Session Q24: Fractional Quantum Hall Effect II

Fractional Quantum Hall Effect of Rydberg-Polaritons

Dark-state-polaritons (DSP) are bosonic quasiparticles arising in the interaction of light with 3-level atoms under conditions of electromagnetically induced transparency (EIT). They can be exposed to artificial magnetic fields, strong enough to enter the lowest Landau level regime [Otterbach et. al., Phys. Rev. Lett. 104 (2010)]. We take into account interactions between the DSPs via Rydberg dipole-dipole interactions and discuss the realization of the $\nu=1/2$-Laughlin state and its anyonic excitations (quasiholes) in such systems. The DSPs can be prepared in the correct total angular-momentum subspace by using orbital angular momentum light beams. A numerical and semi-analytical evaluation of the quasihole-gap is presented.   

Compressible and incompressible phases in lattice fractional quantum Hall systems

We study lattice fractional quantum Hall (FQH) systems in the presence of local potential traps using the exact diagonalization technique. By implementing an array of local potential traps, we show that the system undergoes a series of phase transitions. As the strength of potential traps is increased, the FQH state turns into a compressible metallic state, and then into a topologically trivial insulator. We present the phase diagram as well as convincing numerical evidences which we use to identify these phases and phase transitions, including the energy spectrum, the fidelity metric, the Chern number, and the entanglement spectrum. In addition, we also compare the topological trivial insulator observed in our systems with Anderson insulators, which are expected in ordinary fractional quantum Hall systems in 2D electron gases in the presence of strong impurities.   

Tailoring Fabry-Perot Interferometers for Fragile Fractional Quantum Hall States

Depending on the relevance of Coulomb interactions, electronic Fabry-Perot interferometers can exhibit two qualitatively different types of interference, each of which can shed light on the unique physics of quantum Hall systems. Long observed in the integer quantum Hall (IQH) regime, the so-called ``Coulomb-dominated'' interference has only recently been confirmed in the fractional quantum Hall (FQH) regime, where its observation has remained limited to the simplest and most robust FQH states. Building on our recent observation and analysis of this type of interference at several fractional filling factors, we report on interferometer design improvements yielding greater visibility, most notably for weaker FQH states. We find that parameters such as the distance from the gates defining the interferometer to the 2DEG, gate layout, and wafer structure affect the visibility much more in the FQH regime than in the IQH regime. High sensitivity to such parameters is also a characteristic of the second type of interference, believed to arise from a pure Aharonov-Bohm effect, which has been clearly observed only in the IQH regime; we discuss efforts to observe this behavior in the FQH regime.   

Multipartite wavefunctions for the second Landau level FQHE

We study the multipartite wave functions of composite fermions, in which the composite fermions within each partition are correlated differently than those across partitions. These include the Pfaffian wave function at 5/2 and the Rezayi-Read wave function at 13/5. Neutral and charged excitations of this state are modeled as neutral and charged excitations created in the individual partitions. We investigate how accurate these wave functions are for certain model three and four body interactions, and whether they are adiabatically connected to the Coulomb solutions. In particular, 5/2 state for an odd number of particles, which contains at least one unpaired composite fermion, will be considered. We also test how multiple degeneracy arises in this model for quasiparticles and quasiholes.   

Excitons in FQH states

Fractional quantum Hall (FQH) states are the first experimentally realized systems that exhibit topological order. For a full understanding of these systems it is crucial to be able to describe not only their ground states and quasihole excitations, but in fact the full low-energy sector. The lowest energy excitations are believed to be neutral quasihole-quasielectron pairs. We present a description of these excitations for a wide range of FQH states. Our method allows us to variationally change the model states without changing any of their topological properties.   

Geometrical Description of fractional quantum Hall quasiparticles

We examine a description of fractional quantum Hall quasiparticles and quasiholes suggested by a recent geometrical approach (F. D. M. Haldane, Phys. Rev. Lett. 108, 116801 (2011)) to FQH systems, where the local excess electric charge density in the incompressible state is given by a topologically-quantized ``guiding-center spin'' times the Gaussian curvature of a ``guiding-center metric tensor'' that characterizes the local shape of the correlation hole around electrons in the fluid. We use a phenomenological energy function with two ingredients: the shear distortion energy of area-preserving distortions of the fluid, and a local (short-range) approximation to the Coulomb energy of the fluctuation of charge density associated with the Gaussian curvature. Quasiparticles and quasiholes of the 1/3 Laughlin state are modeled as ``punctures'' in the incompressible fluid which then relax by geometric distortion which generates Gaussian curvature, giving rise to the charge-density profile around the topological excitation.   

