Bilayer Fractional Quantum Hall Research Articles

Principal component analysis for fractional quantum Hall states in bilayer systems

We initiate an unsupervised machine learning (ML) study with the principal component analysis (PCA) for two example bilayer fractional quantum Hall (FQH) systems at filling factors ν=4/5 and 2/3 where the interlayer tunneling effect and Coulomb interaction are considered. Some common features in PCA are exploited to recognize the transition and boundary between two competing ground state (GS) phases even without a complete softening in the spectrum. We also provide the numerical evidence to confirm a bilayer system as an analogy to its single-layer counterpart at large interlayer tunneling limit. The general approaches with PCA have been applied to determine the collapsing boundaries for phases at the strongly-correlated bilayer limit, the decoupled bilayer limit, and the strong interlayer tunneling limit, which lead to the qualitatively similar phase diagrams as previous numerical studies but do not rely on explicit knowledge of the states. Thus, our PCA study is extendable for other unfamiliar bilayer systems.

Quench Dynamics of Collective Modes in Fractional Quantum Hall Bilayers.

We introduce different types of quenches to probe the nonequilibrium dynamics and multiple collective modes of bilayer fractional quantum Hall states. We show that applying an electric field in one layer induces oscillations of a spin-1degree of freedom, whose frequency matches the long-wavelength limit of the dipole mode. On the other hand, oscillations of the long-wavelength limit of the quadrupole mode, i.e., the spin-2 graviton, as well as the combination of two spin-1 states, can be activated by a sudden change of band mass anisotropy. We construct an effective field theory to describe the quench dynamics of these collective modes. In particular, we derive the dynamics for both the spin-2 and the spin-1 states and demonstrate their excellent agreement with numerics.

Numerical investigation of gapped edge states in fractional quantum Hall-superconductor heterostructures

Fractional quantum Hall-superconductor heterostructures may provide a platform towards non-abelian topological modes beyond Majoranas. However their quantitative theoretical study remains extremely challenging. We propose and implement a numerical setup for studying edge states of fractional quantum Hall droplets with a superconducting instability. The fully gapped edges carry a topological degree of freedom that can encode quantum information protected against local perturbations. We simulate such a system numerically using exact diagonalization by restricting the calculation to the quasihole-subspace of a (time-reversal symmetric) bilayer fractional quantum Hall system of Laughlin ν = 1/3 states. We show that the edge ground states are permuted by spin-dependent flux insertion and demonstrate their fractional 6π Josephson effect, evidencing their topological nature and the Cooper pairing of fractionalized quasiparticles. The versatility and efficiency of our setup make it a well suited method to tackle wider questions of edge phases and phase transitions in fractional quantum Hall systems.

Numeric and Analytic Investigation on Phase Diagrams and Phasetransitions of the ν = 2/3 Bilayer Fractional Quantum Hall Systems

The phase diagrams and phase transitions of a typical bilayer fractional quantum Hall (QH) system with filling factor ν = 2/3 at the layer balanced point are investigated theoretically by finite size exact-diagonalization calculations and an exactly solvable model. We find some basic features essentially different from the bilayer integer QH systems at ν = 2, reflecting the special characteristics of the fractional QH systems. The degeneracy of the ground states occurs depending on the difference between intralayer and interlayer Coulomb energies, when interlayer tunneling energy (ΔSAS) gets close to zero. The continuous transitions of the finite size systems between the spin-polarized and spin-unpolarized phases are determined by the competition between the Zeeman energy (ΔZ) and the electron Coulomb energy, and are almost not affected by ΔSAS.

Universal Quantum Computation with Gapped Boundaries.

This Letter discusses topological quantum computation with gapped boundaries of two-dimensional topological phases. Systematic methods are presented to encode quantum information topologically using gapped boundaries, and to perform topologically protected operations on this encoding. In particular, we introduce a new and general computational primitive of topological charge measurement and present a symmetry-protected implementation of this primitive. Throughout the Letter, a concrete physical example, the Z_{3} toric code [D(Z_{3})], is discussed. For this example, we have a qutrit encoding and an abstract universal gate set. Physically, gapped boundaries of D(Z_{3}) can be realized in bilayer fractional quantum Hall 1/3 systems. If a practical implementation is found for the required topological charge measurement, these boundaries will give rise to a direct physical realization of a universal quantum computer based on a purely Abelian topological phase.

