Session P24: Fractional Quantum Hall Effect I

Exploring the possibility of Universal Edge Physics in the Fractional Quantum Hall States

The edge of a fractional quantum Hall (FQH) droplet is described by the chiral Luttinger liquid theory which predicts a universal power-law behavior in the current-voltage ($I$-$V$) characteristics when electrons tunneling into the FQH edge through a barrier, e.g., from a three-dimensional Fermi liquid. However, this university has not been unambiguously observed in transport experiments in two-dimensional electron gases based on GaAs/GaAlAs heterostructures or quantum wells. One plausible cause is reconstruction of the fractional quantum Hall edge, which introduces non-chiral edge modes. The coupling between counterpropagating edge modes can modify the exponent of the $I$-$V$ characteristics. By comparing the fractional quantum Hall states at the filling factor $\nu=1/3$ in modulation-doped semiconductor devices and in graphene devices, we show that the GaAs-based FQH experiments are always in the edge reconstruction regime, whereas graphene-based systems have an experimental accessible parameter region where edge reconstruction can be avoided. This regime offers the possibility of the exploration of the universal edge tunneling exponent predicted by the chiral Luttinger liquid theory.   

Exploration of the Limits to Mobility in Two-Dimensional Hole Systems in C-Doped (001) GaAs/AlGaAs Quantum Wells

We report on the growth of a series of high mobility two-dimensional hole systems (2DHSs) in 20 nm (001) oriented GaAs/AlGaAs quantum wells and the analysis of possible scattering mechanisms. The hole density was controlled by changing the delta-doping setback and Al mole fraction and was measured at low temperature (T = 300 mK) after illumination with a red LED. We varied the density over a range from 2.0 x 10$^{10}$ cm$^{-2}$ to 1.9 x 10$^{11}$ cm$^{-2}$, and the mobility was observed to peak at an intermediate density of 6.5 x 10$^{10}$ cm$^{-2}$ where we report a new record T = 300 mK mobility of 2.3 x 10$^{6}$ cm$^{2}$/Vs . We find that even when the density dependent effective mass is taken into account, remote and background impurity scattering cannot qualitatively explain the behavior of the mobility, in contrast with comparable 2DEGs. We discuss possible mechanisms leading to the observed non-monotonic density dependence of the mobility and the factors leading to our new record mobility.   

Effective Field Theory of Fractional Quantized Hall Nematics

We present a Landau-Ginzburg theory for a fractional quantized Hall nematic state and the transition to it from an isotropic fractional quantum Hall state. This justifies Lifshitz-Chern-Simons theory -- which is shown to be its dual -- on a more microscopic basis and enables us to compute a ground state wave function in the symmetry-broken phase. In such a state of matter, the Hall resistance remains quantized while the longitudinal DC resistivity due to thermally-excited quasiparticles is anisotropic. We interpret recent experiments by Xia et al. (cond-mat/1109.3219) at Landau level filling factor $\nu =7/3$ in terms of our theory.   

Dipolar Bogolons: From Superfluids to Pfaffians

We study neutral fermionic `Bogolons' which are quasiparticle excitations of gapped phases that arise due to fermion (BCS) pairing, such as superfluids, superconductors, and paired quantum Hall states. As we demonstrate, a na\"{i}ve construction of a quasiparticle wavepacket by solving the mean-field BCS equations leads to a contradiction: there is a net electrical current {\it even when the group velocity vanishes}. Resolution of this paradox requires the computation of supercurrents in the wavepacket state, typically a complicated exercise in self-consistency. In this Letter we demonstrate that these corrections may be approximately calculated from correlations in the mean-field ground state, and lead to a divergence-free, dipolar current pattern associated with the quasiparticle. When Maxwell electrodynamics is included, as appropriate to a superconductor, this pattern is confined over a penetration depth. For paired quantum Hall states of composite fermions, the Maxwell term is replaced by a Chern-Simons term, which leads to a dipolar {\it charge} distribution, paralleling Read's observation that composite fermions are neutral dipoles.   

Coulomb Oscillations in Antidots in the Integer and Fractional Quantum Hall Regimes

We present measurements of Coulomb oscillations as a function of both top gate and magnetic field in gate-defined, micron-scale antidots in the integer and fractional quantum Hall regimes. We find resistance oscillations at filling factors $\nu=2,\nu=1,\nu=2/3,$ and $\nu=1/3$. At $\nu=1$, we find the tunneling charge to be $e$ and the presence of one edge. At $\nu=2$, we also find the tunneling charge to be $e$ and the presence of two edges. A generalized picture of Coulomb oscillations in the fractional quantum Hall regime suggests the presence of one charged edge at both $\nu=1/3$ and $\nu=2/3$. We find the tunneling charge at $\nu=1/3$ to be $e/3$ but unexpectedly find the tunneling charge at $\nu=2/3$ to be $(2/3)e$.   

