Session T24: Fractional Quantum Hall Effect III

Evolution of 7/2 fractional quantum Hall state in two subband systems

We report the evolution of the fractional quantum Hall state (FQHS) at total Landau level (LL) filling factor $\nu=7/2$ in wide GaAs quantum wells in which electrons occupy two electric subbands. The data reveal subtle and distinct evolutions as a function of density, magnetic field tilt-angle, or symmetry of the charge distribution. At intermediate tilt angles, for example, we observe a strengthening of the $\nu=7/2$ FQHS. Moreover, in a well with asymmetric change distribution, there is a developing FQHS when the LL filling factor of the symmetric subband $\nu_S$ equals 5/2 while the antisymmetric subband has filling $1<\nu_A<2$.   

Observation of reentrant quantum Hall states in the lowest Landau level

Measurements in very low disorder two-dimensional electrons confined to relatively wide GaAs quantum well samples with tunable density reveal reentrant $\nu=1$ integer quantum Hall states in the lowest Landau level near filling factors $\nu=4/5$ and 6/5. These states are not seen at low densities and become more prominent with increasing density and in wider wells. Our data suggest that these reentrant phases are (bubble) Wigner crystal states, stabilized here in the lowest Landau level thanks to the large electron layer thickness.   

Spin waves in the second Landau level: Probing the spin-polarization enigma

The physics in the second Landau level (SLL) is governed by competing phases resulting in striking phenomena. We use resonant inelastic light scattering experiments to explore collective excitation modes with the focus on low lying spin excitation modes in the SLL. The intensity of the small momentum spin-wave at the bare Zeeman energy (E$_{Z})$ collapses for filling factors away from integer filling factor $\nu <$3 and are dominated by a continuum of modes. We find that at the fractional filling factors 14/5, 8/3, 5/2, 7/3, 11/5 the continuum coexists with a weak but distinct signal at E$_{Z}$, a long wavelength spin-wave that suggests a degree of spin polarization. In addition, at 5/2 an intriguing well developed sharp mode is observable below E$_{Z}$, which is unique for the even-denominator filling factor. Modes at energies larger than E$_{Z}$ merge additionally at odd-denominator states in ILS spectra, most pronounced for 7/3, in a manner that is similar to that of the 1/3 state. This observation could be evidence that the CF framework could be applicable to these states.   

Ballistic transport of (001) GaAs two-dimensional holes and hole-flux composite fermions through a lateral strain-induced superlattice

The addition of a unidirectional superlattice on the surface of high-mobility two-dimensional carrier systems leads to a potential modulation signaled by the observation of magnetoresistance commensurability oscillations. We present measurements in high-mobility, two-dimensional (001) GaAs systems patterned with strain-induced surface superlattices of periodicity 100 to 250 nm. The data exhibit pronounced commensurability oscillations of holes near zero magnetic field and of hole-flux composite fermions near filling factor v = 1/2, allowing us to probe the Fermi contours of these quasi-particles.   

Microscopic Disorder-Based model for non-Abelian Quasi-Particles in $\nu=5/2$ FQH states

The detection of non-Abelian quasiparticles remains an outstanding experimental problem in the $\nu=\frac{5}{2}$ fractional quantum Hall (FQH) state. The presence of non-Abelian statistics would lead to additonal low energy states in the system, and hence an additional low-temperature entropy. One approach to test for non-Abelian quasiparticle statistics uses thermodynamic measurements to detect this entropy contribution~[1,2]. We present a microscopic model for quasiparticles in the $\nu=\frac{5}{2}$ FQH state with a disorder potential that fluctuates on the order of several magnetic lengths, and attempt to determine the feasibility of the experiments proposed in~[1], based on local probe measurements of incompressibility~[3]. \\ \ [1] Cooper, N.R., Stern, A. PRL. {\bf 102}, 176807 (2009). \\ \ [2] Yang, K., Halperin, B.I. PRB {\bf 79}, 115317 (2009). \\ \ [3] Venkatachalam, V., Yacoby, A., Pfeiffer, L., West, K. Nature {\bf 469}, 185 (2011).   

Correlating scattering times with the strength of the $\nu $=5/2 fractional quantum Hall state

There is widespread interest in the fractional quantum Hall effect at $\nu $=5/2. Theory predicts that the state at $\nu $=5/2 may possess non-Abelian braiding statistics. Experimental interrogation remains difficult due to the fragility of the excitation gaps requiring both high quality samples and examination at low temperatures. Mounting evidence suggests that the strength of the most fragile fractional quantum Hall states in the 2$^{nd}$ Landau level including $\nu $=5/2 are poorly correlated with the scattering time extracted from zero-field mobility measurements at higher temperatures. It is also unclear if the quantum scattering time derived from analysis of the low-field Shubnikov de-Haas oscillations provides any additional information relevant to prediction of the strengths of the observed fractional states. We report on a systematic attempt to correlate the T=0.3K behavior of the mobility lifetime, quantum scattering time, and an effective high field mobility lifetime evaluated at $\nu $=5/2 with the measured activation gap. We will present results from a number of heterostructure designs over a wide span of zero-field mobility ranging from $\sim $10x10$^{6}$cm$^{2}$/Vs to greater than 20x10$^{6}$cm$^{2}$/Vs.   

