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Physical review. B./Physical review. B

Role of the defect-core orbital in fractional Chern insulators

Ai‐Lei He

Published Mar 22, 2021• 8 citations
Publisher

Abstract

Fractional Chern insulator (FCI) states have been proposed in singular lattices with arbitrary $n$-fold rotational symmetry. Particles occupy non-defect-core orbitals in previously reported FCIs. However, the role of defect-core orbitals in the singular-lattice FCI states has seldom been discussed. In this paper, we report a series of singular-lattice FCI states with one particle occupying the defect-core orbital. This singular-lattice FCI state can be identified with the aid of the edge excitations, the angular momentum of the ground state (GS), and the optimal trial wave functions. Edge excitations and GS angular momentum are obtained using exact diagonalization. Optimal trial wave functions are constructed on the basis of the generalized Pauli principle, the Jack polynomials, and single-particle states. The present FCI state is identified as a $\ensuremath{\nu}=1/2$ FCI with high values of wave function overlap, where one particle occupies the low-energy defect-core orbital. Our studies reveal the intriguing interplay between the singular-lattice FCI and the defect-core orbital and provide an effective approach for future researches on FCIs with various defects or disorders.

DOI
10.1103/physrevb.103.115138

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Fractional Chern insulator (FCI) states have been proposed in singular lattices with arbitrary nnn-fold rotational symmetry. Particles occupy non-defect-core orbitals in previously reported FCIs. However, the role of defect-core orbitals in the singular-lattice FCI states has seldom been discussed. In this paper, we report a series of singular-lattice FCI states with one particle occupying the defect-core orbital. This singular-lattice FCI state can be identified with the aid of the edge excitations, the angular momentum of the ground state (GS), and the optimal trial wave functions. Edge excitations and GS angular momentum are obtained using exact diagonalization. Optimal trial wave functions are constructed on the basis of the generalized Pauli principle, the Jack polynomials, and single-particle states. The present FCI state is identified as a \ensuremathν=1/2\ensuremath{\nu}=1/2\ensuremathν=1/2 FCI with high values of wave function overlap, where one particle occupies the low-energy defect-core orbital. Our studies reveal the intriguing interplay between the singular-lattice FCI and the defect-core orbital and provide an effective approach for future researches on FCIs with various defects or disorders.
Non-Abelian (NA) fractional topological states with quasiparticles obeying NA braiding statistics have attracted intensive attention for both their fundamental nature and the prospect for topological quantum computation. To date, there are many models proposed to realize the NA fractional topological states, such as the well-known Moore-Read quantum Hall states and the non-Abelian fractional Chern insulators (NA-FCIs). Here, we investigate the NA-FCI in disk geometry with three-body hard-core bosons loaded into a topological flat band. This stable $\ensuremath{\nu}=1$ bosonic NA-FCI is characterized by edge excitations and the ground-state angular momentum. Based on the generalized Pauli principle and the Jack polynomials, we successfully construct a trial wave function for the NA-FCI. Moreover, a $\ensuremath{\nu}=1/2$ Abelian FCI state emerges with the increase of the on-site interaction and it can be identified with the help of the trial wave function as well. Our findings not only lead to an optimal wave function for the NA-FCI, but also directly provide an effective approach for future researches on paired topological states.
Fractional Chern insulators (FCIs) have attracted intensive attention for the realization of fractional quantum Hall states in the absence of an external magnetic field. Most FCIs have been proposed on two-dimensional (2D) Euclidean lattice models with various boundary conditions. In this work, we investigate hyperbolic FCIs which are constructed in hyperbolic geometry with constant negative curvature. Through the studies on hyperbolic analogs of kagome lattices with hard-core bosons loaded into topological flat bands, we find convincing numerical evidences of two types of $\ensuremath{\nu}=1/2$ FCI states, i.e., conventional and unconventional FCIs. Multiple branches of edge excitations and geometry-dependent wave functions for both conventional and unconventional $\ensuremath{\nu}=1/2$ FCI states are revealed. Intriguingly, the geometric degree of freedom plays various roles for these two FCIs. Additionally, a center-localized orbital plays a crucial role in the unconventional FCI state.
