Session V9: Fractional QHE

Abelian and non-Abelian Quasielectrons

While quasiholes in the fractional quantum Hall effect are well-understood and their explicit form is uncontroversial, their negatively charged counterpart, the quasielectrons, have proven to be much more elusive. We solve the problem of defining, within conformal field theory, a proper quasielectron operator. The strength of our description lies in its applicability, and we will give several explicit examples of this, such as the condensation of Abelian and non-Abelian quasielectrons.   

Transference of Fermi Surface Anisotropy to Composite Fermions

The question of how the properties of the underlying particles affect the behavior of the emergent particles is of paramount interest in physics. The system of composite Fermions (CFs), complex electron-flux bound states that experience an effective magnetic field, offers a nontrivial example where this question can be addressed in various contexts. The CF formulation provides an elegant and powerful description of interacting two-dimensional (2D) electrons at high perpendicular magnetic fields. In particular, at a half-filled Landau level (LL), where the effective magnetic field vanishes, the CFs exhibit Fermi-liquid-like properties, similar to their zero-field electron counterparts. Interestingly, however, the Fermi energy and the effective mass of CFs, being determined solely by interactions, possess no memory whatever of the Fermi energy and the band mass of the electrons. That raises a fundamental question: Does an anisotropy of the electron Fermi surface survive composite fermionization? Here we provide an experimental resolution of this question by studying the properties of CFs in AlAs quantum wells where the electrons occupy an ellipsoidal Fermi surface with large eccentricity. Through a comparison of the piezo-resistance for electrons and CFs, we show that the CF Fermi surface qualitatively follows the anisotropy of the electron Fermi surface.   

Fibonacci anyons through the 1D coherent state representation

Recent work has shown that a large class of fractional quantum Hall trial wave functions can be uniquely associated with simple strings of integer patterns, either through the thin torus limit, or Jack polynomials, or patterns of zeros. An interesting question is to what extent these patterns contain information about the statistics of the quasi-particles and quasi-holes of the underlying quantum Hall state, in particular for non-Abelian states. Using the thin torus limit together with the notion of adiabatic continuity and a simple coherent state Ansatz, it has been demonstrated that these patterns essentially uniquely determine the statistics of the $\nu=1$ Moore-Read state. Here we show that the very same method is also applicable to the $k=3$ Read-Rezayi state at $\nu=3/2$. We find that within this approach only two representations of the braid group are consistent with the given set of integer patterns, which are identical up to complex conjugation and an overall Abelian phase. One of these solutions agrees completely with the conformal block monodromies associated to the Read-Rezayi trial wavefunctions. [References: A. Seidel, Phys. Rev. Lett. 101, 196802 (2008), A. Seidel, D.-H. Lee, Phys. Rev. B 76, 155101 (2007).]   

Tilted Field Studies of Competing Phases in the N=1 Landau Level

The N=1 Landau level (LL) exhibits collective electronic phenomena characteristic of both fractional quantum Hall (FQHE) states seen in the lowest LL and pinned, anisotropic states in the higher LLs. Moreover, it has been shown that in tilted magnetic fields, FQHE states at $\nu =5/2$ and $\nu =7/2$ give way to anisotropic states, revealing the close competition between the two phases. To study the energetics of various quantum phases in the N=1 LL, we perform measurements of the activation energy gap and temperature dependence of the resistivity anisotropy in a high mobility and relatively low density 2DES in a GaAs quantum well at $\nu =5/2$, $\nu =7/2$, $\nu =8/3$ and $\nu =7/3$ in tilted magnetic fields. At zero tilt, we observe several re-entrant integer quantum Hall (RIQH) states along with FQHE states in the N=1 LL. As the in-plane magnetic field increases, FQHE activation gaps are reduced at half fillings while strengthened at third fillings. In the meanwhile, transport anisotropy is enhanced at all four fillings with increasing tilt. At high tilt, we observe strong anisotropic states at half fillings in coexistence with FQHE states with accurately quantized Hall plateaus at third fillings.   

