A large class of fractional quantum Hall (FQH) states can be classified according to their pattern of zeros, which describes the way ideal ground state wave functions go to zero as various clusters of electrons are brought together. In this paper we generalize this approach to classify multilayer FQH states. Such a classification leads to the construction of a class of non-Abelian multilayer FQH states that are closely related to $\hat{g}_k$ parafermion conformal field theories, where $\hat{g}_k$ is an affine simple Lie algebra. We discuss the possibility of some of the simplest of these non-Abelian states occuring in experiments on bilayer FQH systems at $ν= 2/3$, 4/5, 4/7, etc.
We discuss the quantum Hall effect of bilayer graphene with finite gate voltage where the Fermi energy exceeds the interlayer hopping energy. We calculated magnetic susceptibility, diagonal and off-diagonal conductivities in finite-magnetic-field formalism, and observed crossover of integer quantum Hall effect from two independent monolayer type system to strongly coupled bilayer systems by changing the ratio of interlayer hopping energy and the gate voltage. We also discuss the case of multilayer systems with Bernal stacking
We theoretically study the correlated insulator states, quantum anomalous Hall (QAH) states, and field-induced topological transitions between different correlated states in twisted multilayer graphene systems. Taking twisted bilayer-monolayer graphene and twisted double-bilayer graphene as examples, we show that both systems stay in spin polarized, $C_{3z}$-broken insulator states with zero Chern number at 1/2 filling of the flat bands under finite displacement fields. In some cases these spin polarized, nematic insulator states are in the quantum valley Hall phase by virtue of the nontrivial…
This is a Reply to a Comment by Tanaka and Machida. We provide some details of the derivation of the effective field theory for integer quantum Hall transitions using the non-Abelian chiral anomaly.
We report on a theoretical study of one-dimensional (1D) states localized at few-layer graphene system ribbon edges, and at interfaces between few-layer graphene systems with different valley Hall conductivities. These 1D states are topologically protected when valley mixing is neglected. We address the influence on their properties of stacking arrangement, interface structure, and external electric field perpendicular to the layers. We find that 1D states are generally absent at multilayer ribbon armchair direction edges, but present irrespective of crystallographic orientation at any interna…
In ultra-thin film of topological insulator, the hybridization between the top and bottom surfaces opens an energy gap and forms two degenerate quantum anomalous Hall states, which give rise to a quantum spin Hall state. In this paper, we demonstrate that a Weyl semimetal can be realized in an ultra-thin film of topological insulator heterostructure in a similar way to that of the surface state of a strong three-dimensional (3D) topological insulator studied by Burkov and Balents . We find that the system realizes both 3D quantum anomalous Hall phase and 3D quantum spin Hall phase, and the Wey…
The current proposals for producing non-Abelian anyons and Majorana particles, which are neither fermions nor bosons, are primarily based on the realization of topological superconductivity in two dimensions. We show theoretically that the unique Landau level structure of bilayer graphene provides a new possible avenue for achieving such exotic particles. Specifically, we demonstrate the feasibility of a "parton" fractional quantum Hall (FQH) state, which supports non-Abelian particles without the usual topological superconductivity. Furthermore, we advance this state as the fundamental explan…
The Landau level spectra and the quantum Hall effect of ABA-stacked multilayer graphenes are studied in the effective mass approximation. The low-energy effective mass Hamiltonian may be partially diagonalized into an approximate block-diagonal form, with each diagonal block contributing parabolic bands except, in a multilayer with an odd number of layers, for an additional block describing Dirac-like bands with a linear dispersion. We fully include the band parameters and, taking into account the symmetry of the lattice, we analyze their affect on the block-diagonal Hamiltonian. Next-nearest …
Domain walls, topological defects that define the frontier between regions of different stacking in multilayer graphene, have proved to host exciting physics. The ability of tuning these topological defects in-situ in an electronic transport experiment brings a wealth of possibilities in terms of fundamental understanding of domain walls as well as for electronic applications. Here, we demonstrate through a MEMS (micro-electromechanical system) actuator and magnetoresistance measurements the effect of domain walls in multilayer graphene quantum Hall effect. Reversible and controlled uniaxial s…
