Faithful tight-binding models and fragile topology of magic-angle bilayer graphene

Correlated insulators and superconductivity have been observed in “magic-angle” twisted bilayer graphene, when the nearly flat bands close to neutrality are partially filled. While a momentum-space continuum model accurately describes these flat bands, interaction effects are more conveniently incorporated in tight-binding models. We have previously shown that no fully symmetric tight-binding model can be minimal, in the sense of capturing just the flat bands, so extended models are unavoidable. Here, we introduce a family of tight-binding models that capture the flat bands while simultaneously retaining all symmetries. In particular, we construct three concrete models with five, six, or ten bands per valley and per spin. These models are also faithful, in that the additional degrees of freedom represent energy bands further away from neutrality, and they serve as optimal starting points for a controlled study of interaction effects. Furthermore, our construction demonstrates the “fragile topology” of the nearly flat bands; i.e., the obstruction to constructing exponentially localized Wannier functions can be resolved when a particular set of trivial bands is added to the model.

Figure 1

Real-space orbitals. (a) Lattices and conventions. The shaded region indicates the relative positions of the sites assigned to the same unit cell. (b) The constructed quasiorbital $h_{p_{+}}^{\left(B\right)}$ in the real space. $\zeta =e^{i2\pi /6}$ and $\omega ={\zeta }^2$. Going from top to bottom, the entries in the three-component vectors attached to the triangular sites denote the amplitude for the $p_z$, $p_{+}$, and $p_{-}$ orbitals; those attached to the kagome sites denote the amplitude of their associated $s$ orbital.

Figure 2

Band structures. (a, b) Bands from the ten-band Hamiltonian $\widehat{H}(t_0,\delta )$. For both panels, we choose $t_0\equiv 130\mathrm{meV}$, and the wave-function parameters $(a,b,c,d)=(0.110,0.033,0.033,0.573)$. We set $\delta =0$ in (a) and 1 in (b). (c) Bands obtained from the continuum theory [4, 12] for twisted bilayer graphene with a twist angle of $\theta =1.{05}^{\circ }$, using the parameters described in Ref. [45]. The ten bands around charge neutrality are highlighted. (d–f) A zoom-in of the two bands at charge neutrality for the corresponding panels in (a)–(c). The three-dimensional plots in (e) and (f) are plotted over the first Brillouin zone centered at $\mathrm{\Gamma }$, showing the presence of exactly two Dirac points pinned to $\mathrm{K}$ and ${\mathrm{K}}^{\prime }=-\mathrm{K}$. Note that (e) is generated from our tight-binding model, whereas (f) is generated from the continuum model.

Figure 3

Band structures from a six-band model. (a) Color code for the orbital characters. (b) The broad energetic features can be set up using only the intraorbital dispersion. (c) Band structure from the full model, with parameters detailed in Appendix pp2-s1.

Figure 4

Real-space orbitals for the five-band model. (a) On each of the triangular sites (filled yellow circles), we consider the three $p$ orbitals $p_z$ and $p_{\pm }\equiv p_x\pm i{p}_y$; on each of the honeycomb sites (open circles), we consider an $s$ orbital. The centers of the nearest-neighbor bonds between the honeycomb sites form a kagome lattice. (b) The constructed quasiorbital ${\rho }_s^{\left(1\right)}$ centered at a kagome site, indicated by a cross. Going from top to bottom, the entries in the three-component vectors attached to the triangular sites denote the amplitude for the $p_z$, $p_{+}$, and $p_{-}$ orbitals; those attached to the honeycomb sites denote the amplitude of their associated $s$ orbital. The other two quasiorbitals labeled by $l=2,3$ can be obtained through symmetries.

Figure 5

Band structure from the five-band model. The following parameters are used: $(\tilde{a},\tilde{b},\tilde{c},\tilde{d})=(0.25,0.2,0.1,0.67)$, $({\mu }_{p_z}^{\prime },{\mu }_{p_{\pm }}^{\prime },{\mu }_{\eta }^{\prime })=(-0.043,0,0.05)t_0^{\prime }$, and $t_0^{\prime }=80\mathrm{meV}$. (a) The full spectrum. (b) A zoom-in for the two nearly flat bands.

Figure 6

Deformation to atomic limits. (a) The band gaps $\mathrm{\Delta }$ below and above the two nearly flat bands stay open for all values of $\mu \in [-{\mu }_0,{\mu }_0]$ in Eq. (7). More than $4\times {10}^4$ momenta are sampled in the Brillouin zone in determining $\mathrm{\Delta }$. We choose ${\mu }_0=t_0=130\mathrm{meV}$. (b–d) Schematic band diagrams at various limits of the deformation. (b) When $\mu =-{\mu }_0$, the lowest four bands arise solely from the $(\eta ,p_{\pm })$ orbitals and are therefore strictly atomic. Correspondingly, the upper six bands, altogether, are also in a strict atomic limit. (Dashed boxes indicate strictly atomic bands.) The same is true for the case of $\mu ={\mu }_0$ in (d), but with the role of the lowest and highest bands exchanged. Since the band gaps are maintained throughout the entire deformation, we can infer that all the (light and dark) purple blocks correspond to trivial, atomic bands. The nontrivial nearly flat bands at charge neutrality, therefore, must feature fragile topology.

Figure 7

Coupling terms in the six-band model. The full coupling terms consist of the indicated hopping together with their Hermitian conjugates. We always take the center site to be a triangular site in the “home” unit cell, and indicate the unit-cell coordinates $l{a}_1+m{a}_2$ of connected sites by $\left(lm\right)$ with $\overline{l}\equiv -l$. For the kagome sites in (e) and (f), we further specify their sublattice indices. The strength of the terms in panels (a)–(f) are, respectively, denoted by $t_{p_{\pm }{p}_{\pm }}^{+}$, $t_{p_{\pm }{p}_{\pm }}^{-}$, $t_{p_{\pm }{p}_z}^{+}$, $t_{p_{\pm }{p}_z}^{-}$, $t_{\kappa p_{\pm }}^{+}$, and $t_{\kappa p_{\pm }}^{-}$.

Figure 8

The leading perturbation to the ten-band model, which breaks the undesirable $\tilde{\mathcal{T}}$ invariance. Each circle denotes a honeycomb site, with the orbital $p_{\pm }$ indicated.

Figure 9

Evolution of the band gaps above and below the two nearly flat bands in the deforming Hamiltonians for (a) the six-band model in Eq. (C1) and (b) the five-band model in Eq. (C2). This establishes an adiabatic deformation of the complementary bands in these models to a strict atomic insulator. More than $4\times {10}^4$ momenta are sampled in the BZ in determining $\mathrm{\Delta }$.

Figure 10

Wilson loop spectra computed following the scheme described in Ref. [67], for the following set of bands in the ten-band model: (a) the two nearly flat bands at charge neutrality, (b) the lowest four bands, and (c) the six bands of (a) and (b) combined. Note that a nontrivial spectral flow, which forbids any atomic description, is found only in (a). This is consistent with the fragile nature of the band topology.

Figure 11

Wilson loop spectra computed following the scheme described in Ref. [67], for the following set of bands in $H_{\mu =0}^{\left(5\right)}$: (a) the two nearly flat bands at charge neutrality, (b) the lowest three bands, and (c) the five bands of (a) and (b) combined.