Tunnel transport and interlayer excitons in bilayer fractional quantum Hall systems

In a bilayer system consisting of a composite-fermion Fermi sea in each layer, the tunnel current is exponentially suppressed at zero bias, followed by a strong peak at a finite bias voltage $V_{\rm max}$. This behavior, which is qualitatively different from that observed for the electron Fermi sea, provides fundamental insight into the strongly correlated non-Fermi liquid nature of the CF Fermi sea and, in particular, offers a window into the short-distance high-energy physics of this state. We identify the exciton responsible for the peak current and provide a quantitative account of the value of $V_{\rm max}$. The excitonic attraction is shown to be quantitatively significant, and its variation accounts for the increase of $V_{\rm max}$ with the application of an in-plane magnetic field. We also estimate the critical Zeeman energy where transition occurs from a fully spin polarized composite fermion Fermi sea to a partially spin polarized one, carefully incorporating corrections due to finite width and Landau level mixing, and find it to be in satisfactory agreement with the Zeeman energy where a qualitative change has been observed for the onset bias voltage [Eisenstein et al., Phys. Rev. B 94, 125409 (2016)]. For fractional quantum Hall states, we predict a substantial discontinuous jump in $V_{\rm max}$ when the system undergoes a transition from a fully spin polarized state to a spin singlet or a partially spin polarized state.

Bilayer fractional quantum Hall states with dipoles

Using the example of dysprosium atoms in an optical lattice, we show how dipolar interactions between magnetic dipoles can be used to obtain fractional quantum Hall states. In our approach, dysprosium atoms are trapped one atom per site in a deep optical lattice with negligible tunneling. Microwave and spatially dependent optical dressing fields are used to define an effective spin-$\frac{1}{2}$ or spin-1 degree of freedom in each atom. Thinking of spin-$\frac{1}{2}$ particles as hard-core bosons, dipole-dipole interactions give rise to boson hopping, topological flat bands with Chern number 1, and the $\ensuremath{\nu}=\frac{1}{2}$ Laughlin state. Thinking of spin-1 particles as two-component hard-core bosons, dipole-dipole interactions again give rise to boson hopping, topological flat bands with Chern number 2, and the bilayer Halperin (2,2,1) state. By adjusting the optical fields, we find a phase diagram, in which the (2,2,1) state competes with superfluidity. Generalizations to solid-state magnetic dipoles are discussed.

Fibonacci anyons from Abelian bilayer quantum Hall states.

The possibility of realizing non-Abelian statistics and utilizing it for topological quantum computation (TQC) has generated widespread interest. However, the non-Abelian statistics that can be realized in most accessible proposals is not powerful enough for universal TQC. In this Letter, we consider a simple bilayer fractional quantum Hall system with the 1/3 Laughlin state in each layer. We show that interlayer tunneling can drive a transition to an exotic non-Abelian state that contains the famous "Fibonacci" anyon, whose non-Abelian statistics is powerful enough for universal TQC. Our analysis rests on startling agreements from a variety of distinct methods, including thin torus limits, effective field theories, and coupled wire constructions. We provide evidence that the transition can be continuous, at which point the charge gap remains open while the neutral gap closes. This raises the question of whether these exotic phases may have already been realized at ν=2/3 in bilayers, as past experiments may not have definitively ruled them out.

Synthetic Topological Qubits in Conventional Bilayer Quantum Hall Systems

The idea of topological quantum computation is to build powerful and robust quantum computers with certain macroscopic quantum states of matter called topologically ordered states. These systems have degenerate ground states that can be used as robust "topological qubits" to store and process quantum information. However, a topological qubit has not been realized since the proposed systems either require sophisticated topologically ordered states that are not available yet, or require complicated geometries that are too difficult to realize. In this paper, we propose a new experimental setup which can realize topological qubits in a simple bilayer fractional quantum Hall (FQH) system with proper electric gate configurations. Compared to previous works, our proposal is accessible with current experimental techniques and only involves well-established topological states. Our system can realize a large class of topological qubits, generalizing the Majorana zero modes studied in the recent literature to more computationally powerful possibilities. We propose three tunneling and interferometry experiments to detect the existence and non-local topological properties of the topological qubits.