We study the effect that the electron-electron interaction has on the properties of a multilayer electron system. We consider the case corresponding to filling factor unity in each layer. We find that as a function of the sample parameters the system has ferromagnetic, canted antiferromagnetic or paramagnetic interlayer spin correlations. These three ground states are QHE phases, because of the existence of a finite activation energy. In the ferromagnetic phase the gap is due to the intrawell exchange energy, whereas in the paramagnetic phase the gap appears due to the spatial modulation of th…
The quantum Hall phase diagram of the half-filled bilayer system in the second Landau level is studied as a function of tunneling and layer separation using exact diagonalization. We make the striking prediction that bilayer structures would manifest two distinct branches of incompressible fractional quantum Hall effect (FQHE) corresponding to the Abelian 331 state (at moderate to low tunneling and large layer separation) and the non-Abelian Pfaffian state (at large tunneling and small layer separation). The observation of these two FQHE branches and the quantum phase transition between them w…
Three-Dimensional Non-Foliated Fractional Quantum Hall Phases with Irrational Anyons in Twisted van der Waals Multilayers
Pohang University of Science and Technology（浦项科技大学）；Daegu Gyeongbuk Institute of Science and Technology (DGIST)（大邱庆北科学技术院）；Korea Advanced Institute of Science and Technology（韩国科学技术院）；Institute for Basic Science（基础科学研究院）
A Zener diode is a paradigmatic device in semiconductor-based electronics that consists of a pn junction where an external electric field induces a switching behavior in the current-voltage characteristics. We study Zener tunneling in HgTe quantum wells and graphene multilayers. We find that the tunneling transition probability depends asymmetrically on the parallel momentum of the carriers to the barrier. In HgTe quantum wells the asymmetry is the opposite for each spin, whereas for graphene multilayers it is the opposite for each valley degree of freedom. In both cases, a spin/valley current…
We present a new class of non-abelian spin-singlet quantum Hall states, generalizing Halperin's abelian spin-singlet states and the Read-Rezayi non-abelian quantum Hall states for spin-polarized electrons. We label the states by (k,M) with M odd (even) for fermionic (bosonic) states, and find a filling fraction $ν=2k/(2kM+3)$. The states with M=0 are bosonic spin-singlet states characterized by an SU(3)_k symmetry. We explain how an effective Landau-Ginzburg theory for the SU(3)_2 state can be constructed. In general, the quasi-particles over these new quantum Hall states carry spin, fractiona…
We show in this paper that thermopower is enhanced in non-Abelian quantum Hall liquids under appropriate conditions. This is because thermopower measures entropy per electron in the clean limit, while the degeneracy and entropy associated with non-Abelian quasiparticles enhance entropy when they are present. Thus thermopower can potentially probe non-Abelian nature of the quasiparticles, and measure their quantum dimension.
The so called quantum spin Hall phase is a topologically non trivial insulating phase that is predicted to appear in graphene and graphene-like systems. In this work we address the question of whether this topological property persists in multilayered systems. We consider two situations: purely multilayer graphene and heterostructures where graphene is encapsulated by trivial insulators with a strong spin-orbit coupling. We use a four orbital tight-binding model that includes the full atomic spin-orbit coupling and we calculate the $Z_{2}$ topological invariant of the bulk states as well as th…
We experimentally image the magnetic precession phase of patterned spin Hall multilayer samples to study the rf driving field vector using time-resolved anomalous Nernst effect (TRANE) microscopy. Our ferromagnetic resonance (FMR) measurements quantify the phase and amplitude for both the magnetic precession and the electric current, which allows us to establish the total driving field orientation and the strength of spin Hall effect. In a channel of uniform width, we observe spatial variation of the FMR phase laterally across the channel. We interpret our findings in the context of electrical…
We show that non-abelian quantum Hall states can be identified by experimental measurements of the temperature-dependence of either the electrochemical potential or the orbital magnetization. The predicted signals of non-abelian statistics are within experimental resolution, and can be clearly distinguished from other contributions under realistic circumstances. The proposed measurement technique also has the potential to resolve spin-ordering transitions in low density electronic systems in the Wigner crystal and strongly-interacting Luttinger liquid regimes.
