Twisted bilayer graphene. III. Interacting Hamiltonian and exact symmetries

We derive the explicit Hamiltonian of twisted bilayer graphene (TBG) with Coulomb interaction projected into the flat bands and study the symmetries of the Hamiltonian. First, we show that all projected TBG Hamiltonians can be written as positive semidefinite Hamiltonians, an example of which was found in work by Kang and Vafek [Phys. Rev. Lett. 122, 246401 (2019)]. We then prove that the interacting TBG Hamiltonian exhibits an exact U(4) symmetry in the exactly flat band (nonchiral-flat) limit. We further define, besides a first chiral limit where the AA stacking hopping is zero, a second chiral limit where the AB/BA stacking hopping is zero. In the first chiral-flat limit (or second chiral-flat limit) with exactly flat bands, the TBG is enhanced to have an exact U(4)$\times$U(4) symmetry, whose generators are different between the two chiral limits. While in the first chiral limit and in the nonchiral case these symmetries have been found in work by Bultinck et al. [Phys. Rev. X 10, 031034 (2020)], for the eight lowest bands, we here prove that they are valid for projection into any $8n_{\text{max}}$ particle-hole symmetric TBG bands, with $n_{\text{max}}>1$ being the practical case for small twist angles $<1^{\circ }$. Furthermore, in the first or second chiral-nonflat limit without flat bands, an exact U(4) symmetry still remains. We also elucidate the link between the U(4) symmetry presented here and the similar but different U(4) of Kang and Vafek [Phys. Rev. Lett. 122, 246401 (2019)]. Furthermore, we show that our projected Hamiltonian can be viewed as the normal-ordered Coulomb interaction plus a Hartree-Fock term from passive bands, and exhibits a many-body particle-hole symmetry which renders the physics symmetric around charge neutrality. We also provide an efficient parametrization of the interacting Hamiltonian. The existence of two chiral limits with an enlarged symmetry suggests a possible duality of the model yet undiscovered.

Figure 1

Illustration of the relation between the graphene BZs of two layers and the moiré BZ (MBZ). Blue solid and red empty circles represent ${\mathcal{Q}}_{+}$ and ${\mathcal{Q}}_{-}$, respectively.

Figure 2

The single-valley TBG band structure at $\theta =1.{05}^{\circ }$ (with exact ph symmetry $P$) for (a) the nonchiral-nonflat limit with $w_0=0.8w_{\text{BM}}$ and $w_1=w_{\text{BM}}$, (b) the first chiral limit with $w_0=0$ and $w_1=w_{\text{BM}}$, and (c) the second chiral limit $w_0=w_{\text{BM}}$ and $w_1=0$, where $w_{\text{BM}}=110$ meV is the hopping in the original Bistritzer-MacDonald TBG model [1]. In particular, in the second chiral limit, the band structure is a perfect metal where all bands are connected (proof given in Ref. [108]).

Figure 3

The relations between the symmetries of projected Hamiltonian within any set of ph symmetric bands of full spin-valley flavors in various limits. The arrows point along the directions along which the symmetry groups are enhanced into a larger one.

Figure 4

The interaction $V\left(\mathbf{q}\right)$ as a function of $\xi q$ given by Eq. (C3).