Alternative $ν+ν$-picture of bosonic fractional Chern insulators at high filling factors in multiple flat-band systems

Abstract

Most fractional quantum Hall states have been traditionally identified within a single energy band, such as the lowest Landau level or topological flat band. As more particles are introduced, they inevitably populate higher energy bands. Whether the inclusion of multiple topological bands leads to new physics remains an open question. Here, we propose a universal picture applicable at higher filling factors $ν\geq 1$ in bosonic systems: the occupied bands tend to coalesce into an effective single topological band characterized by a total Chern number $\vert C\vert$, the sum of the Chern number of all occupied lower topological flat bands. Using a Kekulé lattice model with two lower flat bands featuring a total Chern number $C=1$, regardless of their specific configurations, we identify the emergence of a $\frac{1}{2}$ fractional Chern insulator (FCI) state at integer filling factor $ν=1$, followed by the Jain sequence states $\frac{2}{3}$ and $\frac{3}{4}$ at filling $ν=\frac{4}{3}$ and $\frac{6}{4}$. That is a $ν+ν$ picture, rather than the generally expected $1+ν^{\prime}$ picture, where $ν^{\prime}$ is the permitted FCI filling factor in the single second topological flat band. Our findings deepen the understanding of FCI states and open avenues for discovering exotic fractional topological phases in multiband systems.

Figures (4)

• Figure 1: (a) Schematic structure of the Kekulé lattice model. $\vec{a}_1$ and $\vec{a}_2$ are lattice vectors, and the six atomic orbitals in the unit cell (dotted rhombus) are labeled from "1" to "6". Various hopping terms are illustrated by lines with distinct colors (arrows indicate staggered fluxes), and the right panel displays the associated hopping terms. (b) The first Brillouin zone of the Kekulé lattice. The high-symmetry path $\Gamma$-$K$-$M$-$\Gamma$ are marked by red dash line.
• Figure 2: The energy spectrum of multiband with distinct Chern number combinations on the cylinder geometry, bulk and edge states are represented by blue solid lines and red dashed lines, respectively. (a) multiband with Chern number $\{C_1,C_2\}=\{0,-1\}$. (b) $\{C_1,C_2\}=\{1,0\}$. (c) $\{C_1,C_2\}=\{-1,2\}$. (d) $\{C_1,C_2\}=\{2,-1\}$. The detailed parameters are shown Supplementary Materials (SM).
• Figure 3: Topological features at $\nu=1$ for $\{C_1,C_2\}=\{0,-1\}$ combination. (a) Low-energy spectra with $V_1 = 8.0$ for two lattice sizes $N_s =6\times 3\times2=36$ and $N_s=6\times 4\times 2=48$. (b) Low-energy spectrum versus $\theta_x$ at a fixed $\theta_y = 0$ in the $N_s=36$ lattice. (c) Total Berry curvature of the ground states at $20\times 20$ mesh points, indicating a total Chern number $C=1$. (d) Charge pumping after two flux quanta insertion on a cylinder with $L_y=4$ and $\chi=800$. (e) Entanglement spectrum evolution as a function of flux, four charge sectors $\Delta N$ are marked by blue, red, green and orange. (f) Momentum-resolved entanglement spectrum on a cylinder with $L_y=4$ and $\chi=1800$, revealing one chiral edge mode with counting sequence (1,1,2,3,5,7,$\cdots$).
• Figure 4: Topological features at $\nu=\frac{4}{3}$ with $N_s=6\times3\times 2 =36$(left panels) and at $\nu=\frac{6}{4}$ with $N_s=6\times 2\times 2 =24$ (right panels) for $\{ C_1, C_2 = 1, 0\}$ combination. (a) and (d) Low-energy spectrum; (b) and (e) Charge pumping after flux quanta threading on cylinder geometry with $L_y=3$ and $\chi=600$ ((b)) and $L_y=3$ and $\chi=800$ ((d)); and (c) and (f) Entanglement spectrum evolution as functions of flux insertion.