Twisted bilayer graphene. II. Stable symmetry anomaly

We show that the entire continuous model of twisted bilayer graphene (TBG) (and not just the two active bands) with particle-hole symmetry is anomalous and hence incompatible with lattice models. Previous works [e.g., Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, Phys. Rev. Lett. 123, 036401 (2019); J. Ahn, S. Park, and B.-J. Yang, Phys. Rev. X 9, 021013 (2019); H. C. Po, L. Zou, T. Senthil, and A. Vishwanath, Phys. Rev. B 99, 195455 (2019); J. Kang and O. Vafek, Phys. Rev. X 8, 031088 (2018); M. Koshino, N. F. Q. Yuan, T. Koretsune, M. Ochi, K. Kuroki, and L. Fu, Phys. Rev. X 8, 031087 (2018); J. Liu, J. Liu, and X. Dai, Phys. Rev. B 99, 155415 (2019); L. Zou, H. C. Po, A. Vishwanath, and T. Senthil, Phys. Rev. B 98, 085435 (2018)] found that the two flatbands in TBG possess a fragile topology protected by the $C_{2z}T$ symmetry. Song et al. [Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, Phys. Rev. Lett. 123, 036401 (2019)] also pointed out an approximate particle-hole symmetry ($\mathcal{P}$) in the continuous model of TBG. In this paper, we numerically confirm that $\mathcal{P}$ is indeed a good approximation for TBG and show that the fragile topology of the two flatbands is enhanced to a $\mathcal{P}$-protected stable topology. This stable topology implies $4l+2$ ($l\in \mathbb{N}$) Dirac points between the middle two bands. The $\mathcal{P}$-protected stable topology is robust against arbitrary gap closings between the middle two bands and the other bands. We further show that, remarkably, this $\mathcal{P}$-protected stable topology, as well as the corresponding $4l+2$ Dirac points, cannot be realized in lattice models that preserve both $C_{2z}T$ and $\mathcal{P}$ symmetries. In other words, the continuous model of TBG is anomalous and cannot be realized on lattices. Two other topology related topics, with consequences for the interacting TBG problem, i.e., the choice of Chern band basis in the two flatbands and the perfect metal phase of TBG in the so-called second chiral limit, are also discussed.

Figure 1

The lattice, symmetry anomaly, band structures, and Wilson loop bands of TBG. (a) The moiré unit cell, where the blue sheet and the red sheet represent the top and bottom layers, respectively. In the AA, AB, and BA regions, the A sublattice of the top layer is located above the A sublattice, the B sublattice, and the hexagon center of the bottom layer, respectively. (b) The moiré Brillouin zone. Left: The gray and yellow hexagons represent the moiré Brillouin zone for the graphene valleys $K$ and $K^{\prime }$, respectively. Right: The reciprocal lattices and the high-symmetry momenta of the moiré Brillouin zone in graphene valley $K$. (c) $4l+2$ ($l\in \mathbb{N}$) Dirac points cannot be realized in lattice models with $C_{2z}T$ and $\mathcal{P}$ symmetries. (d, e) The band structure and Wilson loop bands of the middle two bands (shaded) at $\theta =1.{05}^{\circ }$. The crossings in Wilson loop bands are protected by $C_{2z}T$ and/or the approximate $\mathcal{P}$. Each Wilson loop operator is integrated along ${\mathbf{b}}_{M2}$ and the spectrum is plotted along ${\mathbf{b}}_{M1}$. (f, g) The band structure and Wilson loop bands of the middle ten bands (shaded) at $\theta =0.7^{\circ }$. The crossings at $\lambda =0,\pi$ in the Wilson loop bands are protected by $C_{2z}T$ and/or by the approximate $\mathcal{P}$; the double degeneracies with $\lambda \ne 0,\pi$ at $k_1=0,\pi$ are protected by the approximate $\mathcal{P}$. These double degeneracies guarantee Wilson loop flow for any bands with $4n+2$ Dirac nodes at zero energy. In fact, we have kept the $\mathcal{P}$-breaking term $i\theta v_F{\tau }_z{\partial }_{\mathbf{r}}\times \mathit{\sigma }$ [Eq. (1)] in the calculations used to generate this plot, which would split the double degeneracies in principle. However, the splittings are almost invisible by eye in the plot, implying that the $\mathcal{P}$ symmetry is a good approximation. The degeneracies are exact when $P$ is exact. The parameters of the Hamiltonian used in (d)–(g) are $v_F=5.944\mathrm{eV}\mathring{\mathrm{A}}, \left|K\right|=1.703{\mathring{\mathrm{A}}}^{-1}, w_1=110\mathrm{meV}, w_0=0.7w_1$.

