Session Q11: Fractional Quantum Hall Effect II

Unexpected Roles for Spin Degrees of Freedom in Competing Phases of the Second Landau Level

Competing liquid and solid ground states as well as intriguing quantum Hall fluids such as that at $\nu $=5/2 create great current interest in the N=1 Landau level. The spin degrees of freedom in quantum phases of the 2nd Landau level is probed by resonant light scattering. The long wavelength spin wave mode, which monitors the degree of spin polarization, is at the Zeeman energy in the spin polarized state at $\nu $=3. At lower filling factors the intensity of the Zeeman mode collapses indicating loss of spin polarization. At filling factors slightly lower the intensity of the spin wave attenuates and a broad continuum of low-lying excitations emerges - sharp and broad modes coexist. While the coexistence of spectral features has not been explained, the observation could manifest the presence of mixed quantum phases and some loss of spin polarization. A continuum of low-lying excitations emerges that dominates near $\nu $=8/3 and $\nu $=5/2. Resonant Rayleigh scattering reveals that the quantum fluids away from $\nu $=3 break up into robust domains. It is conceivable that these domains could comprise both spin polarized and depolarized quantum fluids. While the state at $\nu $=5/2 is considered to be polarized, these results reveal unprecedented roles for spin.   

Electron teleportation via Majorana Bound States in a Mesoscopic Superconductor

Majorana fermions are non-Abelian anyons in 5/2 fractional quantum Hall states and superconductors, which can store quantum information in an inherently nonlocal way. We describe a phase-coherent electron transport phenomena through two spatially separated Majorana bound states in a mesoscopic superconductor. This striking nonlocal effect arises from the interplay between topological order, superconducting order parameter and mesoscopic effects. We discuss its implications for experimental detection of Majorana fermions and topological quantum computation. Ref: Liang Fu, Phys. Rev. Lett. 104, 056402   

Entanglement Entropy as a Function of the Aspect Ratio in the First and Second Landau Level

Entanglement entropy as a function of aspect ratio has been studied by direct diagonalization in the first and second Landau levels. The torus geometry is used and spin polarized electrons interact via long range Coulomb interaction . As previously noted by Haque et al. (N J Phys 12, 2010 075004), in the first Landau level there is very smooth behavior as a function of aspect ratio making it possible to obtain the topological entanglement entropy. In the second Landau level, the entanglement entropy is much less regular, with possible signatures of quantum phase transitions.   

The phase diagram and particle-hole asymmetry of the reentrant integer quantum Hall states of the second Landau level

The second Landau level of a two-dimensional electron gas reveals a rich set of competing ground states. Besides an increasing number of fractional quantum Hall states, there are also eight reentrant integer quantum Hall states observed. These reentrant integer states are currently not understood, although they are believed to be collective insulators akin to the field induced Wigner solid with one or more electrons per site. These states are strongly affected by tilt in magnetic field and carrier density but surprisingly there is very limited data on their temperature dependence. We present a detailed study of the melting of the reentrant integer quantum Hall states of the second Landau level from which we extract the phase diagram in the temperature versus filling factor plane. We find that the melting temperatures of the various reentrant integer states violate the particle-hole symmetry. We also report that as the temperature is lowered the magnetoresistance deviates from an activated dependence.   

The even denominator fractional quantum Hall states at large Landau level mixing

We present a study of the energy gaps of the even denominator fractional quantum Hall states of the second Landau level in a two-dimensional electron gas with a record low density of n = 8.2x10$^{10}$ cm$^{-2}$. These measurements are motivated by the expectation that Landau level mixing present in samples of low densities breaks the degeneracy of the Pfaffian and its particle-hole conjugate anti-Pfaffian. Cooling the electron gas in our Helium-3 immersion cell to 5mK reveals at filling factor 5/2 a fully quantized Hall plateau and a vanishingly small magnetoresistance. Because of the low density of our sample, the 5/2 fractional state is observed at the highest degree of Landau level mixing reported to date. We have measured the energy gaps of the 5/2 and 7/2 fractional quantum Hall states. The intrinsic gap deduced in the limit of no disorder will be compared to previously reported values for samples with higher densities.   

