Stability, phase transitions, and numerical breakdown of fractional Chern insulators in higher Chern bands of the Hofstadter model

The Hofstadter model is a popular choice for theorists investigating the fractional quantum Hall effect on lattices, due to its simplicity, infinite selection of topological flat bands, and increasing applicability to real materials. In particular, fractional Chern insulators in bands with Chern number $\left|C\right|>1$ can demonstrate richer physical properties than continuum Landau level states and have recently been detected in experiments. Motivated by this, we examine the stability of fractional Chern insulators with higher Chern number in the Hofstadter model, using large-scale infinite density matrix renormalization group (iDMRG) simulations on a thin cylinder. We confirm the existence of fractional states in bands with Chern numbers $C=1,2,3,4,5$ at the filling fractions predicted by the generalized Jain series [Phys. Rev. Lett. 115, 126401 (2015)]. Moreover, we discuss their metal-to-insulator phase transitions, as well as the subtleties in distinguishing between physical and numerical stability. Finally, we comment on the relative suitability of fractional Chern insulators in higher Chern number bands for proposed modern applications.

Figure 1

Band flatness in the Hofstadter model. (a) Single-particle band structure for the Hofstadter model at $n_{\varphi }=3/5$, as a function of momentum $k$. The Chern number of each band $C$, as well as the width $W$ and gap $\mathrm{\Delta }$ of the lowest band are labeled. (b) Band gaps, widths, and gap-to-width ratios against $p<10$ for the first five Chern numbers. (c) Standard deviation of the Berry curvature for the lowest band ${\sigma }_{\mathcal{B}}$ in units of the mean Berry curvature $\overline{\mathcal{B}}$.

Figure 2

FCIs in $C=1$ bands. Expectation value of charge pumped on the left half of the cylinder $\langle Q_{\text{L}}\rangle$ as a function of adiabatic flux insertion through the cylinder ${\mathrm{\Phi }}_x$. All of the computations are performed with cylinder circumference $L_y=2s$, interaction strength $V=10$, bond dimension $\chi =250$, and flux interval $\mathrm{\Delta }{\mathrm{\Phi }}_x/2\pi =0.1$. The flux densities corresponding to $p<10$ are considered, where the lowest band is gapped.

Figure 3

FCIs in $C=2,3$ bands. Charge pumping for FCIs in [(a)–(e)] $C=2$ and [(f)–(h)] $C=3$ bands. All of the computations are performed with interaction strength $V=10$ and flux interval $\mathrm{\Delta }{\mathrm{\Phi }}_x/2\pi =0.1$. The flux densities corresponding to $p<10$ are considered, where the lowest band is gapped.

Figure 4

FCIs in $C=4,5$ bands. Charge pumping for FCIs in [(a), (b)] $C=4$ bands with (a) $n_{\varphi }=6/23$, (b) $n_{\varphi }=4/15$, and [(c), (d)] $C=5$ bands with (c) $n_{\varphi }=5/24$, and (d) $n_{\varphi }=4/19$. The bond dimension coloring is the same as in Fig. 3. All of the computations are performed with interaction strength $V=10$ and flux interval $\mathrm{\Delta }{\mathrm{\Phi }}_x/2\pi =0.1$. The flux densities corresponding to $p<10$ are considered, where the lowest band is gapped. Note that, unlike in Figs. 2 and 3, both columns correspond to negative $r$.

Figure 5

Metal-to-FCI phase transitions in $C=1$ bands. (Top panels) Correlation length $\xi$, von Neumann entanglement entropy $S_{\mathrm{vN}}$, interaction energy $\langle \widehat{V}\rangle$, and entanglement spectrum $\left\{{\epsilon }_i\right\}$, as a function of interaction strength $V$, for $\nu =1/3$, $L_y=6$ FCIs in $C=1$ bands with (a) $n_{\varphi }=1/4$ and (b) $n_{\varphi }=1/6$. The band widths ($W$) and gaps ($\mathrm{\Delta }$) are marked with blue- and red-dotted lines, respectively. The entanglement energies are additionally colored corresponding to their $U\left(1\right)$ charge eigenvalues $Q_{\text{L},i}$. (Bottom panel) Momentum-resolved entanglement spectrum at $V=0.6$. We select the energy scale and rotate the momentum eigenvalues, such that $\tilde{K}=(K+c)mod{L}_y$ where $c$ is a constant, to emphasize the edge states. The counting of the edge states in the zeroth charge sector is also labeled.