Lande's g values and the quantum Hall effect

It is reported that the Lande's formula for the g values of the electron as a function of L, S and J does not have the particle-hole symmetry needed for the understanding of high magnetic field data of quantum Hall effect. Hence it is modified to yield the particle-hole symmetry by means of the two signs before S in J= L$\pm $S. The correct g value is then given by g=(2J+1)/(2L+1). Since the Bohr magneton involves the charge of the particles, the corrected g value formula explains the data of quantum Hall effect. The value L/2L+1 gives 1/3 and ( L+1)/(2L+1) gives 2/3. The consideration of Landau levels gives many values in agreement with the data. In electron clusters the spin need not be $\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} $. Hence a lot of data is explained by means of spin greater than $\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} $. Some of the clusters show the formation of spin waves so that there is a finite spin deviation which is characteristic of electron lattices. It is found that Laughlin's wave function is the ground state of $\nabla ^2\delta (r_i -r_j )$ type Hamiltonian which is not equivalent to Coulomb's Hamiltonian. K. N. Shrivastava, Intl. J. Mod. Phys. B 25, 1301-1357(2011).   

Numerical Study of Realistic Models of the $\nu=5/2$, 7/3, 8/3 Hamiltonians: Effects of Landau-level mixing and Finite Well-width

We construct a realistic effective Hamiltonian for electrons in the first excited Landau level, taking into account the effects of both Landau-level mixing and the finite width of the GaAs quantum well. The latter includes both short-distance softening of the Coulomb interaction as well as sub-band mixing. Through exact diagonalization, we find a rich phase diagram as a function of the Landau level mixing parameter $\kappa$ and quantum well width $d$. In particular, small changes in either parameter can drive phase transitions between states in the universality classes of the Moore-Read Pfaffian, anti-Pfaffian, and exotic compressible metallic states.   

Thermopower and the Fractional Quantized Hall Effect in the N=1 Landau Level

Having recently eliminated an issue involving long thermal time constants [1], we are now able to resolve diffusion thermopower deep into the fractional quantized Hall effect (FQHE) regime. In this talk we report measurements of thermopower in the first excited (N=1) Landau level as a continuous function of magnetic field down to temperatures as low as 30mK. Above 50mK we can clearly resolve the $\nu$ = 5/2 as well as $\nu$ = 7/3, 8/3, and 14/5 FQHEs in both the electrical and thermoelectrical transport. Below 50mK a prominent feature of the electrical transport in the first excited Landau level is the Re-entrant Integer Quantized Hall Effect (RIQHE) which is associated with insulating collective phases [2]. In this temperature regime the thermopower exhibits a series of intriguing sign reversals that are as yet not fully understood. We will conclude with a brief discussion of the connection between thermopower and the entropy of the 2D electron system. This connection is invoked by a recent prediction [3] of the thermopower at $\nu$ = 5/2, which assumes the ground state is the non-Abelian Moore-Read paired composite fermion state.\\[4pt] [1] Chickering, Phys. Rev. B 81, 245319 (2010)\\[0pt] [2] Eisenstein, Phys. Rev. Lett. 88, 076801 (2002)\\[0pt] [3] Yang, Phys. Rev. B 79, 115317 (2009)   

Tilt Magnetic Field Dependence of the 12/5 Fractional Quantum Hall State

The 12/5 state has attracted growing interest due to its superior potential in performing universal topological quantum computation. Up to date, except for the observation of a well developed quantum Hall plateau at this filling, much less experimental work has been carried out and there is no experimental evidence to support this state being a paraferminoic or non-Abelian state. Here, we present our tilt magnetic field dependence results in examining its spin-polarization. It was observed that the diagonal resistance R$_{xx}$ at $\nu $=12/5 shows a non-monotonic dependence on tilt angle ($\theta )$. It first increases sharply with increasing $\theta $, reaches a maximal value of $\sim $ 60 $\Omega $ around $\theta \sim $14$^{o}$, and then decreases with $\theta $ further increased. Correlated with this R$_{xx}$ dependence, the 12/5 activation energy gap ($\Delta _{12/5})$ also shows a non-monotonic $\theta $ dependence. $\Delta _{12/5 }$first decreases. Around 14$^{o}$, R$_{xx}$ becomes non-activated and a true activation energy gap is not obtainable. With further increasing $\theta $, R$_{xx}$ becomes activated again and $\Delta _{12/5 }$increases with $\theta $. This tilt dependence in R$_{xx}$ and $\Delta _{12/5 }$is similar to the composite fermion states at $\nu $=2/5 and 8/5 in the lowest Landau level, which was interpreted as a spin transition. Our results thus call for more investigations on the nature of the 12/5 ground state.   

