Session D5: Fractional Quantum Hall Effect III

Position-Momentum Duality, Geometrical Description and Ideal Host Lattices for Fractional Chern Insulators

The recent discovery of fractional quantum Hall (FQH) physics in flat-band Chern insulators without external magnetic field presents a profound theoretical challenge to understand the interplay of universal long-wavelength physics of the FQH effect and short-wavelength physics determined by the host lattice. Here, we present a first quantization description of fractional Chern insulators that is the dual of the anisotropic FQH problem, with the roles of position and momentum interchanged. The fundamental guiding-center geometry of the problem emerges from the interplay of lattice and interaction metrics that act as momentum-space duals of Haldane's geometrical description of the anisotropic FQH effect. We introduce a novel broad class of ideal C>1 Chern insulator lattice models that are duals of the isotropic limit of the conventional FQH effect. These models afford a particularly elegant analytical framework and act as parent Hamiltonians for lattice FQH states with emergent guiding-center and SU(C) symmetry. Resulting microscopic insight into stabilization of FQH states on the lattice provides a foundation for future analyses of non-Abelian phases and fractional topological insulators.   

Projective construction of the $Z_k$ Read-Rezayi fractional quantum Hall states and their excitations on the torus geometry

Multilayer Abelian fractional quantum Hall states can reproduce the non-Abelian states of the $Z_k$ Read-Rezayi series upon symmetrization over the layer index. This property initially shown by Cappelli et. al. (Nuclear Phys. B 2001) yields all the $Z_k$ states on the sphere, but provides only part of the ground state manifold on the torus. We propose to remedy this issue by introducing topological defects connecting the layers before performing the symmetrization. We give a comprehensive account of this construction for bosonic states, and numerically show that the full ground state and quasihole manifolds are recovered for all computationally accessible system sizes. We analyze the neutral excitation mode above each of the states of the Read-Rezayi series on the torus through an extensive exact diagonalization study. We show numerically that our construction can be used to obtain excellent approximations of low energy neutral excitation modes above the $Z_k$ Read-Rezayi states. We generalize the new symmetrization scheme to the sphere geometry.   

Configuration interaction matrix elements for the quantum Hall effect

In the spherical model of the quantum Hall system, the two-body matrix elements and pseudopotentials can be found analytically in terms of a general scalar pair interaction potential by expressing the pair interaction as a weighted sum over Legendre polynomials. For non-infinite systems, only a finite set of terms in the potential expansion contribute to the interactions; the contributing terms define an effective spatial potential for the system. The connection between the effective spatial potential and the pseudopotential is one-to-one for finite systems, and any completely defined model pseudopotential can be analytically inverted to give a unique corresponding spatial potential. This technique of inverting the pseudopotential to derive effective spatial potentials may be of use for developing accurate model spatial potentials for quantum Monte Carlo simulations. We demonstrate the technique and the corresponding spatial potentials for a few example model pseudopotentials.\footnote{R. Wooten, J. Macek, arXiv:1408.5379}   

Magneto-transport characteristics of a 2D electron system driven to negative magneto-conductivity by microwave photoexcitation

Negative diagonal magneto-conductivity/resistivity is a spectacular- and thought provoking- property of driven, far-from-equilibrium, low dimensional electronic systems. The physical response of this exotic electronic state is not yet fully understood since it is rarely encountered in experiment. The microwave-radiation-induced zero-resistance state in the high mobility GaAs/AlGaAs 2D electron system is believed to be an example where negative magneto-conductivity/resistivity is responsible for the observed phenomena. Here, we examine the magneto-transport characteristics of this negative conductivity/resistivity state in the microwave photo-excited two-dimensional electron system (2DES) through a numerical solution of the associated boundary value problem. The results suggest, surprisingly, that a bare negative diagonal conductivity/resistivity state in the 2DES under photo-excitation should yield a positive diagonal resistance with a concomitant sign reversal in the Hall voltage.   

Non-Abelian two dimensional topological phases constructed from coupled wires and connections to exceptional lie algebras

Non-abelian anyons exhibit exotic braiding statistics which can be utilized to realize a universal topological quantum computer. In this work we focus on Fibonacci anyons which occur in $Z_3$ Read Rezayi fractional quantum hall states. Traditionally they have been constructed using $su(2)_3/u(1)$ coset theories. We introduce conformal field theories(CFTs) of exceptional and non-simply laced Lie Algebras at level 1, for example $G_2,F_4$ which host Fibonacci anyons. We realize these CFT's concretely on the 1d gapless edge of an anisotropic 2d system built out of coupled, interacting Luttinger wires. Interactions are introduced within a bundle of wires to fractionalize the original chiral bosons into different sectors. Next, we couple these sectors to get the desired topological phase in the bulk. The 2d bulk of the stack is gapped by backscattering terms between counterpropagating modes on different bundles. The emergence of this topological phase can be interpreted using techniques of anyon condensation . We also explicitly construct the Kac Moody algebra on the edge CFT using original bosonic degrees of freedom.We acknowledge support from NSF CAREER DMR-1351895(TH) and Simons Foundation (JT).   