We propose an improved scheme to construct many-body trial wave functions for fractional Chern insulators (FCI), using one-dimensional localized Wannier basis. The procedure borrows from the original scheme on a continuum cylinder, but is adapted to finite-size lattice systems with periodic boundaries. It fixes several issues of the continuum description that made the overlap with the exact ground states insignificant. The constructed lattice states are translationally invariant, and have the correct degeneracy as well as the correct relative and total momenta. Our prescription preserves the (possible) inversion symmetry of the lattice model, and is isotropic in the limit of flat Berry curvature. By relaxing the maximally localized hybrid Wannier orbital prescription, we can form an orthonormal basis of states which, upon gauge fixing, can be used in lieu of the Landau orbitals. We find that the exact ground states of several known FCI models at $\ensuremath{\nu}=1/3$ filling are well captured by the lattice states constructed from the Laughlin wave function. The overlap is higher than 0.990.990.99 in some models when the Hilbert space dimension is as large as $3\ifmmode\times\else\texttimes\fi{}{10}^{4}$ in each total momentum sector.
Recently, fractional Chern insulators (FCIs), also called fractional quantum anomalous Hall (FQAH) states, have been theoretically established in lattice systems with topological flat bands. These systems exhibit similar fractionalization phenomena to the conventional fractional quantum Hall (FQH) systems. Using the mapping relationship between the FQH states and the FCI/FQAH states, we construct the many-body wave functions of the fermionic FCI/FQAH states in disk geometry with the aid of the generalized Pauli principle (GPP) and Jack polynomials. Compared with the ground state by the exact diagonalization method, the wave-function overlap is higher than 0.97, even when the Hilbert space dimension is as large as 3 × 10 6 . We also use the GPP and the Jack polynomials to construct edge excitations for the fermionic FCI/FQAH states. The quasi-degeneracy sequences of fermionic FCI/FQAH systems reproduce the prediction of the chiral Luttinger liquid theory, complementing the exact diagonalization results with larger lattice sizes and more particles.
We investigate a fractional Chern insulator (FCI) candidate arising from moir\'e bands with higher Chern number C=2C=2C=2 on a magic-angle twisted bilayer checkerboard lattice (MATBCB). There are two nearly flat low lying bands in the single-particle energy spectrum under the first magic angle $\ensuremath{\phi}\ensuremath{\approx}1.{608}^{\ensuremath{\circ}}$ and chiral limit. We find that MATBCB hosts a nearly uniform Berry curvature distribution and exhibits tiny violation of quantum geometric trace condition in the first moir\'e Brillouin zone (mBZ), indicating that there is a nearly ideal quantum geometry in MATBCB in single-particle level. Turning on projected Coulomb interactions, we perform exact diagonalization and find a 10-fold ground-state quasidegeneracy in many-body energy spectrum with filling fraction $\ensuremath{\nu}=1/5$. The 10-fold quasidegenerate ground states further show spectra flow under flux pumping. By diagnosing the particle entanglement spectrum (PES) of the ground states, we obtain a clear PES gap and quasihole state counting consistent with Halperin spin singlet generalized Pauli principle, suggesting that a C=2C=2C=2 Abelian fractional Chern insulator is realized in this system.
A flat band with Chern number C = 0 , well isolated from the rest of Hilbert space by a gap much larger than interaction strength, is an unexpected context for fractional quantum Hall physics. In this work, we demonstrate the emergence of the fractional Chern insulator (FCI) states in such a trivial flat band using large-scale exact diagonalization and infinite density matrix renormalization group simulations. Hosted by an anisotropic lattice, this band has a sharp peak for the quantum geometry. At 1/3 hole filling, we find the unexpected FCI with 3-fold ground-state degeneracy and σ H = − 1 / 3 ( e 2 / h ) . The momentum-space carrier distribution shows that the quantum geometry peak tends to push the interacting holes away, toward other regions of the Brillouin zone with the nearly "ideal" quantum geometry, underlying the formation of the FCI in the C = 0 flat band. In comparison, we find only trivial ground states when the quantum geometry peak becomes less avoidable. Our work not only shows that FCl states could exist without the single-particle topology, relying on the more intrinsic quantum geometry, but also suggests a possibly generic scenario to explore FCIs in otherwise trivial flat bands.