Figure 2

Errors of the approximate symmetries $\mathcal{P}$ (a) and $C$ (b) on the wave functions [as defined in Eqs. (9) and (10)] in TBG as functions of $w_0/w_1$. Here we change $w_0$ while keeping $w_1$ fixed (110 meV). The errors are shown for different values of the twist angle $\theta =1.{05}^{\circ },1.5^{\circ },2^{\circ }$.

Figure 3

Comparison of Wilson loop windings protected by $\mathcal{P}$ and $C_{2z}T$. (a–c) The Wilson loop bands with $\mathcal{P}$. The crossings at $k_1=0,\pi$ are Kramers pairs protected by $\mathcal{P}$. The ${\mathbb{Z}}_2$ invariant (a) equals to 1 if the Wilson loop bands form a zigzag connection between $k_1=0$ and $\pi$ and (b) equals to zero otherwise. The $\mathcal{P}$-protected topology is stable against adding trivial bands: Coupling the nontrivial Wilson loop bands (a) to trivial Wilson loop bands (b) yields nontrivial Wilson loop bands (c). (d–f) The Wilson loop bands with $C_{2z}T$. The crossings at $\lambda =0,\pi$ are protected by $C_{2z}T$. For a two-band system, the topology is nontrivial if the Wilson loop bands wind (d) and is trivial otherwise (e). The $C_{2z}T$-protected topology is fragile: Coupling the nontrivial Wilson loop bands (d) to trivial Wilson loop bands (f) yields trivial Wilson loop bands (f).

Figure 4

The path used to define the ${\mathbb{Z}}_2$ invariant protected by $\mathcal{P}$. $\mathcal{B}=[-\pi ,\pi ]\otimes [-\pi ,0]$ is half of the Brillouin zone. Its boundary $\partial \mathcal{B}$ is invariant under the particle-hole symmetry $\mathcal{P}$ ($\mathbf{k}\to -\mathbf{k}$). With the $C_{2z}T$ symmetry, one Dirac point in $\mathcal{B}$ contributes to a $\pi$ Berry phase along $\partial \mathcal{B}$.

Figure 5

The distribution of Dirac points in the moiré Brillouin zone at different twisting angles. (a) The gap between the middle two bands in the moiré Brillouin zone at $\theta =0.7^{\circ }, 0.8^{\circ }, 0.9^{\circ }, 1.0^{\circ }, 1.1^{\circ }$, where the Dirac points along $K_M{\mathrm{\Gamma }}_M$ are marked by the white circles. (b) The corresponding band structures at these twisting angles, where the Dirac points are marked by the red circles. For $\theta \ge 0.9^{\circ }$, there are two Dirac points located at $K_M$ and $K_M^{\prime }$, respectively, in the moiré Brillouin zone. For $\theta \le 0.8^{\circ }$, there are 14 Dirac points in the moiré Brillouin zone.

Figure 6

Perfect metal phase of twisted bilayer graphene in the second chiral limit ($w_1=0$). (a) The band structure at $\theta =1.{05}^{\circ }$ with the parameters $v_F=5.944\mathrm{eV}\mathring{\mathrm{A}}, \left|K\right|=1.703{\mathring{\mathrm{A}}}^{-1}, w_1=0, w_0=77\mathrm{meV}$. (b) The Brillouin zone of the twisted bilayer graphene. The solid black lines represent the $C_{2x}$ axis and its conjugations under $C_{3z}$; the dashed black lines represent the effective mirror symmetry $M_x=C_{2x}I$ and its conjugations under $C_{3z}$. The red dots represent Dirac points in the mirror lines. The blue dots represent Dirac points at generic momenta.

Figure 7

The $\mathbf{Q}$ lattice for the momentum space Hamiltonian of twisted bilayer graphene. (a) The blue and red hexagons represent the Brillouin zones of the top layer and the bottom layer, respectively. The blue and red dots represent the positions of Dirac points of the two layers in the graphene valley $K$, respectively. (b) The $\mathbf{Q}$ lattice formed by adding ${\mathbf{q}}_{1,2,3}$ iteratively. At each blue dot a plane-wave state from the top layer is assigned, and at each red dot a plane-wave state from the bottom layer is assigned.

Figure 8

Significant particle-hole asymmetry in the energy spectrum when the $\mathbf{k}$ dependence of the interlayer coupling is considered. The parameters used in the Hamiltonian are $v_F=5.944\mathrm{eV}\mathring{\mathrm{A}}, \left|K\right|=1.703{\mathring{\mathrm{A}}}^{-1}, w_1=110\mathrm{meV}, w_0=0.7w_1, \lambda =250\mathrm{meV}, \theta =1.{05}^{\circ }$.