Numerical and analytical tests of topological nature of fractional quantum Hall edge

We obtain the thermodynamic dispersions for the elementary single boson excitations at filling factors $\nu=1/3$ and $\nu=2/5$ using the accurate framework of composite fermion diagonalization. At $\nu=2/5$ we also consider excitations in which composite fermions change their $\Lambda$ level index, which can sometimes be important even at low temperatures. In addition, we show how non-Fermi liquid behavior arises from projecting the electron creation operator into the low energy subspace of composite fermion edge states. Our study allows a microscopic evaluation of the edge spectral function, and an investigation of the regime of validity of the linear approximation of the chiral Luttinger liquid description of the FQHE edge. Possible experimental signatures of the ``edge roton minimum'' are investigated.   

Composite fermion valley polarization energies: Evidence for particle-hole asymmetry

In an ideal two-component two-dimensional electron system, particle-hole symmetry dictates that the fractional quantum Hall (FQH) states around nu = 1/2 are equivalent to those around nu = 3/2. We demonstrate that composite fermions (CFs) around nu = 1/2 in AlAs possess a valley degree of freedom like their counterparts around nu = 3/2. We valley polarize these CFs by applying an in-plane uniaxial strain. Normalized to the Coulomb energy, the energies required to completely valley-polarize the CFs around nu = 1/2 and 3/2 should be identical. Surprisingly, we find that it takes much less energy to completely valley polarize the CFs around nu = 1/2 compared to the CFs around 3/2. We investigate the FQH states at nu = 2/3 and 4/3 for a wide range of 2D electron density and conclude that particle-hole symmetry is violated in our system.   

``Hall viscosity'', edge-state dipole moments and incompressibility of FQHE fluids

The dissipationless ``Hall viscosity'' described for the inte- ger QHE by Avron \textit{et al.} (1995) and for the FQHE by Read (2009) describes the stress-tensor response to the gradient of the electric field, and is distinct from the Hall conductivity. Previous work assumed rotational invariance and an extrinsic metric; removing this unnecessary assumption (broken by ``tilting'' B) clarifies the relations between Hall viscosity and incompressibility. New properties of FQHE fluids emerge:(1) they have no hydrostatic pressure; (2) (unreconstructed) edges have a universal electric dipole moment given by the Hall viscosity, which (3) has \textit{two} distinct sources, the ``smearing'' of electron density relative to guiding center density and non-trivial behavior of guiding-center occupations near an edge. (4) The second term vanishes in the integer QHE, is odd under particle-hole symmetry, and is expressed in terms of a modified ``shift'' (per flux, as opposed to per particle, as in Read 2009), plus (5) an intrinsic metric that arises from incompressibility itself. (6) Its absolute value provides a lower bound to the coefficient of $Q^4$ behavior of the the guiding-center structure function as $Q \rightarrow 0$, (and is an equality for model states such as Laughlin, Moore-Read.) These properties are related through the $SO$(2,1) algebra of guiding-center deformations.   

Link between the hierarchy of fractional quantum Hall states and Haldane's conjecture for quantum spin chains

We study a strong coupling expansion of the $\nu=1/3$ fractional quantum Hall state away from the Tao-Thouless limit and show that the leading quantum fluctuations lead to an effective spin-1 Hamiltonian that lacks parity symmetry. By analyzing the energetics and discrete symmetries of low-lying excitations, we demonstrate that the $\nu=1/3$ fractional quantum Hall state is adiabatically connected to both Haldane and large-$D$ phases. This result indicates a close relation between the Haldane conjecture for spin chains and the fractional quantum Hall effect.   

Interference of $e/3$ quasiparticles encircling $2/5$ fractional quantum Hall island

We report experiments in a large, 2.5 micron Fabry-Perot interferometer fabricated from a GaAs/AlGaAs heterostructure. Device is defined by etch trenches; front gates deposited in the trenches allow to tune the device. Tunneling in the two constrictions closes an Aharonov-Bohm path around the 2D electron island. Quantized plateaus in $R_{XX}$ and $R_{XY}$ allow to find out both: the bulk and the constriction filling. Etch trench depletion is such that in the fractional quantum Hall regime we obtain the situation when 1/3 chiral edge channels pass through the constrictions and encircle an island of the 2/5 FQH fluid. In this regime the magnetic field oscillation period is 5.4$\pm$0.3 of the integer filling 1 period. In this large device magnetic field period well approximates the flux period. We thus conclude that the flux period is $5h/e$, and the corresponding back-gate period is $2e$. These results agree with our previous reports of these superperiods in smaller size devices [1]. The experimental superperiods are interpreted as imposed by the anyonic statistics of the fractionally charged $e/3$ and $e/5$ quasiparticles. [1] F. E. Camino et al., PRB 72, 075342 (2005); W. Zhou et al., PRB 73, 245322 (2006).   

