Fractional quantum Hall states for moiré superstructures in the Hofstadter regime

We study the transition of $\nu =1/3$ and $2/5$ fractional quantum Hall states of the honeycomb Hofstadter model as we tune to a two-orbital moiré superlattice Hamiltonian, motivated by the flat bands of twisted bilayer graphene in a perpendicular magnetic field. In doing so, we address the extent to which these states survive in moiré systems and analyze the nature of the transition. Through the use of a Peierls substitution, we determine the Landau-level splitting for the moiré Hamiltonian, and study the structure of the Chern bands for a range of magnetic flux per plaquette. We identify topological flat bands in the spectrum at low energies, with numerically tractable lattice geometries that can support the fractional quantum Hall effect. As we tune the model, we find that the orbital-polarized $\nu =1/3$ and $2/5$ states corresponding to the honeycomb Hofstadter model survive up to $\approx 30\%$ of typical moiré superlattice parameters, beyond which they transition into an insulating phase. We present evidence for this through density matrix renormalization group calculations on an infinite cylinder, by verifying the charge pumping, spectral flow, entanglement scaling, and conformal field theory edge counting. We conclude that fractional quantum Hall states from the Hofstadter model can persist up to hopping amplitudes of the same order as those typical for moiré superlattice Hamiltonians, which implies generally that fractional states for moiré superstructures can be discerned simply by analyzing the dominant terms in their effective Hamiltonians.

Figure 1

(a) Real-space lattice in units of $(x,y)=(mb,nc)$, where $b\equiv a/2, c\equiv (1/3)(\sqrt{3}a/2)$, and $a\equiv 1$ is the lattice constant of the triangular sublattice or, equivalently, the length of the hexagonal basis vector. The coordinates of the nearest neighbors (${⎔}_1$) are shown in blue, and the coordinates of the fifth-nearest neighbors (${⎔}_5$) are shown in red. The two conjugate sets of three for the fifth-nearest neighbors are distinguished by dots and crosses. The unit cell is outlined in green and an example of a $q=2$ magnetic unit cell is highlighted in yellow. (b) Sketch of Brillouin zone using the coordinates $x=k_{x}a/2$ and $y=\sqrt{3}{k}_{y}a/2$, with reciprocal-lattice vectors ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$. The green outline shows the Brillouin zone (effective reciprocal unit cell) over which the Chern number calculations are performed, and the yellow region shows the $q=2$ magnetic unit cell corresponding to (a).

Figure 2

Energy spectrum of Eq. (1) as a function of flux density $n_{\varphi }=p/q$. $p$ and $q$ are coprime integers with $q=199$ and $0\le p\le 198$. We consider the cases of (a) ${⎔}_1$ hopping with $(t_1,t_2,t_2^{\prime })=(1,0,0)$ meV, (b) ${⎔}_5$ hopping with $(t_1,t_2,t_2^{\prime })=(0,1,0)$ meV, and (c) typical hopping parameters for the model: $(t_1,t_2,t_2^{\prime })=(1,-0.025,0.1)$ meV [11, 14, 20] with $\kappa =0.3$. Since the bandwidths are narrow at the flux densities considered, we plot the points at $(k_x,k_y)=(0,0)$ only. The dashed lines at $n_{\varphi }=1/3$ and $2/3$ indicate the high-symmetry points in the ${⎔}_5$ Hamiltonian.

Figure 3

Band structure of Eq. (1) at $\kappa =0.3$ with $n_{\varphi }=0$ and $1/3$. (a) Zero-field band structure. The inset shows the Brillouin zone and energy contour plot for the $\xi =+$ valley. (b) The Landau-level splitting of the lower-half energy bands, labeled by their corresponding Chern numbers. The right panel shows the density of states (DOS), based on a ${100}^2$ grid with 100 bins per energy range. (c) The 3D band structure of the three lowest bands in (b). (d) The product of Berry phases, ${\theta }_{\mathrm{B}}$, around Wilson loops in the $k_y$ direction for the bands shown in (c).

Figure 4

Example initial MPS unit-cell geometries for a honeycomb (emergent moiré) lattice in a perpendicular magnetic field with flux density $n_{\varphi }=1/3$. Empty (full) sites are indicated by white (black) discs and the $p$ orbitals at each site are not drawn unless needed. The MPS filling and geometry are chosen such that the (a) $\nu =1/3$ and (b) $\nu =2/5$ states may be recovered in the moiré Hamiltonian.

