Session L51: Fractional QHE: Bilayer, & Many-Body Effects

Meron deconfinement in the quantum Hall bilayer at intermediate distances

Quantum Hall bilayer phase diagram with respect to interlayer distance bears a remarkable similarity with phase diagrams of strongly correlated systems as a function of doping, with magnetic ordering on the one end and Fermi-liquid-like behavior on the other. Moreover, it has been suggested [1] that a BCS correlated state of composite fermions with p-wave pairing may exist in the intermediate region. We discuss features of this state using the composite boson point of view, and exact diagonalization calculations on the torus. Furthermore, we argue that in the same state there is a possibility for meron deconfinement, i.e., the deconfinement of the vortex excitations of the magnetically ordered phase. [1] G. Moller, S. H. Simon, and E. H. Rezayi, Phys. Rev. Lett. 101, 176803 (2008); G. Moller, S. H. Simon, and E. H. Rezayi, Phys. Rev. B 79, 125106 (2009).   

Magnetotransport of a Quantum Hall Double Layer at Landau Level Filling 1/2 $+$3/2

The effect of interlayer-tunneling on electron transport in quantum Hall double layers in the regime of exciton condensation state at Landau level filling factor one (1/2$+$1/2) has been well established, in that the interlayer coherence promotes a huge zero-bias conductance peak due to resonantly-enhanced tunneling (e.g., Phys. Rev. Lett. 84, 5808, 2000) . Consequently, an in-plane magnetic field is found to suppress this tunneling. Recent theoretical work, on a similar system consisting of two layers with fillings 1/2$+$3/2, suggests that here the resonant-enhanced tunneling would be suppressed and an in-plane magnetic field, conversely, would play a promoting role in tunneling. We investigate this regime in high-mobility GaAs/AlGaAs bilayers of suitable parameters and with individually contacted layers. Preliminary results and a brief discussion will be presented.   

Probing a Wigner Crystal via Composite Fermion Commensurability Oscillations in an Adjacent Layer

At high magnetic fields and low temperatures, two-dimensional electrons form a composite fermion (CF) Fermi sea with a well-defined Fermi wave vector when the Landau level fillings factor ($\nu$) is near 1/2. In contrast, when $\nu\ll1$, the Wigner crystal (WC) is the favored ground state. We report measurements of the magneto-resistance in a bilayer electron system with unequal layer densities at high magnetic fields. One layer has a very low density and is in the WC regime ($\nu\ll1$), while the other (“probe”) layer is near $\nu=1/2$ and hosts a CF sea. As the magnetic field is swept away from $\nu=1/2$ of the CF layer, the CFs feel the periodic electric potential of the WC in the other layer and exhibit magneto-resistance maxima whenever their cyclotron orbit encircles certain integer number of the WC lattice points. Via measuring the temperature dependence of strength of these commensurability features, we probe the melting of the WC.   

Spin Transition of Composite Fermion Solids in Wide Quantum Wells Observed with Microwave Spectroscopy

Within a narrow range of Landau filling ($\nu$) near $1$, a resonance in the microwave spectrum in high mobility two-dimensional electron systems is known to occur [1]. The resonance is understood as due to a pinning mode of a Wigner solid of quasicarriers and is present in the $\nu$-region of vanishing diagonal resistance. In microwave spectroscopy an abrupt jump in the resonance frequency, $f_{pk}$, upon decreasing $\nu$ from $1$ was observed in wide quantum wells [2]. This jump was interpreted as a transition between two solid states: S1, which occurred closer to $\nu=1$, and S2 (with enhanced-$f_{pk}$), which occurred farther from $\nu=1$. In this talk we discuss microwave measurements using variable carrier density and in plane magnetic field. Typical for a spin-related transition, tilting the sample at fixed $n$ results in effects similar to those found on increasing $n$ without tilt. Taken together, the dependencies of the resonance on $n$ and the tilt angle are consistent with a ground state spin transition between different solids. We discuss our results in terms of interacting two-flux composite fermions. [1] Chen et al., Phys. Rev. Lett. 93, 206805 (2004). [2] Hatke et al., Nat. Commun. 5, 4154 (2014).   