Model wavefunctions for fractional quantum Hall collective modes

We examine the collective modes of primary fractional quantum Hall states which can be represented by Jack polynomial wavefunctions, in particular the $\nu$ = 1/3 Laughlin and the $\nu$ = 1/2 (or 5/2) Moore-Read states. Using the extension of Jack Polynomial states (B. A. Bernevig and F. D. M. Haldane, Phys. Rev. Lett. 102, 066802 (2009)) to describe states with excited quasiparticles as well as quasiholes, we model the collective mode as a dipole formed by the combination of a single elementary quasiparticle with a single quasihole. In the Laughlin case, this neutral collective excitation is bosonic, while in the Moore-Read case, it has two forms, one bosonic and one fermionic. For small electric dipole moment (also small momentum and wavenumber) the (variational) energy of this mode lies above the threshold of the continuum of roton-pair (Laughlin) or neutral-fermion-pair (Moore-Read) excitations. In the long-wavelength limit the bosonic mode is a ``spin-2'' excitation that has an analogy to the ``graviton'' suggested by a recent geometric approach (F. D. M. Haldane, Phys. Rev. Lett. 108, 116801 (2011)) to FQH systems, while the neutral fermionic mode (present if the ``odd-denominator rule'' is violated) has ``spin-3/2'', and has a possible analogy to the ``gravitino''.   

Determination of counter-propagating edge modes in the $\nu$ = 5/2 fractional quantum Hall state

Determining the wavefunction that describes the fractional quantum Hall state at $\nu$ = 5/2 remains an unresolved question. Two main candidates are the Pfaffian and anti-Pfaffian states. A major difference between the two is the chirality of their neutral Majorana fermion modes, which in the former run parallel to the charged modes and in the latter, anti-parallel. We consider the recent experiment [Bid, A., et al. Nature, 466, 585-590 (2010)], in which counter-propagating, neutral edge modes in the $\nu$ = 5/2 state were detected as a change in shot noise at an inter-edge quantum point contact (QPC) when current was injected at a point downstream of the QPC. We present a theoretical description of this experiment. We model the injection by coupling one edge to an external field and determine that the change in noise is incompatible with a parallel-propagating neutral mode. We also consider the injection as heat transfer to the neutral mode and reach the same conclusion. In agreement with experiment, these results are strong evidence in favor of any state with counter-propagating edge modes, such as the anti-Pfaffian, as a model for the $\nu$ =5/2 state.   

Geometry of the fractional quantum Hall effect

Unlike the integer effect, the incompressible electron fluid that exhibits the fractional effect is {\it not} invariant under ``area-preserving diffeomorphisms'' of the guiding-center degrees of freedom. Instead (F. D. M. Haldane, Phys. Rev. Lett. 108, 116801 (2011)), it has a shear modulus that characterizes the energy cost of distortions of the correlation hole around the electrons, and a ``guiding-center metric tensor'' that exhibits quantum zero-point fluctuations around a preferred shape. In a simple (one-component) fluid, electronic charge-density fluctuations relative to the background set by the magnetic flux density are given by $\delta \rho$ = $(e^*/2\pi)\bar s K$, where $e^*$ is the elementary fractional charge, $\bar s$ is an integer or half-integer ``guiding-center spin'' that is topologically quantized by the Gauss-Bonnet theorem, and $K$ is the local Gaussian curvature of the guiding-center metric. These results provide a simple explanation of the seminal 1985 results of Girvin, MacDonald and Platzman on the FQH structure factor and collective mode, which remained unexplained in previous proposed narrative explanations of FQH incompressibility (Ginzburg-Landau Chern-Simons theory, composite fermions, and non-commutative Chern-Simons field theory).   

Microscopic nature of the backward propagating neutral edge modes of the ``negative'' flux FQHE states

It is believed that FQHE states that require antiparallel vortex attachment, e.g. 2/3, have neutral modes propagating in the backward direction. Recent experiments have observed signatures of such modes. We study the edge excitations of the fully spin polarized as well as spin singlet 2/3 state both from exact diagonalization and from the microscopic composite fermion (CF) theory, to gain insight into the microscopic nature of the neutral edge modes. We investigate the validity of the CF theory for the edge modes, and also study the dependence on the form of the interaction and the background potential. We further evaluate the spectral weights and compare them with the predictions from the effective bosonic description.   