``Perfect'' Coulomb Drag in a Bilayer Quantum Hall System

We report Coulomb drag measurements in Corbino geometry which reveal that equal but oppositely directed electrical currents can freely propagate across the insulating bulk of the bilayer quantized Hall state at $\nu_T =1$ even when the two 2D layers are electrically isolated and interlayer tunneling has been heavily suppressed by an in-plane magnetic field. This effect, which we dub ``perfect'' Coulomb drag, reflects the transport of charge neutral excitons across the bulk of the 2D system. The equal magnitude of the drive and drag currents is lost at high current and when either the temperature or effective separation between the two 2D layers is increased. In each of these cases, ordinary quasiparticle charge transport across the annulus has grown to dominate over exciton transport.   

Topological order, quasi-particle statistics and braiding from ground state entanglement

Topologically ordered phases are gapped states, defined by the properties of excitations when taken around each other. By calculating Topological Entanglement Entropy (TEE) of a disc shaped partition using a Monte Carlo technique, we establish the existence of topological order in $SU(2)$ symmetric gapped spin liquids and lattice Laughlin states obtained by the Gutzwiller projection technique. On the other hand, the TEE of partitioning the torus into two cylinders is generally different and depends on the chosen ground state. We demonstrate a method to extract the statistics and braiding of excitations, given just the set of ground state wave-functions on a torus. Central to our scheme is the identification of groundstates with minimum entanglement entropy, which reflect the quasi-particle excitations. We demonstrate our method by extracting the modular$\mathcal{S}$ matrix of an $SU(2)$ symmetric chiral spin liquid, and prove that its quasi-particles obey semionic statistics. This method offers a route to a nearly complete determination of the topological order in several cases.   

Paired Quantum Hall States at Weak Coupling: Phenomenology

Paired quantum Hall states such as the Pfaffian exhibit a weak-coupling regime much like that of BCS superconductivity. In this regime their lowest energy excitations are neutral fermions -- Bogoliubov quasiparticles constructed from the composite fermions -- and not the charged vortices which generally govern the behavior of quantum Hall states. We discuss a rich set of phenomena which follow from this observation. At finite temperatures of order the pairing scale these include (i) an almost sharp phase transition (ii) a new finite-temperature length scale for the penetration of longitudinal electric fields, and (iii) the existence of a new collective excitation in paired QH states which is a cousin to the well known Artemenko-Volkov-Carlson-Goldman-Schmid-Schon mode in conventional superconductors. At lower temperatures, we find (i) a proximity effect between the paired states and their ancestor metals, which in turn mediates (ii) `Josephson' couplings between paired QH droplets separated by metallic regions and leads to (iii) a distinctive response of such states to disorder; and finally, we also comment on (iv) an analog of Andreev reflection in these systems.   

From Luttinger liquid to non-Abelian quantum Hall states

We formulate a theory of non-Abelian fractional quantum Hall states by considering an array of coupled interacting one dimensional wires, each described by a Luttinger liquid theory. This coupled wire construction provides a solvable Hamiltonian formulated in terms of electronic degrees of freedom, and provides a direct route to characterizing the quasiparticles and edge states in terms of conformal field theory. It also leads to a simple interpretation of the coset construction of conformal field theory, which is a powerful method for describing non-Abelian states. The level-$k$ Read-Rezayi state at filling $\nu=k/(2+qk)$ is constructed by an uneven but periodic magnetic field configuration that organizes the wires into bundles. Gapless degrees of freedom in each bundle are decomposed into conformal sectors, which acuire energy gaps independently by intra-bundle and time reversal breaking inter-bundle interactions.   

A comparative study of the reentrant integer quantum Hall states in the second and third Landau levels

In the two-dimensional electron gas competing electron-electron interactions and disorder effects give rise to many-body ground states such as the fractional quantum Hall and the reentrant integer quantum Hall states (RIQHS). The latter are not yet well understood, but they are believed to be Wigner crystal-like electron solids with one or more electrons at each lattice nodes. We have recently shown that for the RIQHS in the second Landau level one can extract their onset temperature from the temperature-dependent magnetoresistance. We report similar studies of the RIQHS in the third Landau level. To our surprise, the onset temperatures of the RIQHSs in the third Landau level are about a factor of 3 larger than those in the second Landau level. This result clearly shows that the RIQHSs in the second and third Landau level have vastly different cohesion energies and may indicate different internal symmetries for these states. This work was supported by the DOE grant DE-SC0006671.   