Braiding statistics and congruent invariance of twist defects in bosonic bilayer fractional quantum Hall states

We describe the braiding statistics of topological twist defects in fractional quantum Hall (FQH) states. In particular, we study Abelian bosonic $(mmn)$ FQH states with bilayer symmetries. Twist defects are staircases that lift orbiting quasiparticles from one layer to another. These point defects exhibit semiclassical non-Abelian characteristics. We develop a procedure to evaluate consistent basis transformations ($F$ symbols) of the defect fusion theory. Unlike quantum anyonic excitations of a topological phase, the exchange and braiding statistics of twist defects is not characterized by modular transformations, but instead a subcollection known as congruent transformations. We also characterize the projective nature of unitary braiding operations and relate it to the global ${\mathbb{Z}}_{2}$ bilayer symmetry.

Lattice construction of pseudopotential Hamiltonians for fractional Chern insulators

Fractional Chern insulators are new realizations of fractional quantum Hall states in lattice systems without orbital magnetic field. These states can be mapped onto conventional fractional quantum Hall states through the Wannier state representation (Phys. Rev. Lett. 107, 126803 (2011)). In this paper, we use the Wannier state representation to construct the pseudopotential Hamiltonians for fractional Chern insulators, which are interaction Hamiltonians with certain ideal model wavefunctions as exact ground states. We show that these pseudopotential Hamiltonians can be approximated by short-ranged interactions in fractional Chern insulators, and that their range will be minimized by an optimal gauge choice for the Wannier states. As illustrative examples, we explicitly write down the form of the lowest pseudopotential for several fractional Chern insulator models including the lattice Dirac model and the checkerboard model with Chern number 1, and the d-wave model and the triangular lattice model with Chern number 2. The proposed pseudopotential Hamiltonians have the 1/3 Laughlin state as their groundstate when the Chern number C=1, and a topological nematic (330) state as their groundstate when C=2. Also included are the results of an interpolation between the d-wave model and two decoupled layers of lattice Dirac models, which explicitly demonstrate the relation between C=2 fractional Chern insulators and bilayer fractional quantum Hall states. The proposed states can be verified by future numerical works, and in particular provide a model Hamiltonian for the topological nematic states that have not been realized numerically.

Coherent states on the Grassmannian U(4)/U(2)2: oscillator realization and bilayer fractional quantum Hall systems

Bilayer quantum Hall (BLQH) systems, which underlie a U(4) symmetry, display unique quantum coherence effects. We study coherent states (CS) on the complex Grassmannian , orthonormal basis, U(4) generators and their matrix elements in the reproducing kernel Hilbert space of analytic square-integrable holomorphic functions on , which carries a unitary irreducible representation of U(4) with index . A many-body representation of the previous construction is introduced through an oscillator realization of the U(4) Lie algebra generators in terms of eight boson operators. This particle picture allows us to make a physical interpretation of our abstract mathematical construction in the BLQH jargon. In particular, the index λ is related to the number of flux quanta bound to a bi-fermion in the composite fermion picture of Jain for fractions of the filling factor ν = 2. The simpler, and better known, case of spin-s CS on the Riemann–Bloch sphere is also treated in parallel, of which Grassmannian -CS can be regarded as a generalized (matrix) version.

Theoretical approach to ground states of the ν=2/3 bilayer fractional quantum Hall systems

The ground states and phase diagrams of the ν=2/3 bilayer fractional quantum Hall systems at the layer balanced point are investigated theoretically by finite size exact-diagonalization calculations and an exactly solvable model. The degeneracy of the ground states occurs depending on the difference between intralayer and interlayer Coulomb energies, when interlayer tunneling energy (ΔSAS) gets close to zero. The transitions between the spin-polarized and spin-unpolarized phases are mainly determined by the competition between the Zeeman energy (ΔZ) and the electron Coulomb energy. The phase diagrams in ΔSAS–ΔZ planes are given for the first time. We also compare the numerical results with the experimental data and acquire a good agreement.