While the internal structure of Abelian topological order is well understood, how to characterize the non-Abelian topological order is an outstanding issue. We propose a distinctive scheme based on the many-body Chern number matrix to characterize non-Abelian multicomponent fractional quantum Hall states. As a concrete example, we study the many-body ground state of two-component bosons at the filling faction $ν=4/3$ in topological flat band models. Utilizing density-matrix renormalization group and exact diagonalization calculations, we demonstrate the emergence of non-Abelian spin-singlet fr…
We study the tunable quantum Hall effects in a non-Abelian honeycomb optical lattice which is a many-Dirac-points system. We find that the quantum Hall effects present different features as change as relative strengths of several perturbations. Namely, a gauge-field-dressed next-nearest-neighbor hopping can induce the quantum spin Hall effect and a Zeeman field can induce a so-called quantum anomalous valley Hall effect which includes two copies of quantum Hall states with opposite Chern numbers and counter-propagating edge states. Our study extends the borders of the field of quantum Hall eff…
We examine the relation between different electronic transport phenomena in a Fabry-Perot interferometer in the fractional quantum Hall regime. In particular, we study the way these phenomena reflect the statistics of quantum Hall quasi-particles. For two series of states we examine, one abelian and one non-abelian, we show that the information that may be obtained from measurements of the lowest order interference pattern in an open Fabry-Perot interferometer is identical to the one that may be obtained from the temperature dependence of Coulomb blockade peaks in a closed interferometer. We a…
We study continuous quantum phase transitions that can occur in bilayer fractional quantum Hall (FQH) systems as the interlayer tunneling and interlayer repulsion are tuned. We introduce a slave-particle gauge theory description of a series of continuous transitions from the (ppq) Abelian bilayer states to a set of non-Abelian FQH states, which we dub the orbifold FQH states, of which the Z4 parafermion (Read-Rezayi) state is a special case. This provides an example in which Z2 electron fractionalization leads to non-Abelian topological phases. The naive "ideal" wave functions and ideal Hamilt…
We discuss electric transport and orbital magnetism of multilayer graphenes in a weak-magnetic field using the matrix decomposition technique. At zero temperature, the minimum conductivity is given by that of the monolayer system multiplied by the layer number $N$, independent of the interlayer hopping $t$. When the interlayer hopping satisfies the condition $t\gg \hbar/τ$ with $τ$ being collision time of impurity scattering, $[N/2]$ kinks and $[N/2]+1$ plateaux appear in the Fermi-energy (gate voltage) dependence of the conductivity and the Hall conductivity, respectively. These behaviors are…
We consider the fractional quantum Hall effect at the filling factor $ν=4/11$, where two independent experiments have observed a well-developed and quantized Hall plateau. We examine the Abelian state described by the "$4\bar{2}1^{3}$" parton wave function and numerically demonstrate it to be a plausible candidate for the ground state at $ν=4/11$. We work out the low-energy effective theory of the $4\bar{2}1^{3}$ edge and make predictions for experimentally measurable properties of the state.
We evidenced unconventionnal Anomalous Hall Effects (AHE) in 3d/5d (Co0.2nm/Ni0.6nm)N multilayers grown on a thin Pt layer or thin Au:W alloy. The inversion observed on AHE originates from the opposite sign of the spin-orbit coupling of Pt compared to Ni. Via advanced simulations methods for the description of the spin-current profiles based on the spin-dependent Boltzmann formalism, we extracted the spin Hall angle (SHA) of Pt and (Co/Ni) as well as the relevant transport parameters. The extracted SHA for Pt, +20%, is opposite to the one of (Co/Ni), giving rise to an effective AHE inversion f…
Anomalous Hall Crystals in Rhombohedral Multilayer Graphene I: Interaction-Driven Chern Bands and Fractional Quantum Hall States at Zero Magnetic Field
The orbital structure of the quantum Hall ferromagnetic states in the zero-energy Landau level in chiral multilayer graphene (AB, ABC, ABCA, etc.\ stackings) is determined by the exchange interaction with all levels, including deep-lying states in the Dirac sea. This exchange field favors orbitally coherent states with a U(1) orbital symmetry if the filling factor $ν$ is not a multiple of the number of layers. If electrons fill the orbital sector of a fixed spin/valley component to one-half, e.g., at $ν=\pm3,\pm1$ in the bilayer and at $ν=\pm2,\pm6$ in the ABCA four-layer, there is a transitio…
The electron-electron interaction in the Landau levels of bilayer graphene is markedly different from that of conventional semiconductors such as GaAs. We show that in the zeroth Landau level of bilayer graphene, in the orbital which is dominated by the non-relativistic second Landau level wave function, by tuning the magnetic field a topological quantum phase transition from an Abelian composite fermion to a non-Abelian parton fractional quantum Hall state can be induced at filling factors $1/2, ~2/5$ and $3/7$. The parton states host exotic anyons that can potentially be utilized to store an…
The recent measurement of a half-integer thermal conductance for the $ν=5/2$ fractional quantum Hall state has confirmed its non-Abelian nature, making the question of the underlying topological order highly intriguing. We analyze the shot noise at the edge of the three most prominent non-Abelian candidate states. We show that the noise scaling with respect to the edge length can, in combination with the thermal conductance, be used to experimentally distinguish between the Pfaffian, anti-Pfaffian, and particle-hole-Pfaffian edge structures.
