations, using the Keldysh technique. In this Hanbury-Brown and Twiss geometry, at Laughlin filling factors nu = 1/3, the real time noise correlator exhibits oscillations which persist over larger time scales than that of an uncorrelated Hall fluid. The zero frequency noise correlations are negative at filling factor 1/3 as for bare electrons (antibunching), but are strongly reduced in amplitude. These correlations become positive (bunching) for nu < or = 1/5, suggesting a tendency towards bosonic behavior.

Observation of multiple magnetorotons in the fractional quantum Hall effect.

Magnetorotons in the dispersions of collective gap excitation modes of fractional quantum Hall liquids are measured in resonant inelastic light scattering experiments. Two deep magnetoroton minima are observed at nu = 2/5, while a single deep minimum is resolved at nu = 1/3. The observations are the first evidence of multiple roton minima in gap excitations of the quantum liquids. The results support Chern-Simons and composite fermion calculations that predict multiple roton minima for states with nu>1/3.

Evolution of quasiparticle charge in the fractional quantum hall regime

The charge of quasiparticles in a fractional quantum Hall (FQH) liquid, tunneling through a partly reflecting constriction with transmission t, was determined via shot noise measurements. In the nu = 1/3 FQH state, a charge smoothly evolving from e(*) = e/3 for t(1/3) congruent with 1 to e(*) = e for t(1/3)<<1 was determined, agreeing with chiral Luttinger liquid theory. In the nu = 2/5 FQH state the quasiparticle charge evolves smoothly from e(*) = e/5 at t(2/5) congruent with 1 to a maximum charge less than e(*) = e/3 at t(2/5)<<1. Thus it appears that quasiparticles with an approximate charge e/5 pass a barrier they see as almost opaque.