Spin-Polarized Nematic Order, Quantum Valley Hall States, and Field-Tunable Topological Transitions in Twisted Multilayer Graphene Systems

We theoretically study the correlated insulator states, quantum anomalous Hall (QAH) states, and field-induced topological transitions between different correlated states in twisted multilayer graphene systems. Taking twisted bilayer-monolayer graphene and twisted double-bilayer graphene as examples, we show that both systems stay in spin-polarized, $C_{3z}$-broken insulator states with zero Chern number at $1/2$ filling of the flat bands under finite displacement fields. In some cases these spin-polarized, nematic insulator states are in the quantum valley Hall (QVH) phase by virtue of the nontrivial band topology of the systems. The spin-polarized insulator state is quasidegenerate with the valley polarized state if only the dominant intravalley Coulomb interaction is included. Such quasidegeneracy can be split by atomic on-site interactions such that the spin-polarized, nematic state become the unique ground state. Such a scenario applies to various twisted multilayer graphene systems at $1/2$ filling, thus can be considered as a universal mechanism. Moreover, under vertical magnetic fields, the orbital Zeeman splittings and the field-induced change of charge density in twisted multilayer graphene systems would compete with the atomic Hubbard interactions, which can drive transitions from spin-polarized zero-Chern-number states to valley-polarized QAH states with small onset magnetic fields.

Figure 1

(a) The illustrations of twisted bilayer-monolayer graphene. The blue and red hexagonal lattice represent bilayer graphene, and brown lattice is monolayer graphene. (b) The noninteracting energy bands of 1.25°-twisted bilayer-monolayer graphene under $0.4\mathrm{V}/\mathrm{nm}$ electric displacement field. The solid (dashed) lines are the energy bands of the $K$ ($K^{\prime }$) valley. (c) and (d) The Chern numbers of the highest valence band and lowest conduction band under different electric displacement field $U_d$ ($-0.067-0.067\mathrm{eV}$) and varying twist angle $\theta$.

Figure 2

Hartree-Fock phase diagrams: (a) $U_d=-0.0536$ ($D=0.4\mathrm{V}/\mathrm{nm}$), (b) $U_d=0.0536\mathrm{eV}$ ($D=-0.4\mathrm{V}/\mathrm{nm}$) at $1/2$ filling of TBMG with $\theta =1.25°$. The color coding indicates Chern numbers, and Hubbard $U_0=2\mathrm{eV}$ in these calculations. The Hartree-Fock bandstructures of TBMG at $1/2$ filling with (c) $U_d=-0.0536$ and (d) $U_d=0.0536\mathrm{eV}$ ($\epsilon =9.6$ and $\kappa =0.005{\AA }^{-1}$). The blue and red lines represent the energy bands from different valleys.

Figure 3

The charge density distribution in real space at half filling for (a) $U_d=-0.0536$, (b) $U_d=0.0536$ in TBMG, and (c) $U_d=0.04\mathrm{eV}$ in TDBG. (d) The current loops from $K$ valley at the valley Hall phase in TBMG. The color coding represents the strength of magnetic field generated by the current loops, in units of Gauss. The current loops from the $K^{\prime }$ valley are opposite due to time-reversal symmetry.

Figure 4

The Hartree-Fock phase diagram around filling 2 in the parameter space of on-site Hubbard interaction and out-of-plane magnetic fields ($U_0$, $B_z$) in TBMG with $\epsilon =9.6$, $\kappa =0.005{\AA }^{-1}$, $\theta =1.25°$, and $U_d=0.0536\mathrm{eV}$: (a) the occupation is fixed at $n(C=0)=2$, and (b) with the occupation fixed at $n(C=2)=2+2B{\mathrm{\Omega }}_{M}e/h$. The color coding denotes the energy difference between VP and SP states $\mathrm{\Delta }E=E_{\mathrm{VP}}-E_{\mathrm{SP}}$ in (a)–(b). The shaded block in (a) and (b) indicate that the SP and VP states are nearly degenerate at the origin. We also present the calculated gapped states remarked by red lines under magnetic fields with (c) $U_0=0$, (d) $U_0=1.5$, and (e) $U_0=3\mathrm{eV}$ in the TBMG system.