Transport Signatures of Dirac Composite Fermions in the 1/2-filled Landau level

The half-filled state plays a special role among the variety of different fractional quantum Hall effect states. Halperin, Lee, and Read (HLR) predicted the existence of compressible state, where gapless excitations consists of electrons bound to a pair of vortices with emergent Fermi surface. There exists ample experimental evidence for this emergent Fermi surface, despite the presence of the strong magnetic field. However, the role of particle hole symmetry of the half-filled Landau level has remained a puzzle. Recently, an alternative, explicitly particle-hole symmetric state was proposed, where composite fermions fill a single Dirac-cone, analogous to the surface state of a topological insulator. These composite Dirac fermions have a quantized $\pi$ Berry phase due to their spin-1/2, their electric dipole moment, locked to their momentum. In this talk I will consider the experimental probes that can distinguish between HLR and composite Dirac fermions. In particular, I will address the signatures of particle-hole symmetry and Berry phase in the thermoelectric response. In addition, I will discuss other transport experiments which can probe the Fermi surface topology.   

Dirac Composite Fermi Liquid in the Half-filled Landau level

Quantum Hall fluids at filling fraction one-half exhibit a compressible phase known as the `composite Fermi liquid' (CFL) We use infinite-cylinder density matrix renormalization group to numerically determine that this phase is the ground state of a half-filled Landau level with Coulomb interactions. We find evidence for a Fermi surface of composite fermions, while also probing the non-Fermi liquid character of the phase. It has been recently realized that the traditional theory used to describe the CFL breaks particle-hole symmetry, while the lowest-Landau level projected Hamiltonian does not. We find that the composite Fermi liquid has particle-hole symmetry, inconsistent with the traditional theory but consistent with a recent theory proposed by Son [Phys. Rev. X 5, 031027]. Our results show the Dirac nature of the composite fermions. We also observe the suppression of certain kinds of backscattering processes of the composite fermions, similar to the suppression in topological insulator surface states.   

Spontaneous polarization of composite fermions in the $n=1$ Landau level of graphene

Motivated by experiments that reveal expansive fractional quantum Hall states in the $n=1$ graphene Landau level and suggest a nontrivial role of the spin degree of freedom [Amet {\em et al.}, Nat. Commun. {\bf 6}, 5838 (2014)], we perform accurate quantitative study of the the competition between fractional quantum Hall states with different spin polarizations in the $n=1$ graphene Landau level. We find that the fractional quantum Hall effect is well described in terms of composite fermions, but the spin physics is qualitatively different from that in the $n=0$ Landau level. In particular, for the states at filling factors $\nu=s/(2s\pm 1)$, $s$ integer, a combination of exact diagonalization and the composite fermion theory shows that the ground state is fully spin polarized and supports a robust spin wave mode even in the limit of vanishing Zeeman coupling. Thus, even though composite fermions are formed, a mean field description that treats them as weakly interacting particles breaks down, and the exchange interaction between them is strong enough to cause a qualitative change in the behavior by inducing full spin polarization. We also find that the fully spin polarized composite fermion Fermi sea has lower energy than the paired Pfaffian state at the relevant half fillings.   

Anisotropic Composite Fermions and Fractional Quantum Hall Effect

We study the role of Fermi sea anisotropy on the transport properties of composite Fermions near Landau level filling factor $\nu=1/2$ in two-dimensional hole systems confined to GaAs quantum wells. By applying a parallel magnetic field, we tune the Fermi sea anisotropy and monitor the relative change of the transport scattering time along its principal directions. Interpreted in a simple Drude model, our results suggest that the scattering time is longer along the longitudinal direction of the Fermi sea. Furthermore, we find that the measured energy gap for the fractional quantum Hall state at $\nu=2/3$ decreases when anisotropy becomes significant.   

Spectral properties of center-of-mass conserving two-body Hamiltonians

We study the low energy spectral properties of positive center-of-mass conserving two-body Hamiltonians as they arise in models of fractional quantum Hall states. We explore what general constraints can be obtained for such interactions, both in the presence and absence of particle-hole symmetry. We find general bounds that constrain the evolution of the ground state energy with particle number, and in particular constrain the chemical potential at T = 0. Special attention is given to Hamiltonians with zero modes, in which case similar bounds on the first excited state are also obtained, using a duality property. In this case, in particular an upper bound on the charge gap is also obtained.   