Figure 5

IQH and FQH flux insertion examples from many-body calculations of Eq. (1) at $n_{\varphi }=1/3$ with (a)–(c) $\nu =1/3$ and (d)–(f) $\nu =2/5$, using iDMRG on a thin cylinder. For the IQH examples, the simulation parameters are $\nu =1$ and $U=V=0$, whereas for the FQH examples we use $U=10V=100t_1$ and $0\le \kappa \le 0.5$. The simulations were performed with a circumference of $L_y=6$ for $\nu =1/3$ and $L_y=5$ for $\nu =2/5$, at a bond dimension of $\chi =150$. (a), (d) Charge pumping as we thread a flux ${\mathrm{\Phi }}_x$ through the center of the cylinder (sketch inset). The charge pumping for the FQH state is plotted for a range of $\kappa$, as indicated in the legend. (b), (c), (e), (f) Entanglement spectrum flow for the $(k_x,k_y)=(0,0)$ momentum sector, for the states marked in bold in (a) and (d), where the energy levels are labeled by the $U\left(1\right)$ charge sector. Note that the full entanglement spectra are not shown, in order to emphasize the flow of the low-lying states.

Figure 6

FQH entropy scaling example for the $\nu =1/3$ state defined in Fig. 5. (a) Scaling of the von Neumann entanglement entropy from bisecting the cylinder, $S_{\mathrm{vN}}$, with cylinder circumference in units of the magnetic length $L_y/l_{\text{B}}$. The entropy values are obtained by extrapolating data up to bond dimension $\chi =300$, and the $\gamma =0.549$ asymptote for the $\nu =1/3$ Laughlin state is marked with a dashed line. (b) Momentum-resolved entanglement spectra for the FQH state with $n_{\varphi }=1/3$ at $L_y=6$ and 9 sites, up to a rotation in momentum space. The state for $L_y=9$ is calculated at a higher bond dimension of $\chi =500$. The energy levels are labeled by the $U\left(1\right)$ charge sector, and the dashed lines guide the edge states. Note that the full entanglement spectra are not shown, in order to highlight the edge state counting.

Figure 7

FQH to topologically trivial phase transition with respect to the dimensionless tuning parameter, $\kappa$. Parameters are defined in Fig. 5 with a bond dimension of $\chi =200$. (a) Gap-to-width ratio, $\mathrm{\Delta }/W$, of the lowest bands. (b) Average charge on the left side of the cylinder, $\langle Q_{\mathrm{L}}\rangle$, after a flux insertion of $\mathrm{\Delta }{\mathrm{\Phi }}_x=6\pi$ for the $\nu =1/3$ state and $\mathrm{\Delta }{\mathrm{\Phi }}_x=10\pi$ for the $\nu =2/5$ state. (c) Correlation length, $\xi$. (d) Entanglement spectra for the $(k_x,k_y)=(0,0)$ momentum sector, plotted as a function of $\kappa$ for the $n_{\varphi }=1/3$ system at $\nu =2/5$ filling with circumference $L_y=5$. The value of $\kappa =0.3$, discussed throughout this paper, is marked by the boundary of the phase transition. As before, the energy levels are labeled by their $U\left(1\right)$ charge sector.

Figure 8

Convergence analysis for (a), (b) the full moiré superlattice Hamiltonian given in Eq. (2) and (c), (d) the single-orbital fermionic honeycomb Hofstadter model. (a), (c) Entanglement energy ${\epsilon }_{\alpha }\equiv -ln{\mathrm{\Lambda }}_{\alpha }^2$ plotted against Schmidt value index, $\alpha$, for various bond dimensions $\chi$ indicated by their cutoffs. The system parameters are those for (a) the $n_{\varphi }=1/3, L_y=6$ data point in the scaling plot shown in Fig. 6 and (c) the $n_{\varphi }=1/3, L_y=6$ Hofstadter model corresponding to a single-orbital copy of $H_0$ [Eq. (1)] at $\kappa =0$. (b), (d) Corresponding entanglement entropy $S=-{\sum }_{\alpha }{\mathrm{\Lambda }}_{\alpha }^{2}ln{\mathrm{\Lambda }}_{\alpha }^2$ against inverse bond dimension, with a line of best fit drawn through the last three points.