We examine the quantum phase diagram of the fractional quantum Hall effect (FQHE) in the lowest two Landau levels in half-filled bilayer structures as a function of tunneling strength and layer separation, i.e., we revisit the lowest Landau level filling factor 1/2 bilayer problem and make new predictions involving bilayers in the half-filled second Landau level (i.e., filling factor 5/2). Using numerical exact diagonalization we investigate the important question of whether this system supports a FQHE described by the non-Abelian Moore-Read Pfaffian state in the strong tunneling regime. In th…
Skyrmions are topologically protected, two-dimensional, localized hedgehogs and whorls of spin. Originally invented as a concept in field theory for nuclear interactions, skyrmions are central to a wide range of phenomena in condensed matter. Their realization at room temperature (RT) in magnetic multilayers has generated considerable interest, fueled by technological prospects and the access granted to fundamental questions. The interaction of skyrmions with charge carriers gives rise to exotic electrodynamics, such as the topological Hall effect (THE), the Hall response to an emergent magnet…
Using projective construction, a generalized parton construction, we construct many non-Abelian quantum Hall (QH) states, which include the Pfaffian state at filling fraction $ν=1/2$. The projective construction allows us to calculate the bulk and the edge effective theory for the constructed QH state. We illustrate how to use the bulk effective theory to calculate the ground state degeneracy of non-Abelian QH liquids on torus. We point out that the full description of the effective theory requires both the effective Lagrangian and the definition of electron operators. The latter generates all…
We construct Landau-Ginzburg effective field theories for fractional quantum Hall states -- such as the Pfaffian state -- which exhibit non-Abelian statistics. These theories rely on a Meissner construction which increases the level of a non-Abelian Chern-Simons theory while simultaneously projecting out the unwanted degrees of freedom of a concomitant enveloping Abelian theory. We describe this construction in the context of a system of bosons at Landau level filling factor $ν=1$, where the non-Abelian symmetry is a dynamically-generated SU(2) continuous extension of the discrete particle-hol…
We find a series of possible continuous quantum phase transitions between fractional quantum Hall (FQH) states at the same filling fraction in two-component quantum Hall systems. These can be driven by tuning the interlayer tunneling and/or interlayer repulsion. One side of the transition is the Halperin (p,p,p-3) Abelian two-component state while the other side is the non-Abelian Z4 parafermion (Read-Rezayi) state. We predict that the transition is a continuous transition in the 3D Ising class. The critical point is described by a Z2 gauged Ginzburg-Landau theory. These results have implicati…
A large class of fractional quantum Hall (FQH) states can be classified according to their pattern of zeros, which describes the order of zeros in ground state wave functions as various clusters of electrons are brought together. The pattern-of-zeros approach can be generalized to systematically classify multilayer/spin-unpolarized FQH states, which has led to the construction of a class of non-Abelian multicomponent FQH states. Here we discuss some of the simplest non-Abelian two-component states that we find and the possibility of their experimental realization in bilayer systems at $ν= 2/3$…
We study the electronic structures and topological properties of $(M+N)$-layer twisted graphene systems. We consider the generic situation that $N$-layer graphene is placed on top of the other $M$-layer graphene, and is twisted with respect to each other by an angle $θ$. In such twisted multilayer graphene (TMG) systems, we find that there exists two low-energy flat bands for each valley emerging from the interface between the $M$ layers and the $N$ layers. These two low-energy bands in the TMG system possess valley Chern numbers that are dependent on both the number of layers and the stacking…
Magnetic skyrmions are topologically non-trivial nanoscale objects. Their topology, which originates in their chiral domain wall winding, governs their unique response to a motion-inducing force. When subjected to an electrical current, the chiral winding of the spin texture leads to a deflection of the skyrmion trajectory, characterized by an angle with respect to the applied force direction. This skyrmion Hall angle was believed to be skyrmion diameter-dependent. In contrast, our experimental study finds that within the plastic flow regime the skyrmion Hall angle is diameter-independent. At …
We experimentally investigate the current-induced magnetization reversal in Pt/[Co/Ni]$_3$/Al multilayers combining the anomalous Hall effect and magneto-optical Kerr effect techniques in crossbar geometry. The magnetization reversal occurs through nucleation and propagation of a domain of opposite polarity for a current density of the order of 0.3 TA/m$^2$. In these experiments we demonstrate a full control of each stage: i)the Ørsted field controls the domain nucleation and ii) domain-wall propagation occurs by spin torque from the Pt spin Hall effect. This scenario requires an in-plane magn…
