Fractional Chern insulators with a non-Landau level continuum limit

Recent developments in fractional quantum Hall (FQH) physics highlight the importance of studying FQH phases of particles partially occupying energy bands that are not Landau levels. FQH phases in the regime of strong lattice effects, called fractional Chern insulators, provide one setting for such studies. As the strength of lattice effects vanishes, the bands of generic lattice models asymptotically approach Landau levels. In this paper, we construct lattice models for single-particle bands that are distinct from Landau levels even in this continuum limit. We describe how the distinction between such bands and Landau levels is quantified by band geometry over the magnetic Brillouin zone and reflected in the electromagnetic response. We analyze the localization-delocalization transition in one such model and compute a localization length exponent of 2.57(3). Moreover, we study interactions projected to these bands and find signatures of bosonic and fermionic Laughlin states. Most pertinently, our models allow us to isolate conditions for optimal band geometry and gain further insight into the stability of FQH phases on lattices.

Figure 1

Schematic depiction of Landau levels as the weak-field limit of Harper-Hofstadter bands near the minimum of a periodic potential. As the flux per plaquette is decreased, the elementary MUC—represented by the blue region of the lattice—must increase in size to enclose the same number of flux quanta. In turn, the number of states in each band decreases, and each band comprises states increasingly close to the minimum of the zero-field dispersion.

Figure 2

Energy eigenvalues of the zero-quadratic model, Eq. (17) with $t_1=1$, $t_2=-1/4$, as a function of magnetic flux per elementary lattice plaquette $\varphi =(p/q){\varphi }_0$.

Figure 3

TISM $\langle \mathcal{T}\rangle$ for the lowest-lying band of (17) as a function of $t_2$ with $t_1=1$. For $t_2=0$, this is the Hofstadter model, while $t_2=-1/4$ gives the zero-quadratic model with effective Hamiltonian (20).

Figure 4

TISM $\langle \mathcal{T}\rangle =2\langle a^{†}a\rangle$ for the ground state band of the Hamiltonian $H^{\left(2n\right)}=K_x^{2n}+K_y^{2n}$. When $n=1$, this is the Landau level Hamiltonian for which the lowest-lying band has $\langle \mathcal{T}\rangle =0$ as shown.

Figure 5

RMS fluctuation ${\sigma }_{\mathcal{B}}$ of Berry curvature $\mathcal{B}$ from its average value on the MBZ. The mixed-order anisotropic model is defined in Eq. (49).

Figure 6

Summary of current per orbital response of the bands of $H^{\left(2n\right)}$ to inhomogeneous electric potential $V\left(x\right)={\sum }_p{c}_p{x}^p$. For each value of $n$, we plot the numerical factor multiplying $c_3$ (top) and $c_5$ (bottom) in the current per orbital (61). The dotted lines show the values of these factors for the corresponding Landau levels.

Figure 7

Summary of current density response of the bands of $H^{\left(2n\right)}$ to inhomogeneous electric potential $V\left(x\right)={\sum }_p{c}_p{x}^p$. For each value of $n$, we plot the numerical factor multiplying $c_3$ (top) and $c_5$ (bottom) in the current density (62). The dotted lines show the values of these factors for the corresponding Landau levels.

Figure 8

Renormalized inverse localization length in the disordered zero-quadratic model at flux $\varphi =(1/10)({\varphi }_0/2\pi )$ and disorder strength $s=0.1t_1$ near the center of the lowest level. Points: Renormalized inverse localization-lengths as a function of width. Lines: Fit to $\mathrm{\Gamma }(E,W)$ using Eq. (79) with parameters taken as the average of the values determined from the fits to 100 synthetic data sets $\tilde{\mathrm{\Gamma }}(E,W)=\mathrm{\Gamma }(E,W)+\mathrm{gaussian}(0,\sigma (E,W))$, where $\sigma (E,W)$ is the standard deviation of the data point $\mathrm{\Gamma }(E,W)$ across the 20 realizations. The mean fit value for $\nu$ across the 1000 data sets is $\nu =2.57$ with a standard deviation of 0.03.

Figure 9

Gap between first excited state and twofold degenerate ground state for $N_p=8$ bosons occupying the ground state band of the tight binding model (17) at filling $\nu =1/2$. The interaction between bosons is an onsite, infinite density-density repulsion. We plot the dependence of the gap on the TISM $\langle \mathcal{T}\rangle$ as the NNN hopping parameter $t_2$ is varied. Each panel shows a different value of flux per plaquette $\varphi$: From left to right we have $\varphi =1/16$, $\varphi =1/49$ and $\varphi =1/81$. For each case, the largest gap is observed for the ordinary Hofstadter model with $t_2=0$.

Figure 10

Gap between first excited state and threefold degenerate ground state for $N_p=8$ fermions occupying the ground state band of the tight binding model (17) at filling $\nu =1/3$. The interaction between fermions is nearest neighbor, density-density repulsion. We plot the dependence of the gap on the TISM $\langle \mathcal{T}\rangle$ as the NNN hopping parameter $t_2$ is varied. Each panel shows a different value of flux per plaquette $\varphi$: From left to right we have $\varphi =1/24$, $\varphi =1/54$ and $\varphi =1/96$.

Figure 11

Gap between first excited state and threefold degenerate ground state for $N_p=8$ fermions occupying the ground state band of the tight binding model (17) at filling $\nu =1/3$. The interaction between fermions is a repulsive density-density interaction that falls off as $e^{-r^4}$, as in (88). We plot the dependence of the gap on the TISM $\langle \mathcal{T}\rangle$ as the NNN hopping parameter $t_2$ is varied. Each panel shows a different value of flux per plaquette $\varphi$: From left to right we have $\varphi =1/24$, $\varphi =1/54$ and $\varphi =1/96$. In contrast with the case of NN interactions, for each case the largest gap is observed for the ordinary Hofstadter model with $t_2=0$.

Figure 12

Convergence of $C_0$ with the number of iterations $N$. The exact value $C_0=0.987926$ is marked with a dashed line.

Figure 13

Electronic localization-delocalization critical exponent $\nu$ as dependent on NNN hopping strength $t_2$. We see that $\nu$ is independent of $t_2$ up to the precision of the simulations as all exponents are in the interval (2.51,2.57), and we estimate a precision of $\pm 0.03$ on each exponent determined. The determined critical exponents are in concurrence with recent theoretical predictions of the IQHE localization-delocalization critical exponent.