Pomeranchuk Instability driven by Coulomb interaction in half-filled Landau levels

We study the Coulomb interaction driven Pomeranchuk instability as a mechanism for observed electronic nematic phases at high Landau levels. Such a mechanism will be signaled by the instability of the Fermi surface to quadripolar deformations: $F_2<-1$, where $F_2$ is the Fermi liquid parameter for the angular momentum $L=2$ channel. We compare the Fermi Liquid parameters for the lowest three half-filled Landau levels ($\nu=1/2, 5/2 and 9/2$). We calculate the Fermi liquid parameters by evaluating energies of eight independent particle-hole pair excitation configurations using a quantum Monte Carlo algorithm through correlated sampling. We used composite fermion many-body wave functions for $37$ electrons on a toroidal geometry that are interacting through the Coulomb potential. We find $F_2$ to become increasingly negative as we go to higher Landau levels. This is consistent with experimental observations.   

Bulk-edge correspondence and entanglement spectra of quantum Hall trial wave functions

We construct edge states wave functions for quantum Hall trial states built out of conformal blocks (e.g. Laughlin, Moore-Read or Read-Rezayi wave functions). We then compute the overlaps between these edge states in the thermodynamic limit. The result is that the Hilbert space of the edge theory is isomorphic to the one of the conformal field theory (CFT) which defines the quantum Hall state. This is a microscopic derivation of the bulk/edge correspondence for trial states given by conformal blocks. Our result definitely rules out the use of non-unitary theories for the construction of quantum Hall states. We obtain this result by analysing the quantum Hall droplet in the thermodynamic limit, under the assumption that all correlations are short-range inside the droplet. We argue that one is then left with a CFT in the domain outside the droplet, with a perturbed conformal boundary condition along the edge. We show that the entanglement spectra of these states can be tackled analytically with the same techniques.   

Finite-size studies of the $\nu=5/2$ quantum Hall state in wide quantum wells: the effect of subband mixing and breaking of particle-hole symmetry

A number of theoretical studies have argued that the quantized plateau at half filling of the second Landau level is described by the Pfaffian wavefunction of Moore and Read, or by its particle-hole conjugate, the anti-Pfaffian. The two wavefunctions are difficult to compare in finite-size systems due to their different shifts in the spherical geometry, or because of their high mutual overlap on the torus. Here we propose a way to circumvent these problems by envisioning systems with periodic boundary conditions, for which the Pfaffian and anti-Pfaffian become orthogonal to each other due to their different symmetry properties under discrete rotations. Furthermore, we show that periodic boundary conditions can be used to study the Moore-Read ground state, as well as the collective excitation spectrum, in finite systems in a ``quartered'' Brillouin zone scheme. To demonstrate the utility of this method, we provide a realistic, two-component model of a wide quantum well that can unambiguously distinguish between the Pfaffian and anti-Pfaffian state in finite-sized systems. These results describe the recent experiments that probed the stability of the $\nu=5/2$ state by tuning the mixing between electronic subbands and Landau levels in a wide quantum well.   

Quantitative analysis of the disorder broadening and the intrinsic gap for the $\nu $=5/2 fractional quantum Hall state

We analyze several different methods of extracting intrinsic gaps of fractional quantum Hall states (FQHS) of the second Landau level from experimental data. Because of the discrepancy between these methods, we introduce a new way of estimating the disorder broadening in the second Landau level based on scaling of the gaps of the major odd denominator states. The results of our technique are in good agreement with a previously used method utilizing only the gaps of the even denominator states. We successfully apply this technique to several samples of high quality and find an excellent agreement between the estimated intrinsic gap and results of numerical simulations. We also report, for the first time, the dependence of the intrinsic gap of $\nu $=5/2 FQHS on Landau level mixing. This work was supported by the NSF grant DMR- 0907172.   

Quantum Topology of Lattice Dislocations in Fractional Chern Insulators

An exciting prospect in condensed matter physics is the possibility of realizing fractional quantum Hall (FQH) states in simple lattice models without a large external magnetic field. Here we find a remarkable consequence of the interplay between the lattice translation symmetry and topological properties of these fractional Chern insulators. When the partially filled flat band has a Chern number N, it can be mapped to an N-layer quantum Hall system. We find that lattice dislocations can act as wormholes connecting the different layers and effectively change the topology of the space. Lattice dislocations become defects with non-trivial quantum dimension, topological degeneracy, and non-Abelian statistics, even when the FQH state being realized is a conventional Abelian FQH state.