Nonconventional odd denominator fractional quantum Hall states in the second Landau level

The odd denominator fractional quantum Hall states in the second Landau level of a two-dimensional electron gas are believed to be different from those of the lowest Landau level. While at first sight these states could be part of the composite fermion hierarchy, several recent theoretical works suggest that some might be supporting generalized Pfaffian-like correlations. Recent progress in cooling electrons allowed us to observe a new fractional quantum Hall state at the filling factor 2+6/13. By assuming that the effective mass of the composite fermions does not explicitly depend on the Landau level index we find that energy gaps of the prominent 2+1/3 and 2+2/3 states are consistent with the values predicted by the free composite fermion model. However, the weaker 2+2/5 and 2+6/13 states deviate significantly from the prediction of this model. This deviation constitutes a first demonstration of the nonconventional nature of the latter two odd denominator fractional quantum Hall states.   

Tunneling experiments in the lowest Landau level

Recently, a quasiparticle-tunneling experiment on the 5/2 state [1] led to the unintentional discovery of a process we term annealing. In this experiment top gates are used to bring counter-propagating edge states close enough together for tunneling to occur. By keeping the quantum point contact (QPC) top gates energized for a few days at 4 Kelvin, one can create equal electron densities in the QPC region and the bulk of a GaAs heterostructure. This is a great advantage for studying quasiparticle tunneling in QPCs. Conditions under which annealing has proved effective are presented. In addition, in order to better understand and control quasiparticle tunneling in QPCs, further tunneling experiments have been performed in the lowest Landau level. \\[4pt] [1] Iuliana P. Radu, J. B. Miller, C. M. Marcus, M. A. Kastner, L. N. Pfeiffer, and K. W. West, Science 320, 899 (2008).   

Haldane Statistics in the Finite Size Entanglement Spectra of Laughlin States

We conjecture that the counting of the levels in the orbital entanglement spectra (OES) of finite-sized Laughlin Fractional Quantum Hall (FQH) droplets at filling 1/m is described by the Haldane statistics of particles in a box of finite size. This principle explains the observed deviations of the OES counting from the edge-mode conformal field theory counting and directly provides us with a topological number of the FQH states inaccessible in the thermodynamic limit- the boson compactification radius. It also suggests that the entanglement gap in the Coulomb spectrum in the conformal limit protects a universal quantity- the statistics of the state.   

Polarized Fractional Quantum Hall States at 1/3 and 5/2 Filling: a Density-Matrix Renormalization Group Calculation

In this talk, the density-matrix renormalization group method is employed to investigate the fractional quantum Hall effect (FQHE) at filling numbers $\nu=1/3$ and $5/2$. We present benchmark results for both filling numbers for larger system sizes to show the accuracy as well as the capacity of our numerical algorithm. Furthermore, we demonstrate that by keeping a large number of states, one can also obtain reliable entanglement spectrum at $\nu=5/2$, which characterizes the topological properties of FQHE states. Based on a finite-size scaling analysis, we also confirm that the entanglement gap defined by Li and Haldane for $\nu=5/2$ state with Coulomb interaction remains finite in the thermodynamic limit.   

Paired composite fermion wavefunctions for excitations at 5/2

The Pfaffian wave function, which is thought to be relevant for the ground state at filling fraction $\frac{5}{2}$, represents a paired state of composite fermions. It can be expressed as an antisymmetrized bilayer (331) wave function. This formulation can be extended to construct wave functions for neutral as well as charged excitations of the Pfaffian. The space spanned by the quasihole excitations exactly matches that of the previously known quasihole wave functions. By comparison to exact results with up to 14 particles, we find that our neutral excitations and also the quasiparticle excitations describe well the actual excitations of the model three body interaction for which the Pfaffian ground state wave function is exact. The relevance to the solutions of the second Landau level Coulomb interaction is less conclusive. Also, the counting of states on the quasihole and quasiparticle sides is significantly different. Relation of our wave functions to other ansatz wave functions in the literature will be discussed.   

Suppression of Interlayer Phase Coherence by Gauge Fluctuations in Bilayer Composite Fermi Liquids

The $\nu$ =1/2+1/2 bilayer quantum Hall system exhibits at least two phases as a function of layer spacing, $d$. For $d/l \gg 1$, ($l$ is magnetic length), the system decouples into two $\nu = 1/2$ composite fermion (CF) liquids. For $d/l$ sufficiently small, the system enters an incompressible bilayer quantum Hall state. Recently, Alicea et al. [1] have proposed a state which might exist for intermediate layer spacing ($d \sim l$). In this ``interlayer phase coherent" state, CFs tunnel coherently between layers forming well-defined bonding and antibonding Fermi seas, though there is no actual tunneling of physical electrons. Here we show that scattering from gauge fields in the CF liquids leads to strong layer-dependent fluctuations in the Aharonov-Bohm phases seen by CFs which suppress interlayer phase coherence. This suppression appears as a singular contribution to the correlation energy which inhibits any T=0 phase transition into an interlayer phase coherent state, and drives any such transition first order. Work supported by US DOE.\\[4pt] [1] J. Alicea, O.I. Motrunich, G. Refael, M.P.A. Fisher, PRL 103, 256403 (2009).   

