Interplay of fractional Chern insulator and charge density wave phases in twisted bilayer graphene

We perform an extensive exact diagonalization study of interaction-driven insulators in spin- and valley-polarized moiré flat bands of twisted bilayer graphene aligned with its hexagonal boron nitride substrate. In addition to previously reported fractional Chern insulator phases, we provide compelling evidence for competing charge-density-wave phases at multiple fractional fillings of a realistic single-band model. A thorough analysis at different interlayer hopping parameters, motivated by experimental variability, and the role of kinetic energy at various Coulomb interaction strengths highlight the competition between these phases. The interplay of the single-particle and the interaction-induced hole dispersion with the inherent Berry curvature of the Chern bands is intuitively understood to be the driving mechanism for the ground-state selection. The resulting phase diagram features remarkable agreement with experimental findings in a related moiré heterostructure and affirms the relevance of our results beyond the scope of graphene-based materials.

Figure 1

Schematic TBLG/hBN phase diagram of the identified order tendencies over their observed band filling range $\nu$ for the studied hopping parameters $w_{AB}=90\mathrm{meV}$ and $w_{AB}=110\mathrm{meV}$ in the strongly interacting regime. The classification is based on the results compiled in Sec. 5. We find two fractional Chern insulators (FCIs), a series of Wigner crystals (WCs) locked at specific fillings, as well as a charge-density wave (CDW) phase around $1/3$ filling, with a seemingly finite density range extent.

Figure 2

Overview of the single-particle band structure ${\epsilon }_{\mathbf{k}}$ (top), the Berry curvature $F\left(\mathbf{k}\right)$ (middle), and the interaction-induced hole dispersion $E_h\left(\mathbf{k}\right)$ (bottom) of the $\tau =-$ valence band for various $w_{AB}=90\mathrm{meV}\mathrm{to}110\mathrm{meV}$, organized into columns (a) to (e). More remote bands are separated by energetic gaps from the flat band cluster and lie outside the chosen energy window. Common to all $w_{AB}$ is the minimum (maximum) of the valence (conduction) band dispersion as well as the maximum of $-E_h\left(\mathbf{k}\right)$ at $\mathrm{\Gamma }$. $F\left(\mathbf{k}\right)$ is redistributed from a relatively uniform case in (a) to a sharp peak at $\mathrm{\Gamma }$ in (e).

Figure 3

(a, b) Ground-state energy for various clusters and (c) many-body spectrum on the cluster 36 at $\nu =1/3$ filling for both band parameters. At $w_{AB}=110\mathrm{meV}$ a clear sensitivity to the presence of the $\mathbf{K}$ points is observable, while the $w_{AB}=90\mathrm{meV}$ ground state in (a) is indifferent to this geometric feature. Clusters with aspect ratios far from 1, like 21B and 27B, violate this pattern. For more geometric details see Table 1. Shaded areas in (c) mark the set of identified ground states, whose locations in the MBZ are marked in the inset for the respective hopping amplitude. The $k$-space locations of momenta associated to each orbital index are given in Appendix pp1. (d) Spectral flow of ground-state orbitals on the cluster 21A at $w_{AB}=90\mathrm{meV}$ under the insertion of magnetic flux ${\mathrm{\Phi }}_1$. We incorporated a slight valence band dispersion via $\eta =0.3$ for better separation from excited states. The general effect of kinetic energy is discussed in Sec. 4b.

Figure 4

Structure factor distribution over the MBZ of the cluster 36 and extrapolation to the TDL for $\nu =1/3$ and both $w_{AB}$. The dominant peaks in (b) are strong evidence for $\mathbf{K}$-CDW order and the accompanied tripling of the unit cell, which survives in the TDL in (d). The CDW signatures at $w_{AB}=90\mathrm{meV}$ in (a) and (c) are less pronounced and are expected to vanish in the TDL.

Figure 5

(a) Various commensurate enlargements of the original moiré Wigner-Seitz cell (blue), corresponding to CDW/WCs that break the real-space ${\mathbf{L}}_{1,2}^{\text{M}}$ moiré translational symmetry. The respective fillings are $\nu =1/3$ (red), $\nu =1/4$ (green), and $\nu =1/7$ (magenta). The pattern at $\nu =1/7$ splits into two classes, which are related by an out-of-plane $C_2$ operation. (b) Illustration of the characteristic density-density correlation function ${\chi }_0\left(\mathbf{r}\right)$ as well as the three orthogonal realizations (red, green, blue) of a CDW at $\nu =1/3$ (cf. Appendix pp2 for details).

Figure 6

Orbital occupation at $\nu =1/3$ for both $w_{AB}$ and two convex combination factors $\eta$. While (a) and (c) indicate a smoothed redistribution of $n\left(\mathbf{k}\right)$ with $\eta$, (b) and (d) are nearly identical. The used cluster is 27A.