Magnetic flux superperiods in fractional quantum Hall interferometers

Superperiodic Aharonov-Bohm oscillations in conductance of $e/3$ quasiparticles have been reported in three Fabry-Perot interferometer devices. Superperiods are observed in the FQH regime, when filling 1/3 edge channel encircles an island of 2/5 FQH fluid. Etch trenches define the devices, which consist of a 2D electron island connected to the 2DES bulk via two wide constrictions. An oscillatory signal in the conductance is observed when tunneling occurs in the constrictions. The width of the 1/3 edge channel weakly depends on the size of the device, on the other hand, the enclosed 2/5 island area varies by a factor of ~4. We compare the magnetic field periods in the different size devices and review the evidence that the flux period is $5h/e$. [1] The FQH edge channel structure essentially depends on the 2D electron density profile. We discuss the self- consistent density profile in the device defined by the etch trenches. We also discuss electron depletion due to electric field of front gates, which is not screened efficiently by 2D electrons and thus leads to a smaller gradient of the confining potential than the mesa etch. [1] F. E. Camino et al., PRB 72, 075342 (2005); W. Zhou et al., PRB 73, 245322 (2006); P. V. Lin et al., PRB (in press, 2009).   

Exotic resonant level models in non-Abelian quantum Hall states coupled to quantum dots

We study the coupling between a quantum dot and the edge of a non-Abelian fractional quantum Hall state. We focus on the physics of level degeneracy with electron number on the dot. The physics of such a resonant level is governed by a $k$-channel Kondo model when the quantum Hall state is a Read-Rezayi state at filling fraction $\nu=2+k/(k+2)$ or its particle-hole conjugate at $\nu=2+2/(k+2)$. The $k$-channel Kondo model is channel symmetric even without fine tuning any couplings in the former state; in the latter, it is generically channel asymmetric. The two limits exhibit non-Fermi liquid and Fermi liquid properties, respectively, and therefore may be distinguished. By exploiting the mapping between the resonant level model and the multichannel Kondo model, we discuss the thermodynamic and transport properties of the system. In the special case of $k=2$, our results provide a novel venue to distinguish between the Pfaffian and anti-Pfaffian states at filling fraction $\nu=5/2$. Transport through a double-point contact geometry is possibly governed by an unusual fixed point. arXiv:0911.1799   

Spin-related origin of the magnetotransport feature at filling factor 7/11

Experiments by Pan et al. disclosed quantum Hall (QH) effect-like features at unconventional filling fractions, such as 4/11 and 7/11, not included in the Jain sequence [1]. These features were considered as evidence for a new class of fractional quantum Hall (FQH) states whose origin, unlike ordinary FQH states, is linked to interactions between composite fermions (CFs). However, the exact origin of these features is not well established yet. Here we focus on 7/11, where a minimum in the longitudinal resistance and a plateau-like structure in the Hall resistance are observed at a much higher field, 11.4 T, in a 30-nm quantum well (QW). Our density-dependent studies show that at this field, the FQH states flanking 7/11, viz. the 2/3 and 3/5 states, are both fully spin polarized. Despite of this fact, tilted-field experiments reveal that the 7/11 feature weakens and then disappears upon tilting. Using a CF model, we show that the spin degree of freedom may not be completely frozen in the region between the 2/3 and 3/5 states even when both states are fully polarized. Systematic studies unveil that the exact location of the 7/11 feature depends on the electron density and the QW width, in accordance with the model. Our model can also account for the reported contrasting behavior upon tilting of 7/11 and its electron-hole counterpart 4/11. [1] Pan et al., Phys. Rev. Lett. 90, 016801 (2003).   

Wave functions for hierarchical quantum Hall states

We propose a framework for obtaining microscopic representative wave functions which reflect Wen and Zee's classification of Abelian quantum Hall states. Explicit wave functions are related to coherent state transforms of combinations of chiral and antichiral blocks in simple conformal field theories. The approach reproduces all positive and negative Jain states as special cases, and also encompasses hierarchical states which involve condensates of both quasiholes and quasielectrons. We show that the proposed wave functions reduce to the known ground states in the thin cylinder limit.