We investigate the nature of the fractional quantum Hall (FQH) state at filling factor $ u=13/5$, and its particle-hole conjugate state at $12/5$, with the Coulomb interaction, and address the issue of possible competing states. Based on a large-scale density-matrix renormalization group (DMRG) calculation in spherical geometry, we present evidence that the physics of the Coulomb ground state (GS) at $ u=13/5$ and $12/5$ is captured by the $k=3$ parafermion Read-Rezayi RR state, $text{RR}_3$. We first establish that the state at $ u=13/5$ is an incompressible FQH state, with a GS protected by a finite excitation gap, with the shift in accordance with the RR state. Then, by performing a finite-size scaling analysis of the GS energies for $ u=12/5$ with different shifts, we find that the $text{RR}_3$ state has the lowest energy among different competing states in the thermodynamic limit. We find the fingerprint of $text{RR}_3$ topological order in the FQH $13/5$ and $12/5$ states, based on their entanglement spectrum and topological entanglement entropy, both of which strongly support their identification with the $text{RR}_3$ state. Furthermore, by considering the shift-free infinite-cylinder geometry, we expose two topologically-distinct GS sectors, one identity sector and a second one matching the non-Abelian sector of the Fibonacci anyonic quasiparticle, which serves as additional evidence for the $text{RR}_3$ state at $13/5$ and $12/5$.
The non-Abelian topological order has attracted a lot of attention for its fundamental importance and exciting prospect of topological quantum computation. However, explicit demonstration or identification of the non-Abelian states and the associated statistics in a microscopic model is very challenging. Here, based on density-matrix renormalization group calculation, we provide a complete characterization of the universal properties of bosonic Moore-Read state on Haldane honeycomb lattice model at filling number $ u=1$ for larger systems, including both the edge spectrum and the bulk anyonic quasiparticle (QP) statistics. We first demonstrate that there are three degenerating ground states, for each of which there is a definite anyonic flux threading through the cylinder. We identify the nontrivial countings for the entanglement spectrum in accordance with the corresponding conformal field theory. Through inserting the $U(1)$ charge flux, it is found that two of the ground states can be adiabatically connected through a fermionic charge-$textit{e}$ QP being pumped from one edge to the other, while the ground state in Ising anyon sector evolves back to itself. Furthermore, we calculate the modular matrices $mathcal{S}$ and $mathcal{U}$, which contain all the information for the anyonic QPs. In particular, the extracted quantum dimensions, fusion rule and topological spins from modular matrices positively identify the emergence of non-Abelian statistics following the $SU(2)_2$ Chern-Simons theory.
The discontinuity of guiding-center Hall viscosity (a bulk property) at edges of incompressible quantum Hall fluids is associated with the presence of an intrinsic electric dipole moment on the edge. If there is a gradient of drift velocity due to a non-uniform electric field, the discontinuity in the induced stress is exactly balanced by the electric force on the dipole. The total Hall viscosity has two distinct contributions: a trivial contribution associated with the geometry of the Landau orbits, and a non-trivial contribution associated with guiding-center correlations. We describe a relation between the guiding-center edge-dipole moment and momentum polarization, which relates the guiding-center part of the bulk Hall viscosity to the orbital entanglement spectrum(OES). We observe that using the computationally-more-onerous real-space entanglement spectrum (RES) just adds the trivial Landau-orbit contribution to the guiding-center part. This shows that all the non-trivial information is completely contained in the OES, which also exposes a fundamental topological quantity $gamma$ = $tilde c- u$, the difference between the chiral stress-energy anomaly (or signed conformal anomaly) and the chiral charge anomaly. This quantity characterizes correlated fractional quantum Hall fluids, and vanishes in uncorrelated integer quantum Hall fluids.
The role of Fermi arc surface-quasiparticle states in topological metals (where some Fermi surface sheets have non-zero Chern number) is examined. They act as Fermi-level plumbing conduits that transfer quasiparticles among groups of apparently-disconnected Fermi sheets with non-zero Chern numbers to maintain equality of their chemical potentials, which is required by gauge invariance. Fermi arcs have a chiral tangential attachment to the surface projections of sheets of the bulk Fermi Surface: the total Chern number of each projection equals the net chirality of arc-attachments to it. Information from the Fermi arcs is needed to unambiguously determine the quantized part of the anomalous Hall effect that is not determined at the bulk Fermi surface.
The topological order is equivalent to the pattern of long-range quantum entanglements, which cannot be measured by any local observable. Here we perform an exact diagonalization study to establish the non-Abelian topological order through entanglement entropy measurement. We focus on the quasiparticle statistics of the non-Abelian Moore-Read and Read-Rezayi states on the lattice boson models. We identify multiple independent minimal entangled states (MESs) in the groundstate manifold on a torus. The extracted modular $mathcal{S}$ matrix from MESs faithfully demonstrates the Majorana quasiparticle or Fibonacci quasiparticle statistics, including the quasiparticle quantum dimensions and the fusion rules for such systems. These findings support that MESs manifest the eigenstates of quasiparticles for the non-Abelian topological states and encode the full information of the topological order.
We perform an exact diagonalization study of the topological order in topological flat band models through calculating entanglement entropy and spectra of low energy states. We identify multiple independent minimal entangled states, which form a set of orthogonal basis states for the ground-state manifold. We extract the modular transformation matrices S (U) which contains the information of mutual (self) statistics, quantum dimensions and fusion rule of quasi-particles. Moreover, we demonstrate that these matrices are robust and universal in the whole topological phase against different perturbations until the quantum phase transition takes place.
We study the nature of the u=5/2 quantum Hall state in wide quantum wells under the mixing of electronic subbands and Landau levels. We introduce a general method to analyze the Moore-Read Pfaffian state and its particle-hole conjugate, the anti-Pfaffian, under periodic boundary conditions in a quartered Brillouin zone scheme containing both even and odd numbers of electrons. We examine the rotational quantum numbers on the torus, and show spontaneous breaking of the particle-hole symmetry can be observed in finite-size systems. In the presence of electronic-subband and Landau-level mixing the particle-hole symmetry is broken in such a way that the anti-Pfaffian is unambiguously favored, and becomes more robust in the vicinity of a transition to the compressible phase, in agreement with recent experiments.
Model quantum Hall states including Laughlin, Moore-Read and Read-Rezayi states are generalized into appropriate anisotropic form. The generalized states are exact zero-energy eigenstates of corresponding anisotropic two- or multi-body Hamiltonians, and explicitly illustrate the existence of geometric degrees of in the fractional quantum Hall effect. These generalized model quantum Hall states can provide a good description of the quantum Hall system with anisotropic interactions. Some numeric results of these anisotropic quantum Hall states are also presented.