Braiding statistics of the Gaffnian through the coherent state representation

Certain quantum Hall states have trial wave functions that can be connected to non-unitary conformal field theories, and arguments exist implying that such wave functions cannot describe gapped states. For these trial wave functions, the question arises whether braiding statistics may still be well defined through a formal Berry phase calculation. In essence, this corresponds to assuming an artificial gap to ``non-zero modes'' introduced by non-local terms in the Hamiltonian. The presence of long ranged correlations may still foil the emergence of well defined statistics. However, assuming that this is not the case, the question of what such statistics would be, and how they compare to those defined in terms of conformal block monodromies, can be analyzed using a recently developed coherent state Ansatz based on the thin torus limit. We report pertinent results for the Gaffnian state. Time and/or results permitting, we also present developments on the application of this method to a state by Thomale \emph{et al.}, the associated conformal field theory of which is currently unknown. [References: S. Simon \emph{et al.}, Phys. Rev. B 75, 075317 (2007), J. Flavin and A. Seidel, arXiv:1108.2734v1, N. Read, Phys. Rev. B 79, 045308 (2009)]   

Gapless excitations in the Haldane-Rezayi state: The thin-torus limit

We study the thin-torus limit of the Haldane-Rezayi state. Eight of the ten ground states are found to assume a simple product form in this limit, as is known to be the case for many other quantum Hall trial wave functions. The two remaining states have a somewhat unusual thin-torus limit, where a ``broken'' pair of defects forming a singlet is completely delocalized. We derive these limits from the wave functions on the cylinder, and deduce the dominant matrix elements of the thin-torus hollow-core Hamiltonian. We find that there are gapless excitations in the thin-torus limit. This is in agreement with the expectation that local Hamiltonians stabilizing wave functions associated with non-unitary conformal field theories are gapless. We also use the thin-torus analysis to obtain explicit counting formulas for the zero modes of the hollow-core Hamiltonian on the torus, as well as for the parent Hamiltonians of several other paired and related quantum Hall states. [Reference: A. Seidel, K. Yang, PRB 84, 085122 (2011)]   

Dependence of pinning modes of 2D electron system on short-ranged alloy disorder

A 2D electron system (2DES) in a low Landau filling ($\nu )$ pinned Wigner solid state exhibits a striking resonance in its rf or microwave spectrum. The resonance is understood as a pinning mode, in which the electrons oscillate about their pinned positions, and the frequency (f$_{pk})$ increases for larger disorder. We report on microwave spectroscopy of the low-$\nu $ Wigner solids in Al$_{x}$Ga$_{1-x}$As-Al$_{0.1}$Ga$_{0.9}$As heterojunctions with x=0.4 and 0.8{\%}. The 2DES resides mainly in the dilute Al$_{x}$Ga$_{1-x}$As, and the alloy disorder has been shown to be randomly distributed [1]. We compare the pinning modes of the samples as density ($n$, controlled by backgates) and magnetic field are varied. For example, with densities around $n\sim $6.5$\times $10$^{10 }$cm$^{-2}$ (in sample state with no indication of a 1/5 fractional quantum Hall effect) f$_{pk} \approx $ 5.93 and 8.55 GHz at $\nu \sim $0.2 for x=0.4 and 0.8{\%} respectively.\\[4pt] [1] W. Li \textit{et al}., Appl. Phys. Lett., 83, 2832 (2003).   

Local Thermometry of Neutral Modes on the Quantum Hall Edge

A system of electrons in two dimensions and strong magnetic fields can be tuned to create a gapped 2D system with one dimensional channels along the edge. Interactions among these edge modes can lead to independent transport of charge and heat, even in opposite directions. Measuring the chirality and transport properties of these charge and heat modes can reveal otherwise hidden structure in the edge. Here, we heat the outer edge of such a quantum Hall system using a quantum point contact. By placing quantum dots upstream and downstream along the edge of the heater, we can measure both the chemical potential and temperature of that edge to study charge and heat transport, respectively. We find that charge is transported exclusively downstream, but heat can be transported upstream when the edge has additional structure related to fractional quantum Hall physics.   

Real Space Entanglement Spectrum of Fractional Hall States

The entanglement spectrum has been shown by Li and Haldane to provide a reliable tool to detect the topological order of Hall states. For example, bi-partitioning the system in orbital space produces the signature count of Hall edge states. The spectrum, however, appears to bear no resemblance to the linear spectrum conjectured by Kitaev and Preskill analogous to the actual edge mode dispersion. Here we employ two types of cuts: a real space and a modified particle bi-partitioning. On the sphere, we obtain the entanglement spectrum for the Laughlin, Moore-Read and Read-Rezayi states for both. We also consider the filled Landau level with a real space cut and show that the Laughlin state for $\nu$=1/3 has the same count of levels, which agrees with the chiral CFT (for up to $\Delta M=N/2$, $\Delta M=L_{\rm max}-L_z$). Moreover, the entanglement spectrum of the Laughlin state approaches a linear spectrum, similar to the filled Landau level, as the size of the system increases.