Halperin States Research Articles

Gate-Tunable Fractional Chern Insulators in Twisted Double Bilayer Graphene.

We predict twisted double bilayer graphene to be a versatile platform for the realization of fractional Chern insulators readily targeted by tuning the gate potential and the twist angle. Remarkably, these topologically ordered states of matter, including spin singlet Halperin states and spin polarized states in Chern number C=1 and C=2 bands, occur at high temperatures and without the need for an external magnetic field.

Model states for a class of chiral topological order interfaces

Interfaces between topologically distinct phases of matter reveal a remarkably rich phenomenology. To go beyond effective field theories, we study the prototypical example of such an interface between two Abelian states, namely the Laughlin and Halperin states. Using matrix product states, we propose a family of model wavefunctions for the whole system including both bulks and the interface. We show through extensive numerical studies that it unveils both the universal properties of the system, such as the central charge of the gapless interface mode and its microscopic features. It also captures the low energy physics of experimentally relevant Hamiltonians. Our approach can be generalized to other phases described by tensor networks.

Characterization of quasiholes in two-component fractional quantum Hall states and fractional Chern insulators in |C| =2 flat bands

We perform an exact-diagonalization study of quasihole excitations for the two-component Halperin (221) state in the lowest Landau level and for several $\ensuremath{\nu}=1/3$ bosonic fractional Chern insulators in topological flat bands with Chern number $|C|$=2. Properties including the quasihole size, charge, and braiding statistics are evaluated. For the Halperin (221) model state, we observe isotropic quasiholes with a clear internal structure, and obtain the quasihole charge and statistics matching the theoretical values. Interestingly, we also extract the same quasihole size, charge, and braiding statistics for the continuum model states of $|C|$=2 fractional Chern insulators, although the latter possess a ``color-entangled'' nature that does not exist in ordinary two-component Halperin states. We also consider two real lattice models with a band having $|C|$=2. There we find that a quasihole can exhibit much stronger oscillations of the density profile, while having the same charge and statistics as those in the continuum models.

Matrix product state description of Halperin states

Many fractional quantum Hall states can be expressed as a correlator of a given conformal field theory used to describe their edge physics. As a consequence, these states admit an economical representation as an exact Matrix Product States (MPS) that was extensively studied for the systems without any spin or any other internal degrees of freedom. In that case, the correlators are built from a single electronic operator, which is primary with respect to the underlying conformal field theory. We generalize this construction to the archetype of Abelian multicomponent fractional quantum Hall wavefunctions, the Halperin states. These latest can be written as conformal blocks involving multiple electronic operators and we explicitly derive their exact MPS representation. In particular, we deal with the caveat of the full wavefunction symmetry and show that any additional SU(2) symmetry is preserved by the natural MPS truncation scheme provided by the conformal dimension. We use our method to characterize the topological order of the Halperin states by extracting the topological entanglement entropy. We also evaluate their bulk correlation length which are compared to plasma analogy arguments.

Parent Hamiltonians for lattice Halperin states from free-boson conformal field theories

We introduce a family of many-body quantum states that describe interacting spin one-half hard-core particles with bosonic or fermionic statistics on arbitrary one- and two-dimensional lattices. The wave functions at lattice filling fraction ν=2/(2m+1) are derived from deformations of the Wess–Zumino–Witten model su(3)1 and are related to the (m+1,m+1,m) Halperin fractional quantum Hall states. We derive long-range SU(2) invariant parent Hamiltonians for these states which in two dimensions are chiral t–J–V models with additional three-body interaction terms. In one dimension we obtain a generalisation to open chains of a periodic inverse-square t–J–V model proposed in [25]. We observe that the gapless low-energy spectrum of this model and its open-boundary generalisation can be described by rapidity sets with the same generalised Pauli exclusion principle. A two-component compactified free boson conformal field theory is identified as the low-energy effective theory for the periodic inverse-square t–J–V model.

