Stability of quasi-particle creation and multiband geometry in fractional Chern insulators under magnetic fields

TL;DR

This paper investigates the stability of fractional Chern insulator (FCI) quasiparticles under magnetic fields, focusing on two lattice models with distinct quantum geometry: Kapit-Mueller with $C_s=+1$ (ideal, LLL-like) and checkerboard with $C_s=-1$ (non-ideal). Using exact diagonalization and a multiband geometry framework, including the multiband tensor $oldsymbol{ exteta}^{ab}_{oldsymbol{eta}oldsymbol{eta}}(m{k})$, the trace condition, and Chern invariants, the authors connect geometry to field-tuned stability. They find that Kapit-Mueller supports stable quasiholes and quasielectrons for both signs of $N_{ extphi}$, while the checkerboard model exhibits stability for quasielectrons but not for quasiholes when the field direction is unfavorable, yielding a field-induced non-FCI phase; this is attributed to its non-ideal multiband geometry. Overall, multiband geometry and bandwidth emerge as key determinants of FCI robustness under magnetic fields, offering a predictive framework for lattice FCIs in moiré and related systems.

Abstract

We study creation of quasi-particles in fractional Chern insulators (FCI) under magnetic fields. We consider two representative models, the Kapit-Mueller model and the checkerboard model, which have distinct band properties in terms of the quantum geometry. The former satisfies the so-called ideal condition and well mimics the lowest Landau level, while the latter is not ideal for realization of FCI states. It is found within exact diagonalization that both quasi-holes and quasi-electrons are stably created by the magnetic fields in the Kapit-Mueller model. On the other hand, stability of the quasi-particle creation depends on directions of the magnetic field in the checkerboard model. Although the quasi-electron creation is stable under a magnetic field, the quasi-hole creation and the underlying FCI state are unstable for the opposite field direction, leading to a field-induced non-FCI state. We point out that this difference can be understood based on the multiband quantum geometry in the presence of the magnetic fields.

Figures (16)

• Figure 1: Schematic pictures of (a) the Kapit-Muller model and (b) the checkerboard model under zero magnetic fields. For the Kapit-Mueller model, $\phi_0$ is a model parameter which is distinguished from an applied magnetic field. For the checkerboard model, the direction of the arrows on the NN hopping corresponds to the sign of the phase factor $\psi_{jk}=\pm 1$. The NNN hopping strength $t_1(t_2)$ is specified by solid(dashed) lines. The NNNN hopping strength is $t"$ represented by the dashed curves.
• Figure 2: Expected phase diagrams for (a) the Kapit-Mueller model and (b) the checkerboard model with the pinning potential $V_p\neq0$. $\phi$ is the applied magnetic field and $U$ is the strength of the nearest-neighbor interaction. The strength of the pinning potential is chosen so that the created quasiparticles are well localized. It is assumed that effects of interactions between the localized quasiparticles are negligible in a thermodynamically large system.
• Figure 3: Spectral flow in the Kapit-Mueller model for $N_s=36, N=4, U=1.0$ under the magnetic fields $N_{\phi}=\pm 1$. The flux is $N_{\phi}=+1$ in (a) and (b). The pinning potential is (a) $V_{p}=0.0$ and (b) $V_{p}=1.0$. Similarly, (c) $N_{\phi}=-1, V_p=0$ and (d) $N_{\phi}=-1, V_p=-1.0$. Blue curves in (a) and (c) represent energy levels predicted by the counting-rule, $\#_{\text{qh}}=13$ and $\#_{\text{qe}}=11$. Blue curves in (b) and (d) represent the lowest three energy states corresponding to the topological degeneracy.
• Figure 4: One-plaquette Chern number $\mathcal{C}$ in the Kapit-Mueller model with $U=1.0, N=4$ at (a) $N_{\phi}=+1, V_p=-1.0$ and (b) $N_{\phi}=-1, V_p=+1.0$. (c) Variance $\mathcal{V}[\mathcal{C}]$ for ${|V_p|=1,} U=1\sim8$ and the flux density $N_{\phi}=0,\pm 1$. The yellow region with large $\mathcal{V}[\mathcal{C}]$ corresponds to the non-FCI state around the non-interacting limit $U=0$.
• Figure 5: Spectral flow in the checkerboard model for the system $N_s=36$, $N=6$, and $N_{\phi}=-1$. The parameters are (a) $U=1, V_p=0$, (b) $U=1, V_p=10$, (c) $U=2, V_p=0$, (d) $U=2, V_p=10$, (e) $U=3, V_p=0$, and (f) $U=3, V_p=10$. Blue curves represent in (a), (c), (d) are states expected from the counting rule $\#_{\text{qh}}=19$. Blue curves in (b), (d), (f) are the lowest three energy levels.