Exciton Band Topology in Spontaneous Quantum Anomalous Hall Insulators: Applications to Twisted Bilayer Graphene

We uncover topological features of neutral particle-hole pair excitations of correlated quantum anomalous Hall (QAH) insulators whose approximately flat conduction and valence bands have equal and opposite nonzero Chern number. Using an exactly solvable model we show that the underlying band topology affects both the center-of-mass and relative motion of particle-hole bound states. This leads to the formation of topological exciton bands whose features are robust to nonuniformity of both the dispersion and the Berry curvature. We apply these ideas to recently reported broken-symmetry spontaneous QAH insulators in substrate aligned magic-angle twisted bilayer graphene.

Figure 1

(a) Schematic of the four-band LLL model at $\nu =1$ showing the exchange-splitting $U^{\mathrm{HF}}=\sqrt{(\pi /2)}(e^2/{\ell }_B)$ and the different neutral excitation types. (b) Spin wave energy (blue curve) as a function of $y$-momentum $q$ for $\alpha =0$. Horizontal lines show the momentum-independent valley-flip exciton energies. Both modes saturate to $U^{\mathrm{HF}}$. (c) Intervalley exciton spectrum at $q_x=0$ in the magnetic Brillouin with increasing bandwidth $w=0.01$, 0.02, 0.03. Calculations were performed on a $20\times 20$ momentum-space mesh.

Figure 2

(a) Spin- and valley-flip exciton spectrum of $\nu =+3$ QAH state of TBG using the continuum model [24], shown along the $K_M\text{−}{\mathrm{\Gamma }}_M\text{−}{K}_M$ line in the moiré BZ. (b) As substrate potential is varied, intervalley exciton bands cross at ${\mathrm{\Gamma }}_M$ in a topological transition where the Chern numbers of the lowest bands change as indicated with $|\mathrm{\Delta }{C}_{\mathrm{exc}}|=1$. (c) Berry curvature (multiplied by the moiré BZ area) and exciton energy for the lowest valley-flip band of (a) in the moiré BZ. The origin ${\mathrm{\Gamma }}_M$ and the reciprocal lattice vectors ${\mathbf{b}}_M^{1(2)}=\sqrt{3}{k}_{\theta }(\pm 1/2,\sqrt{3}/2)$ are indicated, where the moiré wave vector $k_{\theta }=(8\pi /3\sqrt{3}a)\sin \frac{\theta }{2}$. (d) Same as (c) but for the $C_{\mathrm{exc}}=-1$ band of the final panel of (b). System size is $20\times 20$.