The temperature-dependences of the dissipative conductance in quantum Hall states

We discuss how to estimate the saddle point gap of quantized Hall states from the temperature dependence of the longitudinal response, $\log \sigma_{xx}$. The dissipative response is assumed to be the result of thermally activated quantum tunneling through saddle points in the long-range impurity potential set up by the ionized donors (see Phys. Rev. Lett. 106, 126804 (2011)). We apply the method to published data on states at $\nu=5/2$, as well data at other integer and fractional filling fractions, and compare the gap estimates and the typical parameters of the saddle points with expectation from microscopic calculations. Even in the case of very weak quantum Hall states, we find that the analysis suggests saddle point gaps which are consistently around 50\% of the gap predicted for the homogeneous system.   

Fractionalization in spontaneous integer quantum Hall systems

We show that electron fractionalization can occur in quantum Hall liquids even in the absence of strong correlations. Focusing on a Kondo lattice model that exhibits spontaneous integer Hall effect due to non-coplanar magnetic ordering, we find that $Z_2$ vortices in the magnetic order parameter can bind fractional quantum numbers. The vortices have anyonic exchange statistics.   

Exact soliton solutions in a many-body system in non-trivial background

Soliton solutions are usually described as lumps of density propagating without changing their shape. They usually occur in integrable systems. We consider the classical Calogero model in an external harmonic potential. Due to the presence of the external potential, momentum in the system is not conserved and the aforementioned description of solitons is inapplicable, nevertheless the system remains integrable. Naturally, the question about the existence and the form of soliton solutions arises. I will explain what a soliton solution of this model is and I will show how to find these solutions in the case of finite number of particles and in the hydrodynamic limit. In the latter limit the model is described by hydrodynamic equations on continuous density and velocity fields. Soliton solutions in this case are finite dimensional reductions of the hydrodynamic model and describe the propagation of lumps of density and velocity in the nontrivial background. These solutions of Calogero model were previously qualitatively described by A. Polychronakos in the context of the quantum Hall effect.   

On the hydrodynamic description of FQHE fluid

A simple classical two-dimensional hydrodynamic model for FQHE fluid has been constructed. Hydrodynamics of this model is Hamiltonian, nonlinear and has many features of fractional quantum Hall states. The effective classical fluid is incompressible and has a correct static structure factor. The model incorporates FQHE relation between the density and the vorticity of the fluid, gives the correct value for the Hall viscosity and for the Hall conductivity at finite wavevectors. The model can serve as a starting point in deriving the boundary theory of FQHE states beyond chiral Luttinger liquid. Indeed, in the leading approximation the model is reduced to chiral waves with nonlinearities defined by a curvature of the confining potential.   

Anomalous resistance across a quantum point contact in the integer and fraction quantum Hall regimes

A $\sim $600 nm quantum point contact (QPC) is studied in various integer and fractional quantum Hall states. An anomalous increase of the differential resistance across the QPC from the expected quantized value is observed at nonzero DC bias. The onset of this increased differential resistance occurs sharply at DC bias values which vary continuously with magnetic field and gate voltage. Dependence of the increased resistance on filling factor, magnetic field, gate voltage, and temperature is presented. Of particular interest is the observation that the onset DC bias shows opposite gate voltage dependence for integer and fractional quantum Hall states.   

Improved Measurements of Quasi-Particle Tunneling in the nu = 5/2 Fractional Quantum Hall State

It is predicted that the nu = 5/2 fractional quantum Hall state may potentially exhibit novel non-abelian quasi-particle statistics, which would make it a candidate for implementation of topological quantum computation. We present measurements of quasi-particle tunneling between edge channels, which provide information about the wave function of the nu = 5/2 state. Weak tunneling is investigated as a function of temperature and DC bias and fit to the theoretical tunneling conductance. We improve on previous quasi-particle tunneling measurements by reducing measurement noise and studying two different quantum point contact (QPC) geometries. For both QPCs the best fits give e*, the quasi-particle effective charge, close to the expected value of e/4 and g , the strength of the interaction between quasi-particles, close to 3/8. Here we show that fits corresponding to the various proposed wave functions, along with qualitative features of the data, strongly favor the abelian 331 state.