Quantum spin models for the [formula omitted] Wess–Zumino–Witten model

We propose 1D and 2D lattice wave functions constructed from the SU(n)1 Wess–Zumino–Witten (WZW) model and derive their parent Hamiltonians. When all spins in the lattice transform under SU(n) fundamental representations, we obtain a two-body Hamiltonian in 1D, including the SU(n) Haldane–Shastry model as a special case. In 2D, we show that the wave function converges to a class of Halperin's multilayer fractional quantum Hall states and belongs to chiral spin liquids. Our result reveals a hidden SU(n) symmetry for this class of Halperin states. When the spins sit on bipartite lattices with alternating fundamental and conjugate representations, we provide numerical evidence that the state in 1D exhibits quantum criticality deviating from the expected behaviors of the SU(n)1 WZW model, while in 2D they are chiral spin liquids being consistent with the prediction of the SU(n)1 WZW model.

Bloch Model Wave Functions and Pseudopotentials for All Fractional Chern Insulators

We introduce a Bloch-like basis in a C-component lowest Landau level fractional quantum Hall (FQH) effect, which entangles the real and internal degrees of freedom and preserves an N(x)×N(y) full lattice translational symmetry. We implement the Haldane pseudopotential Hamiltonians in this new basis. Their ground states are the model FQH wave functions, and our Bloch basis allows for a mutatis mutandis transcription of these model wave functions to the fractional Chern insulator of arbitrary Chern number C, obtaining wave functions different from all previous proposals. For C>1, our wave functions are related to color-dependent magnetic-flux inserted versions of Halperin and non-Abelian color-singlet states. We then provide large-size numerical results for both the C = 1 and C = 3 cases. This new approach leads to improved overlaps compared to previous proposals. We also discuss the adiabatic continuation from the fractional Chern insulator to the FQH in our Bloch basis, both from the energy and the entanglement spectrum perspectives.

Graphene flakes with defective edge terminations: Universal and topological aspects, and one-dimensional quantum behavior

Systematic tight-binding investigations of the electronic spectra (as a function of the magnetic field) are presented for trigonal graphene nanoflakes with reconstructed zigzag edges, where a succession of pentagons and heptagons, that is 5-7 defects, replaces the hexagons at the zigzag edge. For nanoflakes with such reczag defective edges, emphasis is placed on topological aspects and connections underlying the patterns dominating these spectra. The electronic spectra of trigonal graphene nanoflakes with reczag edge terminations exhibit certain unique features, in addition to those that are well known to appear for graphene dots with zigzag edge termination. These unique features include breaking of the particle-hole symmetry, and they are associated with nonlinear dispersion of the energy as a function of momentum, which may be interpreted as nonrelativistic behavior. The general topological features shared with the zigzag flakes include the appearance of energy gaps at zero and low magnetic fields due to finite size, the formation of relativistic Landau levels at high magnetic fields, and the presence between the Landau levels of edge states (the socalled Halperin states) associated with the integer quantum Hall effect. Topological regimes, unique to the reczag nanoflakes, appear within a stripe of negative energies E_b < E < 0, and along a separate feature forming a constant-energy line outside this stripe. The lower bound (E_b) specifying the energy stripe is independent of size. A main finding concerns the limited applicability of the continuous Dirac-Weyl equation, since the latter does not reproduce the special reczag features. (See also the extended abstract in the paper.)

Structure of spinful quantum Hall states: A squeezing perspective

We provide a set of rules to define several spinful quantum Hall model states. The method extends the one known for spin polarized states. It is achieved by specifying an undressed root partition, a squeezing procedure and rules to dress the configurations with spin. It applies to both the excitation-less state and the quasihole states. In particular, we show that the naive generalization where one preserves the spin information during the squeezing sequence, may fail. We give numerous examples such as the Halperin states, the non-abelian spin-singlet states or the spin-charge separated states. The squeezing procedure for the series (k=2,r) of spinless quantum Hall states, which vanish as r powers when k+1 particles coincide, is generalized to the spinful case. As an application of our method, we show that the counting observed in the particle entanglement spectrum of several spinful states matches the one obtained through the root partitions and our rules. This counting also matches the counting of quasihole states of the corresponding model Hamiltonians, when the latter is available.

Bilayer coherent and quantum Hall phases: duality and quantum disorder.