Figure 6

Metal-to-FCI phase transitions in $C=2$ bands. (Top panels) Correlation length $\xi$, von Neumann entanglement entropy $S_{\mathrm{vN}}$, interaction energy $\langle \widehat{V}\rangle$, and entanglement spectrum $\left\{{\epsilon }_i\right\}$, as a function of interaction strength $V$, for FCIs in $C=2$ bands with (a) $\nu =1/5$, $L_y=10$, $n_{\varphi }=6/11$, and (b) $\nu =2/9$, $L_y=9$, $n_{\varphi }=5/9$. The band widths ($W$) and gaps ($\mathrm{\Delta }$) are marked with blue- and red-dotted lines, respectively. The entanglement energies are additionally colored corresponding to their $U\left(1\right)$ charge eigenvalues $Q_{\text{L},i}$. (Bottom panel) Momentum-resolved entanglement spectrum at $V=3$. We select the energy scale and rotate the momentum eigenvalues, such that $\tilde{K}=(K+c)mod{L}_y$ where $c$ is a constant, to emphasize the edge states.

Figure 7

Scaling of $V_{\mathrm{crit}}$ in $C=1,2$ bands. Critical interaction strength $V_{\mathrm{crit}}$ in units of (a) band width $W$ and (b) band gap $\mathrm{\Delta }$ as a function of $p$. Values are shown for the $C=1$ FCIs from Fig. 2 in the main plots and for the $C=2$ FCIs from Fig. 3 in the insets.

Figure 8

Scaling of correlation functions with bond dimension and system size. Scaling of density-density correlation functions $g\left(x\right)=\langle :{\rho }_{0,0}{\rho }_{x,0}:\rangle -\langle {\rho }_{0,0}{\rho }_{\infty ,0}\rangle$, as a function of $x$, the number of sites in the $x$ direction, for the case studies analyzed in (left) Fig. 5 and (right) Fig. 6, with respect to (a) bond dimension and (b) system size. The $x=0$ point has been excluded since trivially $\langle :{\rho }_{0,0}{\rho }_{0,0}:\rangle =0$ by the Pauli exclusion principle. The corresponding correlation lengths are marked with dashed lines

Figure 9

Connected two-point correlation functions for FCIs in $C=1$ bands. Density-density correlation functions for the FCIs in Fig. 2, presented as in Fig. 8. The parameters are the same as those used for the charge pumping. In this case, $\chi =250$ and $L_y=2s$. The corresponding correlation lengths are marked with dashed lines.

Figure 10

Connected two-point correlation functions for FCIs in $C=2,3$ bands. Density-density correlation functions for the FCIs in Fig. 3, presented as in Fig. 8. The parameters are the same as those used for the charge pumping. In this case, (a) $L_y=10$, (b) $L_y=6$, (c) $L_y=9$, (d) $L_y=14$, (e) $L_y=11$, (f) $L_y=14$, (g) $L_y=10$, and (h) $L_y=11$. The corresponding correlation lengths are marked with dashed lines.

Figure 11

Connected two-point correlation functions for FCIs in $C=4,5$ bands. Density-density correlation functions for the FCIs in Fig. 4, presented as in Fig. 8. The parameters are the same as those used for the charge pumping. In this case, (a) $L_y=14$, (b) $L_y=15$, (c) $L_y=18$, and (d) $L_y=19$. The bond dimensions and $p$ values are presented as in Fig. 10. The corresponding correlation lengths are marked with dashed lines.

Figure 12

Numerical instability with decreasing flux density. Comparison of the charge pumping for the $C=1$ FCIs from Figs. 2–2 at the first two allowed cylinder circumferences. The charge pumping is shown for (a) $\nu =3/7$, $L_y=7,14$, (b) $\nu =4/7$, $L_y=7,14$, (c) $\nu =4/9$, $L_y=9,18$, and (d) $\nu =5/9$, $L_y=9,18$.

Figure 13

Numerical instability with increasing bond dimension. Comparison of the charge pumping for the $\nu =3/7$ FCI from Fig. 12 as a function of bond dimension. The charge pumping is shown for the flux densities (a) $n_{\varphi }=1/3$, (b) $n_{\varphi }=1/4$, (c) $n_{\varphi }=1/5$, and (d) $n_{\varphi }=1/6$.