Figure 7

(a) The FCI reaches a stability maximum near $\eta \simeq 0.5$, accompanied by a suppression of $S(\mathbf{q}={\mathbf{K}}_{\pm })$. The stable CDW is degraded with $\eta$ until its signatures vanish close to $\eta \simeq 0.8-0.9$. The used cluster is 27A.

Figure 8

(a) Increasing $\eta$ in the conduction band at $\nu =1+1/3$ leads to a suppression of the FCI for $w_{AB}=90\mathrm{meV}$, stabilizing the CDW according to (c) up until $\eta \simeq 0.7$. The data in (b) and (d) at $w_{AB}=110\mathrm{meV}$ qualitatively replicate the situation of Figs. 7 and 7(d), albeit the CDW order is slightly more stable. The used cluster is 27A.

Figure 9

Evidence for an $\mathbf{M}$-WC at $\nu =1/4$ (a–c) and a $C_6$-WC at $\nu =1/7$ (d–f) for $w_{AB}=110\mathrm{meV}$. Although the signatures are less pronounced than for $\nu =1/3$, they are clearly visible in (b) and (e). The finite-size data in (c) and (f) suggest the order will prevail in the TDL. The displayed discretizations in (b) and (e) are 36 and 49, respectively. For more details on the used clusters, refer to Table 1.

Figure 10

Many-body spectrum at $\nu =2/5$ for (a) $w_{AB}=90\mathrm{meV}$ and (b) $w_{AB}=110\mathrm{meV}$. (c, d) The charge structure factor for both band parameters at $\nu =2/5$ as well as (e) spectral flow of the $w_{AB}=90\mathrm{meV}$ ground states, consistent with an FCI. In (e) $\eta =0.5$ is used for clearer separation of the ground-state manifold. Symbols have been omitted for clarity. The used cluster is 25.

Figure 11

(a) Many-body spectrum at $\nu =1/2$ of the cluster 36 with ground-state orbitals marked in the inset MBZ. The quasi–sixfold-degenerate ground-state manifold of (a) at $w_{AB}=90\mathrm{meV}$ is stabilized by the single-particle dispersion in (b), as indicated by the increased gap to ground-state splitting ratio $\mathrm{\Delta }{E}_{5,6}/\mathrm{\Delta }{E}_{0,5}$. The ground states at $\eta =0$ are marked in violet, red, and blue, while the next higher set of states is green, yellow, and orange. (c–f) The measurement of $S\left(\mathbf{q}\right)$ again signals an increased charge order tendency at $w_{AB}=110\mathrm{meV}$, although no universal order momentum could be identified and the peaks are less pronounced than for $\nu \le 1/3$.

Figure 12

Ground-state orbital occupation of the clusters 36 and 28A at $\nu =1/2$. Similarly to Fig. 6, $w_{AB}=90\mathrm{meV}$ leads to a more uniform occupation across the whole MBZ.

Figure 13

Introduction of a finite single-particle dispersion on the cluster 28A at $\nu =1/2$. The lowest two states at each of the $\mathbf{M}$ points are marked in violet, red, and blue, while the energetically minimal one at $\mathrm{\Gamma }$ is orange.

Figure 14

Structure factor sharpness $\mathcal{R}$ and identified regions of correlated phases over the scanned filling range $\nu$ for both interlayer hopping amplitudes. The appearance of significant signals only at the commensurate fillings $\nu =1/12$, $1/9$, $1/7$, $1/4$ suggests WC-type order (green) for both $w_{AB}$, whereas the extended region near $\nu =1/3$ supports the formation of a more robust $\mathbf{K}$-CDW (blue) at $w_{AB}=110\mathrm{meV}$. Evidence for the FCI (red) at $w_{AB}=90\mathrm{meV}$ was found for $\nu =1/3$ as well as $\nu =2/5$. Data points below $\mathcal{R}=0.25$ are marked in gray. Different symbols represent data from specific clusters. More details on the used clusters is found in Table 1.

Figure 15

Spectrum and location of ground-state orbitals for the cluster 28A at $\nu =1/2$ filling.

Figure 16

Spectrum and location of ground-state orbitals for the cluster 36 at $\nu =1/3$ filling with $\lambda =L^{\text{M}}/6$.

Figure 17

Spectrum and location of ground-state orbitals for the cluster 36 at $\nu =1/4$ filling.

Figure 18

Spectrum and location of ground-state orbitals for the cluster (a) 49 and (b) 28A at $\nu =1/7$ filling. The inset in (a) highlights the substrate-induced splitting of the ground-state manifold into two $C_6$-related sets of orbitals.