Variational study of bosonic phases in two dimensions: fractional Chern insulator, Mott insulator and superfluid

We numerically study the model wave functions for a system of hard core bosons at half filling on a square lattice. The candidate wave functions are based on the projective construction approach [1] where a boson is decomposed into two (slave) fermions, each described by a Chern insulator model. Our results confirm that the wave functions demonstrate the following phases: the superfluid, the Mott insulator and the fractional Chern insulator. In addition, we find that the wave functions can be continuously tuned from one phase to another by varying the parameters of slave particles. We further propose a microscopic Hamiltonian with a rich phase diagram which supports all the aforementioned phases in different regimes of parameters. The critical behavior across the phase boundaries is investigated and the critical exponents are computed. [1] M. Barkeshli and J. McGreevy, Phys. Rev. B 89, 235116 (2014).   

Pairing in half-filled Landau level

Pairing of composite fermions in half-filled Landau level state is reexamined by solving the BCS gap equation with full frequency dependent current-current interactions. Our results show that there can be a continuous transition from the Halperin-Lee-Read state to a chiral odd angular momentum Cooper pair state for short-range contact interaction. This is at odds with the previously established conclusion of first order pairing transition, in which the low frequency effective interaction was assumed for the entire frequency range. We find that even if the low frequency effective interaction is repulsive, it is compensated by the high frequency regime, which is attractive. We construct the phase diagrams and show that $\ell=1$ angular momentum channel is quite different from higher angular momentum channel $\ell \ge 3$. Remarkably, the full frequency dependent analysis applied to the bilayer Hall system with a total filling fraction $\nu=\frac{1}{2}+\frac{1}{2}$ is quantitatively changed from the previously established results but not qualitatively.\\[4pt] Reference: Annals of Physics, 351, 727 (2014).   

What Determines the Fermi Wave Vector of Composite Fermions

We report the observation of a pronounced asymmetry in the magnetic field positions of the commensurability resistance minima of fully spin-polarized composite fermions (CFs) with respect to the field at $\nu \quad =$ 1/2 in two-dimensional (2D) electron and hole systems. The asymmetry is observed across a wide range of 2D densities and modulation periods. We can explain the asymmetry quantitatively if we assume that the CFs are fully spin-polarized and their density is equal to the density of the minority carriers in the lowest, spin-resolved Landau level (LL), namely the density of electrons when $\nu $ \textless 1/2 and of holes when $\nu $ \textgreater 1/2. Our results provide direct evidence that CFs are formed by pairing up of the minority carriers in the lowest spin-resolved LL with flux quanta. They further indicate that the CF commensurability minima are not observed at $\nu $ and (1 - $\nu )$, as expected from a simple particle-hole symmetry principle, pointing to a subtle breaking of this symmetry.   

Continuous preparation of a fractional Chern insulator

We present evidence of a direct, continuous quantum phase transition between a Bose superfluid and the v = 1/2 fractional Chern insulator in a microscopic lattice model. In the process, we develop a detailed field theoretic description of this transition in terms of the low energy vortex dynamics. The theory explicitly accounts for the structure of lattice symmetries and predicts a Landau forbidden transition that is protected by inversion. That the transition is continuous enables the quasi-adiabatic preparation of the fractional Chern insulator in non-equilibrium, quantum optical systems.   

Optical Signatures of Competing Quantum Phases in the Second Landau Level

The fractional quantum Hall states and anisotropic phases of electrons in the N$=$1 Landau Level (LL) have drawn considerable experimental focus in recent years[1-3]. We report evidence that the competition and coexistence of these phases is probed directly by optical recombination from the partially populated N$=$1 LL at dilution refrigerator temperatures. Spectral bands that display striking dependence on perpendicular magnetic field in the full range 3\textgreater nu\textgreater 2 are interpreted as linked to anisotropic phases. Optical recombination thus enables the monitoring of the evolution competing phases as the N$=$1 LL filling is changed. Remarkable changes in the optical recombination reveal that coexistence of distinct quantum phases has a marked dependence on filling of the N$=$1 LL. The signatures of anisotropic phases remain strong at filling factors of the FQHE. This is consistent with recent reports on anisotropic FQHE states in the second Landau level[1-3]. References: [1] J. Xia et al, Phys. Rev. Lett. 105, 176807 (2010). [2] J.$^{\mathrm{\thinspace }}$Xia, J.P. Eisenstein, L.N. Pfeiffer, and K. West, K.W. Nature Phys$.$ 7, 845--848 (2011).$^{\mathrm{\thinspace }}$[3]$^{\mathrm{\thinspace }}$Y. Liu, et al, Phys. Rev. B 88, 035307 (2013).   