We compute the physical properties of non-Abelian Fractional Quantum Hall (FQH) states described by Jack polynomials at general filling $ν=\frac{k}{r}$. For $r=2$, these states are identical to the $Z_k$ Read-Rezayi parafermions, whereas for $r>2$ they represent new FQH states. The $r=k+1$ states, multiplied by a Vandermonde determinant, are a non-Abelian alternative construction of states at fermionic filling $2/5, 3/7, 4/9...$. We obtain the thermal Hall coefficient, the quantum dimensions, the electron scaling exponent, and show that the non-Abelian quasihole has a well-defined propagator f…
We propose a simple realization of the three-dimensional (3D) Weyl semimetal phase, utilizing a multilayer structure, composed of identical thin films of a magnetically-doped 3D topological insulator (TI), separated by ordinary-insulator spacer layers. We show that the phase diagram of this system contains a Weyl semimetal phase of the simplest possible kind, with only two Dirac nodes of opposite chirality, separated in momentum space, in its bandstructure. This particular type of Weyl semimetal has a finite anomalous Hall conductivity, chiral edge states, and occurs as an intermediate phase b…
Significant insights into non-Abelian quantum Hall states were obtained from studying special multi-particle interaction Hamiltonians, whose unique ground states are the Moore-Read and Read-Rezayi states for the case of spinless electrons. We generalize this approach to include the electronic spin-1/2 degree of freedom. We demonstrate that in the absence of Zeeman splitting the ground states of such Hamiltonians have large degeneracies and very rich spin structures. The spin structure of the ground states and low-energy excitations can be understood based on an emergent SU(3) symmetry for the …
We formulate a theory of non-Abelian fractional quantum Hall states by considering an anisotropic system consisting of coupled, interacting one dimensional wires. We show that Abelian bosonization provides a simple framework for characterizing the Moore Read state, as well as the more general Read Rezayi sequence. This coupled wire construction provides a solvable Hamiltonian formulated in terms of electronic degrees of freedom, and provides a direct route to characterizing the quasiparticles and edge states in terms of conformal field theory. This construction leads to a simple interpretation…
Fascinating structures have arisen from the study of the fractional quantum Hall effect (FQHE) at the even denominator fraction of $5/2$. We consider the FQHE at another even denominator fraction, namely $ν=2+3/8$, where a well-developed and quantized Hall plateau has been observed in experiments. We examine the non-Abelian state described by the "$\bar{3}\bar{2}^{2}1^{4}$" parton wave function and numerically demonstrate it to be a feasible candidate for the ground state at $ν=2+3/8$. We make predictions for experimentally measurable properties of the $\bar{3}\bar{2}^{2}1^{4}$ state that can …
In this letter we propose an interferometric experiment to detect non-Abelian quasiparticle statistics -- one of the hallmark characteristics of the Moore-Read state expected to describe the observed FQHE plateau at nu=5/2. The implications for using this state for constructing a topologically protected qubit as has been recently proposed by Das Sarma et. al. are also addressed.
We propose a scheme to construct the most prominent Abelian and non-Abelian fractional quantum Hall states from K-component Halperin wave functions. In order to account for a one-component quantum Hall system, these SU(K) colors are distributed over all particles by an appropriate symmetrization. Numerical calculations corroborate the picture that the proposed scheme allows for a unification of both Abelian and non-Abelian trial wave functions in the study of one-component quantum Hall systems.
We propose a mechanism for the quenching of the Shubnikov de Haas oscillations and the quantum Hall effect observed in epitaxial graphene. Experimental data show that the scattering time of the conduction electron is magnetic field dependent and of the order of the cyclotron orbit period, \textit{i.e.} can be much smaller than the zero field scattering time. Our scenario involves the extraordinary graphene $n=0$ Landau level of the uncharged layers that produces a high density of states at the Fermi level. We find that the coupling between this $n=0$ Landau level and the conducting states of t…
Some theories predict that the filling factor 5/2 fractional quantum Hall state can exhibit non-Abelian statistics, which makes it a candidate for fault-tolerant topological quantum computation. Although the non-Abelian Pfaffian state and its particle-hole conjugate, the anti-Pfaffian state, are the most plausible wave functions for the 5/2 state, there are a number of alternatives with either Abelian or non-Abelian statistics. Recent experiments suggest that the tunneling exponents are more consistent with an Abelian state rather than a non-Abelian state. Here, we present edge-current-tunneli…