Pinning mode of 2D electron system with short-range alloy disorder

At the low Landau filling ($\nu )$ termination of the fractional quantum Hall effect (FQHE) series, a two-dimensional electron system (2DES) becomes an insulator, which is identified in sufficiently low-disorder samples as a form of pinned Wigner solid. The microwave conductivity spectrum of such a solid shows a striking resonance, which is understood as a pinning mode, in which pieces of solid oscillate within the disorder potential. We report on the observation of the pinning mode of a 2DES that resides within Al$_{x}$Ga$_{1-x}$As with x=0.85{\%}. For a carrier density of n= 8.7 x10$^{10}$ /cm$^{2}$, a resonance with a peak frequency (f$_{pk})$ of about 5 GHz appears as $\nu $ goes below the 2/3 FQHE. A local minimum in resonance amplitude vs. $\nu $ occurs around $\nu $ =1/2. We will discuss the contribution of the alloy disorder to f$_{pk}$.   

Properties of the Composite Fermion Wigner Crystal

In two dimensional electron systems at small filling factor the ground state is a Wigner crystal. Wigner crystals can also be observed for systems near integer fillings, where electrons or holes in the partially filled Landau Level form a Wigner crystal. Recent experimental evidence (PRL 105, 126803 (2010)) suggests that a Wigner crystal of composite fermions forms near the filling factor of $v=\frac{1}{3}$. Motivated by these results, we calculate the shear modulus of the composite fermion Wigner crystal in the vicinity of several fillings of the form $v=\frac{1}{3},\frac{2}{5},\frac{3}{7}$, following the procedure of Maki-Zotos, using the effective two-body real space interactions between composite fermions calculated by Lee, Scarola, and Jain. We discuss the differences from the electron Wigner crystal, and also the experimental implications of our results.   

Fabry-Perot Interferometry in the Integer and Fractional Quantum Hall Regimes

We present measurements of electronic Fabry-Perot interferometers in the integer and fractional quantum Hall regimes. Two classes of resistance oscillations may be seen as a function of magnetic field and gate voltage, as we have previously reported. In small interferometers in the integer regime, oscillations of the type associated with Coulomb interaction are ubiquitous, while those consistent with single-particle Aharonov-Bohm interference are seen to co-exist in some configurations. The amplitude scaling of both types with temperature and device size is consistent with a theoretical model. Oscillations are further observed in the fractional quantum Hall regime. Here the dependence of the period on the filling factors in the constrictions and bulk of the interferometer can shed light on the effective charge of the interfering quasiparticles, but care is needed to distinguish these oscillations from those associated with integer quantum Hall states.   

Topological screening and interference of fractionally charged quasi-particles

Interference of fractionally charged quasi-particles is expected to lead to Aharonov-Bohm oscillations with periods larger than the flux quantum $\Phi_0$. However, according to the Byers-Yang theorem, observables of an electronic system are invariant under insertion of a quantum of singular flux. We resolve this paradox by considering a {\em microscopic} model of an electronic interferometer made from quantum Hall edges at filling factor $\nu=1/m$. An approximate ground state of such an interferometer is described by a Laughlin type wave function, and low-energy excitations are incompressible deformations of this state. We construct a low-energy effective theory by projecting the state space onto the space of such deformations. Amplitudes of quasi-particle tunneling in this theory are found to be insensitive to the singular flux. This behavior is a consequence of {\em topological screening} of the flux by the quantum Hall liquid. We describe strong coupling of the edges to Ohmic contacts and the resulting quasi-particle current through the interferometer with the help of a master equation. As a function of the singular magnetic flux, the current oscillates with the period $\Phi_0$. These oscillations are suppressed with increasing system size. When the magnetic flux is varied with a modulation gate, current oscillations have the quasi-particle period $m\Phi_0$ and survive in the thermodynamic limit.