Bilayer quantum Hall phase transitions and the orbifold non-Abelian fractional quantum Hall states

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 Hamiltonians associated with these orbifold states do not in general correspond to incompressible phases, but instead lie at a nearby critical point. We discuss this unusual situation from the perspective of the pattern of zeros/vertex algebra frameworks and discuss implications for the conceptual foundations of these approaches. Due to the proximity in the phase diagram of these non-Abelian states to the (ppq) bilayer states, they may be experimentally relevant, both as candidates for describing the plateaus in single-layer systems at filling fraction 8/3 and 12/5, and as a way to tune to non-Abelian states in double-layer or wide quantum wells.

Excitation properties ofν=2/3bilayer quantum Hall phases investigated by magnetotransport methods

We report on novel excitation properties in the \ensuremath{\nu} $=$ 2/3 bilayer quantum Hall (QH) phases by carrying out magnetotransport measurements using a GaAs$/$AlGaAs double-quantum-well sample. We refer to the areas with different particular excitation properties in a single phase as phase internal areas (pi-areas). The differences of the excited states between the pi-areas in the high- and low-magnetic-field region of the same phase lead to different behavior of thermal activation energy \ensuremath{\Delta} under a tilted magnetic field. A new pi-area appears between the balanced point and the single-layer region when \ensuremath{\Delta}${}_{\mathrm{SAS}}$ is small. A simple theoretical model based on the composite-fermion picture is advanced for bilayer fractional QH systems to interpret the characteristics of these pi-areas. Finite-size exact-diagonalization calculations are also executed to verify the validity of the theoretical model.

Activation energy gap of the layer-imbalanced bilayer [formula omitted] quantum Hall states

We investigate the bilayer fractional quantum Hall state (FQHS) in layer-imbalanced configurations at the total Landau level filling factor ν = 1 3 , which is interpreted as the bilayer ν = 1 integer quantum Hall state (IQHS) of composite fermions (CFs). We carried out detailed magnetotransport experiments, especially focused on measurements of the activation energy gap Δ in the bilayer ν = 1 3 FQHS under tilted magnetic fields. We observed Δ as a function of in-plane magnetic field for various layer density imbalance parameters σ , which is compared with the reported data of the bilayer ν = 1 IQHS. At small σ , the FQHSs collapse by applying large in-plane fields. This behavior is interpreted as the transition from the commensurate (C)-like phase to no-FQHS. At large σ , Δ stays finite even in large in-plane field, which means the incommensurate (IC)-like phase appears also in the ν = 1 3 layer-imbalanced bilayer FQHS. We discuss the interactions among CFs from the phase transition point between the C-like and the IC-like phases in the layer-imbalanced ν = 1 3 FQHS.

Activation study of the bilayer [formula omitted] quantum Hall state

We investigated the bilayer fractional quantum Hall state (FQHS) at total Landau level filling factor ν = 1 3 , which is interpreted as the bilayer ν = 1 integer quantum Hall state (IQHS) of composite fermions. We carried out detailed magnetotransport experiments of the bilayer ν = 1 3 FQHS in tilted magnetic field. The data are compared with reported data of the bilayer ν = 1 IQHS. We found that the commensurate - incommensurate phase transition, which occurs in the bilayer ν = 1 IQHS, is missing in the bilayer ν = 1 3 FQHS.

High-Field Fractional Quantum Hall Effect in Optical Lattices

We consider interacting bosonic atoms in an optical lattice subject to a large simulated magnetic field. We develop a model similar to a bilayer fractional quantum Hall system valid near simple rational numbers of magnetic flux quanta per lattice cell. Then we calculate its ground state, magnetic lengths, fractional fillings, and find unexpected sign changes in the Hall current. Finally we study methods for detecting these novel features via shot noise and Hall current measurements.

Edge state transport of separately contacted bilayer systems in the fractional quantum Hall regime

Hall and diagonal resistances of bilayer fractional quantum Hall systems are discussed theoretically. The bilayers have electrodes attached separately to each layer. They are assumed to be coupled weakly by interlayer tunneling, while the interlayer Coulomb interaction is negligibly small. It is shown that source–drain voltage dependence of the resistances reflects the Luttinger liquid parameter of the edge state.