Characterization and stability of a fermionic $\nu =1/3$ fractional Chern insulator

Using the infinite density matrix renormalization group method on an infinite cylinder geometry, we characterize the $1/3$ fractional Chern insulator state in the Haldane honeycomb lattice model at $\nu =1/3$ filling of the lowest band and check its stability. We investigate the chiral and topological properties of this state through (i) its Hall conductivity, (ii) the topological entanglement entropy, (iii) the $U\left(1\right)$ charge spectral flow of the many-body entanglement spectrum, and (iv) the charge of the anyons. In contrast to numerical methods restricted to small finite sizes, the infinite cylinder geometry allows us to access and characterize directly the metal to fractional Chern insulator transition. We find indications it is first order and no evidence of other competing phases. Since our approach does not rely on any band or subspace projection, we are able to prove the stability of the fractional state in the presence of interactions exceeding the band gap, as has been suggested in the literature. As a by-product we discuss the signatures of Chern insulators within this technique.

Figure 1

(a) Honeycomb lattice on an infinite cylinder with an $L$-site $(L=2L_y)$ circumference implemented in iDMRG. (b) Hopping conventions for the Haldane model [34] used in this work. (c) Band structure for the Haldane model with optimal band flatness ratio of the lower band $(\sim 1/7)$, achieved with $cos\left(\varphi \right)=t_1/\left(4t_2\right)=3\sqrt{3/43}(\varphi =0.65)$ and $m=0$.

Figure 2

(a) Charge pumping after one flux insertion for a half-filled trivial insulator on a cylinder with $L=6$ and $m\ne 0,t_2=0$, and $\chi =200$ (blue upper curve) and a CI with $m=0,\varphi \ne 0,t_2\ne 0$, and $\chi =400$ (lower green curve). The bond dimension was chosen large enough to represent the ground state for this size [52]. (b) Entanglement spectrum evolution as a function of flux for the CI. The spectrum is shifted by a unit charge after one flux quantum has been adiabatically inserted. Different charge sectors $Q^L$ are color coded. In (c) and (d) we show the charge pumping and ES for the $1/3$ filled lower band of the Haldane model after three flux insertions for $L=12$ and $\chi =500$. In this case ${\sigma }_{H;\mathrm{iDMRG}}\sim 0.33e^2/h$ per flux quantum and the ES is shifted by a unit charge only after three flux quanta have been adiabatically inserted.

Figure 3

(a) Finite size scaling of the von Neumann and infinite Renyi entropies as a function of cylinder circumference $L$ for $V_1=t_1$. They extrapolate to the values ${\gamma }_{\mathrm{VN}}=0.50$ and ${\gamma }_{\infty }=0.587$, respectively. The theoretically expected value $\gamma =\frac{1}{2}log3$ is indicated by a horizontal dashed line. (b) Momentum resolved entanglement spectrum of the GS at ${\Phi }_y=2\pi$ for $V_1=t_1$ showing the CFT edge theory counting $\{1,1,2,3,5,\cdots \}$ for each $Q^L$ sector (labeled by different colors). The dashed lines enclose the $Q^L=0$ sector counting for clarity. (c) Entanglement spectrum ${\varepsilon }_{\alpha }$ as a function of $V_1/t_1$ with the same color coding as in (b). The left shaded region is a metallic state (M) while the right unshaded region corresponds to the FCI state. Inset: Correlation length $\xi$ in units of $a$ as a function of $V_1/t_1$ for different $\chi$. The sharp discontinuity at $V_1\sim 2W$ signals a direct M-FCI phase transition.