DMRG Study of a $\nu=1/3+1/3$ Bilayer Fractional Hall System

DMRG Study of a $\nu=1/3+1/3$ Bilayer Fractional Hall System Bilayer quantum Hall systems provide new experimental parameters that generate a rich phase diagram proposed to contain to new non-Abelian phases. We use the density matrix renormalization group (DMRG) and exact diagonalization to study a bilayer quantum Hall system with filling fraction 1/3 per layer. Using this method, we can study the phase diagram in terms of parameters such as interlayer separation, interlayer tunnelling, layer width, and density imbalance between the layers. We identify the possible phases based on their entanglement properties and determine the order of phase transitions. Prospects for stabilizing new non-Abelian phases with small perturbations to the Coulomb interaction will be discussed.   

Microwave spectroscopic observation of multiple phase transitions in the bilayer electron solid in wide quantum wells

The termination of the low Landau filling factor ($\nu$) fractional quantum Hall series for a single layer two dimensional system results in the formation of a pinned Wigner solid for $\nu<1/5$ [1]. In a wide quantum well the system can support a bilayer state in which interlayer and intralayer interactions become comparable, which is measured in traditional transport as an insulating state for $\nu<1/2$ [2]. We perform microwave spectroscopic studies of this bilayer state and observe that this insulator exhibits a resonance, a signature of a solid phase. Additionally, we find that as we increase the density of the well at fixed $\nu$ this bilayer solid exhibits multiple sharp reductions in the resonance amplitude vs $\nu$. This behavior is characteristic of multiple phase transitions, which remain hidden from dc transport measurements. \\[4pt] [1] Lozovik and Yudson, JETP Lett. 22, 11 (1975).\\[0pt] [2] Manoharan et al., Phys. Rev. Lett. 77, 1813 1996).   

Microwave spectroscopic observation of the $\nu=1/2$ fractional quantum Hall Wigner solid

Microwave spectroscopic studies have revealed Wigner solidification of quasiparticles within a narrow range of Landau filling ($\nu$) near the integer quantum Hall effect at $\nu=1$ [1] and the fractional quantum Hall effect at $\nu=1/3$ [2, 3]. Here, an observed resonance is taken as a signature of a pinned Wigner solid in which the quasiparticles oscillate about their pinned positions. In wide quantum wells (and double quantum wells), where interlayer and intralayer correlations can be of comparable magnitude, the even-denominator $\nu=1/2$ state is known to exist [4]. In this talk I will discuss recent microwave spectroscopic studies of the wide quantum well system that demonstrates the formation of a $\nu=1/2$ Wigner solid. Additionally, I will show that this $\nu=1/2$ Wigner solid undergoes a first order phase transition to the recently observed bilayer solid, observable as a region of coexistence of the two solids. \\[4pt] [1] Chen et al., Phys. Rev. Lett. 93, 206805 (2004).\\[0pt] [2] Zhu et al., Phys. Rev. Lett, 105, 126803 (2010).\\[0pt] [3] Archer et al., Phys. Rev. Lett, 111, 146804 (2013).\\[0pt] [4] Manoharan et al., Phys. Rev. Lett. 77, 1813 1996); Suen et al., Phys. Rev. Lett. 68, 1379 (1992); Eisenstein et al., Phys. Rev. Lett. 68 1383 (1992).   

Collapse of the $\nu=1$ quantum Hall effect near a Landau level crossing

We report magneto-resistance measurements of 2D hole systems (density $2.1\times 10^{11}$ cm$^{-2}$) confined to a 40-nm-wide GaAs quantum well as a function of tilted magnetic fields. We observe a strong $\nu=1$ quantum Hall effect (QHE) at zero parallel field ($B_{||}$). The $\nu=1$ QHE disappears at $B_{||}\simeq4.8\,\mathrm{T}$, where we expect a crossing between the lowest two Landau levels. Near this crossing, the energy gap for the $\nu=1$ QHE collapses from 6 K to zero in a very small $B_{||}$ range of 0.3 T. The $\nu=1$ QHE comes back at $B_{||}\simeq8.1\,\mathrm{T}$ and eventually disappears at $B_{||}>17\,\mathrm{T}$ where the system becomes bilayer-like. The sudden collapse of the $\nu=1$ QHE and the fact that it comes back after a large $B_{||}$ range of 3.3 T is intriguing and suggests a pinning of the Landau levels near the crossing.   

Exact diagonalization study of $\nu=1/3+1/3$ bilayer quantum Hall in search of Fibonnacci anyons

Non-abelian states with Fibonnacci anyons that can support universal topological quantum computation have been elusive. Recently it has been proposed that a remarkably simple setting of Abelian quantum Hall bilayer could support an exotic state with Fibonnacci anyons (Vaezi and Barkeshli, arXiv:1403.3383). Here we explore $\nu=1/3+1/3$ bilayer quantum Hall system considering different possibilities for interaction between bilayers using exact diagonalization. We find a sizable region in phase space potentially exhibiting topological degeneracy expected of Fibonnacci anyon states.