Neutral Excitations in the Gaffnian state

The Fractional Quantum Hall Effect (FQHE) is one of the most well-studied systems having topological order. Starting with the pioneering work by Laughlin, the model wave function approach has been shown to provide essential information for understanding topological order in gapped incompressible states. We study a model wave function called the Gaffnian state which is believed to represent a gapless, strongly correlated state that is very different from conventional metals. To understand this exotic gapless state better, we provide a representation in which the pairing structure of the Gaffnian state becomes more explicit. We employ the single-mode approximation of the Girvin-MacDonald-Platzman (GMP) mode, which is a neutral collective exitation mode, in order to have a physical picture of the gaplessness of the Gaffnian state. In particular, we discuss how to extract systematically the relevant physics in the long-distance, large electron number limit of the FQH states using a numerical calculation with relatively few electrons.   

Beyond the Plasma Analogy: Collective Field Theory for Quantum Hall States

We develop a quantum field theory of collective coordinates describing fractional quantum Hall (FQH) states. We show that the familiar properties of Laughlin states are captured by a Gaussian free field theory with a background charge. Gradient corrections to the Gaussian theory arise from ultraviolet regularization, and go beyond the celebrated plasma analogy. They give rise to a gravitational anomaly described by the Liouville theory of 2D quantum gravity. The field theory simplifies the computation of correlation functions in FQH states and makes manifest the effect of quantum anomalies. This talk is based on the preprint arXiv:1412.8716.   

Resistively detected high-order magnetoplasmons in a high-quality 2D electron gas

We report on high-order magnetoplasmon resonances detected in photoresistance in high-mobility GaAs quantum wells. These resonances manifest themselves as a series of resistance extrema in the regime of Shubnikov-de Haas oscillations. Extending to orders above 20, the extrema exhibit alternating strength, being less (more) pronounced at even (odd) order magnetoplasmon modes. The lower magnetoplasmon modes reveal the importance of retardation effects.   

Zero modes, bosonization, and topological quantum order: The Laughlin state in second quantization

We introduce a ``second-quantized'' representation of the ring of symmetric functions to further develop a purely second-quantized approach to the study of zero modes of frustration-free Haldane-pseudopotential--type Hamiltonians, which in particular stabilize Laughlin ground states. We present three applications of this formalism. We start demonstrating how to systematically construct all zero modes of Laughlin-type parent Hamiltonians in a framework that is free of first-quantized polynomial wave functions, and show that they are in one-to-one correspondence with dominance patterns. Second, as a by-product, we make contact with the bosonization method, and obtain an alternative proof for the equivalence between bosonic and fermionic Fock spaces. Finally, we explicitly derive the second-quantized version of Read's nonlocal order parameter for the Laughlin state, extending an earlier description by Stone.   

Structure of Ground state Wave Functions for the Fractional Quantum Hall Effect: A Variational Approach

The internal structure and topology of the ground states for fractional quantum Hall effect (FQHE) are determined by the relative angular momenta between all the possible pairs of electrons. Laughlin wave function is the only known microscopic wave function for which these relative angular momenta are homogeneous (same) for any pair of electrons and depend solely on the filling factor. Without invoking any microscopic theory, considering only the relationship between number of flux quanta and particles in spherical geometry, and allowing the possibility of inhomogeneous (different) relative angular momenta between any two electrons, we develop a general method for determining a closed-form ground state wave function for any incompressible FQHE state. Our procedure provides variationally obtained very accurate wave functions, yet having simpler structure compared to any other known complex microscopic wave functions for the FQHE states. This method, thus, has potential in predicting a very accurate ground state wave function for the puzzling states such as the state at filling fraction 5/2.   

Insulating Behavior of Strongly Interacting 2D Electrons in Si MOSFETs

Experiments on low disorder strongly-interacting 2D electron systems have shown that in the absence of a magnetic field, the temperature dependence of the resistivity changes from metallic-like to insulating behavior as the electron density $n_s$ is reduced below a critical density $n_c$ [1]. It has been shown that a metal to insulator transition also occurs in these systems for fixed electron density $n_s$ at a critical (density-dependent) in-plane magnetic field which results in complete spin polarization of the electrons [2]. Here we report measurements of the temperature dependence of the resistivity in a high mobility Si-MOSFET sample, where in one case the insulating state is reached by reducing the electron density in zero field, and in the other case it is reached by "quenching" the metallic behavior with an in-plane field of 5 T. We find that the resistivity of the insulating state behaves in very similar ways for both cases, exhibiting Efros-Shklovskii variable range hopping regardless of the degree of polarization of the electron spins. [1] S. V. Kravchenko et al., Phys. Rev. B 51, 7038 (1995) [2] D. Simonian et al., Phys. Rev. Lett. 79, 2304 (1997)