We consider a fully spin-polarized quantum Hall system with no interlayer tunneling at total filling factor nu = 1/k (where k is an odd integer) using the Chern-Simons-Ginzburg-Landau theory. Exploiting particle-vortex duality and the concept of quantum disordering, we find a large number of possible compressible and incompressible ground states, some of which may have relevance to recent experiments of Spielman et al. [Phys. Rev. Lett. 84, 5808 (2000)]. We find interlayer coherent compressible states without Hall quantization and interlayer incoherent incompressible states with Hall quantization in addition to the usual (k,k,k) Halperin states, which are both interlayer coherent and incompressible.

Edge dynamics in quantum Hall bilayers: Exact results with disorder and parallel fields

We study edge dynamics in the presence of interlayer tunneling, parallel magnetic field, and various types of disorder for two infinite sequences of quantum Hall states in symmetric bilayers. These sequences begin with the 110 and 331 Halperin states and include their fractional descendants at lower filling factors; the former is easily realized experimentally while the latter is a candidate for the experimentally observed quantum Hall state at a total filling factor of 1/2 in bilayers. We discuss the experimentally interesting observables that involve just one chiral edge of the sample and the correlation functions needed for computing them. We present several methods for obtaining exact results in the presence of interactions and disorder which rely on the chiral character of the system. Of particular interest are our results on the 331 state which suggest that a time-resolved measurement at the edge can be used to discriminate between the 331 and Pfaffian scenarios for the observed quantum Hall state at filling factor 1/2 in realistic double-layer systems.

Tunneling in paired fractional quantum Hall states: Conductance and Andreev reflection of non-abelions

We study the edge transport properties of paired fractional quantum Hall (FQH) states — the Haldane-Rezayi (HR), Moore-Read (Pfaffian) and Halperin (3 3 1) states. The table of exponents is given for the tunneling between the edges of paired FQH states in gated 2D structures and the tunneling into the edge of FQH states from a normal Fermi liquid (N). It is found that HR, Pfaffian and 3 3 1 states have different exponents for quasiparticle tunneling. For the tunneling through a FQH-N junction, we propose unusual Andreev reflection processes that may also probe the non-abelian FQH states.

Josephson Effects in Double-Layer Quantum Hall States

Under quite plausible assumptions on double-layer quantum Hall states with strong interlayer correlation, we show in general framework that coherent tunneling of a single electron between two layers is possible. It yields Josephson effects with unit charge tunneling. The origin is that Halperin states in the quantum Hall states are highly degenerate in electron number difference between two layers in the absence of electrons tunneling.

Exactly soluble model of multi-species anyons and the braid group on a torus

Exactly soluble models of multi-species anyons are studied on a torus and ground-state wave functions are explicitly obtained. These models describe quasi-excitations of recently discovered fractional quantum Hall states in double-layer electron systems. The representation of the braid group on a torus by the multi-species anyons is examined, and its relationship with ground-state wave functions is clarified. The braid group on a torus requires that the anyon wave function must be multi-component. However, the gauge-covariant wave functions are single-component in the present formalism. We find that zero modes of the Chern-Simons (CS) gauge fields play an essentially important role to link the anyon wave function and the braid group on a torus. Elements of the braid group of noncontractible cycles on a torus induce a “fractional” large gauge transformation of the zero modes, and the canonical commutation relation of the CS zero modes induces the braid-group algebra on a torus. The quantum Hall state in anyon systems is also discussed by comparing the anyon wave functions with the Laughlin and Halperin states.

Spinning braid-group representation and the fractional quantum Hall effect

The path-integral approach to representing braid group is generalized for particles with spin. Introducing the notion of charged winding number in the super-plane, we represent the braid-group generators as homotopically constrained Feynman kernels. In this framework, super Knizhnik-Zamolodchikov operators appear naturally in the hamiltonian, suggesting the possibility of spinning nonabelian anyons. We then apply our formulation to the study of fractional quantum Hall effect (FQHE). A systematic discussion of the ground states and their quasi-hole excitations is given. We obtain Laughlin, Halperin and Moore-Read states as exact ground-state solutions to the respective hamiltonians associated to the braid-group representations. The energy gap of the quasi-excitation is also obtainable from this approach.