Domain wall competition in the Chern insulating regime of twisted bilayer graphene

We consider magic-angle twisted bilayer graphene at filling $\nu =+3$, where experiments have observed a robust quantized anomalous Hall effect. This has been attributed to the formation of a valley- and spin-polarized Chern insulating ground state that spontaneously breaks time-reversal symmetry and is stabilized by a hexagonal boron nitride substrate. We identify three different types of domain wall and study their properties and energetic selection mechanisms via theoretical arguments and Hartree-Fock calculations adapted to deal with inhomogeneous moiré systems. We comment on the implications of these results for transport and scanning probe experiments.

Figure 1

Domain wall phase diagram in the twist angle ($\theta$)-substrate gradient (${\mathrm{\Delta }}_{\text{max}}/w$) plane, as determined by the energy difference of the two HF solutions $E_{\text{Chern}}-E_{\text{valley}}$ plotted as contours. Chern DWs have chiral edge modes whereas valley DWs have achiral modes. Valley DWs are favored at small ${\mathrm{\Delta }}_{\text{max}}/w$, which allows more texturing and hence reduces the exchange penalty. Contours for $\theta <1.{05}^{\circ }$ are dashed since the gap to the remote bands becomes small enough that treating the remote bands as inert is a poorer approximation. (We take $N_1=N_2=10,2w=3a_{\text{M}}$.)

Figure 2

Top-down view of DWs and numerical setup. We study systems of $N_{\text{tot}}=N_1{N}_2$ moiré unit cells ($N_1=6,N_2=3$ Chern DW shown for illustration) with periodic boundary conditions. Black dots indicate $AA$ stacking regions, where central bands have largest weight ($AB,BA$ regions are also shown on the right). The substrate potential $\mathrm{\Delta }\left(\mathit{r}\right)$ can vary along ${\mathbf{a}}_{\text{M}}^{\left(1\right)}$ (the sign-changing choice shown can stabilize two Chern or valley walls). Each Chern or intertwined DW hosts two localized copropagating chiral modes, while a valley wall hosts nonchiral gapless modes. Top plots show the order parameter profiles for the Chern and valley walls.

Figure 3

Scaling collapse of width $\xi$ of valley (dots) and Chern (crosses) DWs against substrate gradient ${\mathrm{\Delta }}_{\text{max}}/w$ at twist angle $\theta =1.2^{\circ }$ for different widths $w$ of the $\mathrm{\Delta }\left(\mathbf{r}\right)$ profile. Solid line is the scaling $\xi \sim a_{\text{M}}{[{\rho }_s/(a_{\text{M}}{\mathrm{\Delta }}_{\text{max}}/w)]}^{1/3}$ predicted by Eq. (1). Departures from scaling are seen at large ${\mathrm{\Delta }}_{\text{max}}/w$, where moiré-scale physics is relevant. $N_1=N_2=10$.

Figure 4

Example symmetric set of plane waves (black dots) that are kept when generating the Bloch functions of the CM. The size of the cutoff shown leads to 228 bands per spin and valley. The red and blue stars are the momenta of the Dirac points of layers 1 and 2. The black star is the $\mathit{X}$ vector, which belongs to the ${\mathrm{\Gamma }}^{\text{M}}$ point of the $K$-valley mBZ. The reference grid indicates plane waves that are equivalent to ${\mathrm{\Gamma }}^{\text{M}}$. Here we have chosen $N_1=N_2=4$ for illustration, and have only shown the plane waves belonging to layer 1 and sublattice $A$ in valley $K$. The included plane waves corresponding to the other combinations of layer, sublattice, and valley can be obtained by acting with TRS and the elements of $D_6$.

Figure 5

Comparison of the noninteracting band energies between the fixed Hilbert space (FHS) method with external substrate, and the usual continuum model solution with substrate potential. $\text{max}\left(\right|\delta E\left|\right)$ is defined as the maximum energy difference between the band structures of the two methods over the mBZ.

Figure 6

Comparison of the projected Hilbert spaces between the fixed Hilbert space (FHS) method with external substrate and the usual continuum model solution with substrate potential. Overlap is defined as a normalized $\text{Tr}{P}_{\text{FHS}}{P}_{\text{CM}}$, where $P$ is the projector onto the central bands for the respective method. This quantity is 1 if the subspaces are identical.

Figure 7

Phase diagram of the energy difference between Chern wall and valley wall, $E_{\text{Chern}}-E_{\text{valley}}$. This is the same as Fig. 1 in the main text, however, here we use the subtraction scheme for the interactions with filled central bands, i.e., the reference projector has ${\nu }_0=+4.N_1=N_2=10,2w=3a_{\text{M}}$.

Figure 8

Schematic of energy levels for a sign-changing substrate $\mathrm{\Delta }\left(x\right)$. Consider the spinless theory at $\nu =-1$. On the left, we assume that we are polarized in $KA$. A valley wall involves a Chern-filtered valley rotation from $KA$ to $\overline{K}B$, which is captured by the Ginzburg-Landau order parameter $\mathit{n}$. A Chern wall has global valley polarization but swaps between sublattices (and hence Chern sectors). This can be captured by an effective Ising degree of freedom $m_z$

Figure 9

Breakdown of the energy difference between the Chern wall and valley wall into a direct and exchange part. We do not show the single-particle contributions to the energy since these are comparable for the two DW types. $\theta =1.2^{\circ }$ and $N_1=N_2=10$.

Figure 10

Phase diagram of the energy difference between intertwined wall and uniform solution, $E_{\text{intertwined}}-E_{\text{uniform}}$. $N_1=N_2=10,2w=3a_{\text{M}}$.

Figure 11

(a) HF band structure of the Chern DW organized by momentum parallel to the DW. The substrate ${\mathrm{\Delta }}_2\left(\mathit{r}\right)$ is ${\mathrm{\Delta }}_{\text{max}}=20$ meV in region $L$ and $-{\mathrm{\Delta }}_{\text{max}}$ in region $R$. Points are color-coded according to the real-space localization of the corresponding HF orbitals (cf. Fig. 2 in the main text). Dashed lines are guides indicating gapless chiral modes. (b) Overlap of the HFWQ orbitals [obtained using a uniform $\mathrm{\Delta }={\mathrm{\Delta }}_{\text{max}}$ with the converged DW solution, plotted against the average $x_1$ position (in units of $a_{\text{M}}$) of the orbitals]. $K\text{Left}$, $K\text{Right}$ refer to the $K$-polarized HFWQ basis constructed in a uniform substrate $\pm {\mathrm{\Delta }}_{\text{max}}$, respectively, and analogously for valley $K^{\prime }$. Note that the $L$ and $R$ bases for a given valley are not orthogonal, which explains the moiré-scale oscillations. Bottom shows a schematic of the substrate profile. (c) Line cuts of the spatially resolved spectral function (relevant for STM) at selected energies; only one moiré cell in the ${\mathbf{a}}_{\text{M}}^{\left(2\right)}$ direction is shown as the results are periodic. Darker regions indicate higher weight. Parameters are $N_1=N_2=20,\theta =1.2^{\circ }$, and the dual-gate screened potential is used.

Figure 12

Same as Fig. 11 except for the intertwined wall. It is evident from the HFWQ overlaps and the $E=0\text{meV}$ spatially resolved density of states that the DW prefers to lock to half-integer positions.

Figure 13

Same as Fig. 11 except for the valley wall. At HF level, the counterpropagating modes of the valley DW can hybridize and gap out. However, a gapless mode must reemerge as quantum fluctuations restore $\text{U}{\left(1\right)}_v$ symmetry along the DW. Hence we have omitted plots of the spectral function.