The stability of the $\ensuremath{\nu}=1/3$ fractional Chern insulator (FCI) phase is analyzed on the example of a checkerboard lattice undergoing a transition into a Lieb lattice. The transition is performed by the addition of a second sublattice, whose coupling to the checkerboard sites is controlled by sublattice staggered potential. We investigate the influence of these sites on the many-body energy gap between three lowest energy states and the fourth state. We consider cases with different complex phases acquired in hopping and a model with a flattened topologically nontrivial band. We find that an interaction with the additional sites either open the single-particle gap or enlarge the existing one, which translates into similar effect on the many-particle gap. By looking at Berry curvature flatness we notice its strong correlation with the magnitude of the many-body energy gap, suggesting that the main mechanism of the FCI stabilization by additional atoms is via their influence on the Berry curvature. Evidence of the FCI phase for a region in a parameter space with larger energy gap is shown by looking at momenta of the threefold degenerate ground state, spectral flow, and quasihole excitation spectrum.
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.
We investigate topological phase transitions in Chern insulators within three-band models, focusing on the empty band and lowest band populated by spinless fermions. We consider Lieb and kagome lattices and notice phase transitions driven by the hopping integral between nearest-neighbors, which leads to the change of the lowest band Chern number from
C=1 to
C=−1. In the single-particle picture, different phases are examined by investigating corresponding entanglement spectra and the evolution of the entanglement entropy. Entanglement spectra reveal the spectral flow characteristic of topologically nontrivial systems before and after phase transitions. For the lowest band with
1/3 filling, Fractional Chern insulator (FCI) phases are identified by examining the ground state momenta, the spectral flow, and counting of the entanglement energy levels below the gap in the entanglement spectrum. A quasilinear dependence of the entanglement entropy
α(nA) term is observed for FCI phase, similarly to linear behavior expected for Laughlin phase in the fractional quantum Hall effect. At the topological phase transition, for both empty and partially filled lowest bands, the energy gap closure and a discontinuity in the entanglement entropy are observed. This coincides with the divergence of a standard deviation of the Berry curvature. We notice also a phase transition driven by the nearest-neighbor interaction in the Lieb lattice, where the many-body energy gap closes. The phase transitions are shown to be stable for an arbitrary system size, thus predicted to be present in the thermodynamic limit. While our calculations are performed for an interaction energy far exceeding the gap between the two lowest energy bands, we note that the higher band does not affect the phase transitions, however destabilizes FCI phases.
Fractional Chern insulators have attracted broad interest as lattice analogs of fractional quantum Hall states without Landau levels. However, low-filling fractional Chern insulators are fragile because charge-ordered phases can compete strongly with the fractional topological liquid. Here, we propose a center-decorated kagome model, motivated by geometry-tunable artificial lattices, in which the center-site hopping
t2 provides a direct knob for the quantum geometry of an isolated
C=1 flat band. Here quantum geometry refers to the Berry curvature and Fubini--Study metric, which determine the form factors of interactions projected into the Chern band. Exact diagonalization shows that tuning
t2 away from the flatness-optimized kagome limit reduces the trace-condition deviation, suppresses competing charge order, and enhances the many-body stability at both
ν=1/3 and the more fragile
ν=1/5 filling. At
ν=1/5, this stability-enhanced window persists under nearby interaction profiles, including variations of the dominant third-neighbor repulsion and weak nearest-neighbor admixtures. Low-energy spectra, spectral flow, quasihole and entanglement counting, static structure factors, and the quantized total many-body Chern number
Ctot=1 consistently support Laughlin-like fractional Chern insulators. These results identify quantum-geometry engineering as a route to stabilizing dilute fractional Chern insulators beyond band-flatness optimization alone.
We devise local lattice models whose ground states are \emph{ideal fractional Chern insulators}---Abelian and non-Abelian topologically ordered states characterized by exact ground state degeneracies at any finite size and infinite entanglement gaps. Most saliently, we construct exact parent Hamiltonians for two distinct families of bosonic lattice generalizations of the
Zk parafermion quantum Hall states: (i) color-entangled fractional Chern insulators at band filling fractions
ν=k/(C+1) and (ii) nematic states at
ν=k/2, where
C is the Chern number of the lowest band in our models. In spite of a fluctuating Berry curvature, our construction is partially frustration free: the ground states reside entirely within the lowest band and exactly minimize a local
(k+1)-body repulsion term by term. In addition to providing the first known models hosting intriguing states such as higher Chern number generalizations of the Fibonacci anyon quantum Hall states, the remarkable stability and finite-size properties make our models particularly well-suited for the study of novel phenomena involving e.g. twist defects and proximity induced superconductivity, as well as being a guide for designing experiments.
We investigate a fractional Chern insulator (FCI) candidate arising from moir\'e bands with higher Chern number
C=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=2 Abelian fractional Chern insulator is realized in this system.