Twisted symmetric trilayer graphene. II. Projected Hartree-Fock study

The Hamiltonian of the magic-angle twisted symmetric trilayer graphene (TSTG) can be decomposed into a twisted-bilayer-graphene- (TBG-) like flat band Hamiltonian and a high-velocity Dirac fermion Hamiltonian. We use Hartree-Fock mean field approach to study the projected Coulomb interacting Hamiltonian of TSTG developed in Călugăru et al. [Phys. Rev. B 103, 195411 (2021)] at integer fillings $\nu =-3,-2,-1$, and 0 measured from charge neutrality. We study the phase diagram with $w_0/w_1$, the ratio of $AA$ and $AB$ interlayer hoppings, and the displacement field, which introduces an interlayer potential $U$ and hybridizes the TBG-like bands with the Dirac bands. At small $U$, we find the ground states at all fillings $\nu$ are in the same phases as the tensor products of a Dirac semimetal with the filling $\nu$ TBG insulator ground states, which are spin-valley polarized at $\nu =-3$, and fully (partially) intervalley coherent at $\nu =-2,0$ ($\nu =-1$) in the flat bands. An exception is $\nu =-3$ with $w_0/w_1\gtrsim 0.7$, which possibly becomes a metal with competing orders at small $U$ due to charge transfers between the Dirac and flat bands. At strong $U$ where the bandwidths exceed interactions, all the fillings $\nu$ enter a metal phase with small or zero valley polarization and intervalley coherence. Lastly, at intermediate $U$, semimetal or insulator phases with zero intervalley coherence may arise for $\nu =-2,-1,0$. Our results provide a simple picture for the electron interactions in TSTG systems, and reveal the connection between the TSTG and TBG ground states.

Figure 1

(a) The phase diagram at filling factor $\nu =-3$ obtained on an $8\times 8$ momentum lattice in the $(w_0,U)$ plane. The color represents the valley polarization $N_v/N_M$ of the ground state. (b) The displacement field dependence of other quantities $\mathcal{C},N_v,S^{\pm }$, and $\mathrm{Ch}$ on an $8\times 8$ lattice at $w_0/w_1=0.2$. (c) Similar to (b), the displacement field dependence of these quantities with $w_0/w_1=0.8$. (d) The filling factors for Dirac fermions and TBG fermions as a function of $w_0$. In this plot, the interlayer potential is fixed to be $U=0\mathrm{meV}$.

Figure 2

Some typical HF band structures illustrating the three regions of the phase diagram at filling factor $\nu =-3$ on a $10\times 10$ momentum lattice. (a) The band structure in region I with $w_0/w_1=0.2$ and $U=50\mathrm{meV}$. (b) The band structure in region II with $w_0/w_1=0.2$ and $U=180\mathrm{meV}$. (c) The band structure in region III with $w_0/w_1=1$ and $U=0\mathrm{meV}$. The color of each point represents the valley polarization $v_i\left(\mathbf{k}\right)$ of each single-body state, which is defined in Eq. (35).

Figure 3

(a) The phase diagram at filling factor $\nu =-2$ obtained on an $8\times 8$ momentum lattice in the $(w_0,U)$ plane, and the color represents the intervalley coherence, which is defined in Eq. (41). (b), (c) The displacement field dependence of physical quantities $\mathcal{C},N_v,\mathrm{Ch}$, and $S^{\pm }$ on an $8\times 8$ at fixed $w_0/w_1=0.2$ (b) and $w_0/w_1=0.8$ (c). By considering the different HF parameters and band structures, we can define three different regions in the phase diagram, denoted I, II, and III in (a).

Figure 4

The HF band structure at $w_0/w_1=0.8$ for $U=0$ (a), at $w_0/w_1=0.2$ for $U=100\mathrm{meV}$ (b), at $w_0/w_1=0.8$ for $U=220\mathrm{meV}$ (c), and at $w_0/w_1=0.2$ for $U=180\mathrm{meV}$ (d) on a $10\times 10$ lattice at filling factor $\nu =-2$. The color represents the valley polarization $v_i\left(\mathbf{k}\right)$ of each single-body state defined in Eq. (35).

Figure 5

(a) The phase diagram at filling factor $\nu =-1$ obtained on an $8\times 8$ momentum lattice in the $(w_0,U)$ parameter space. The color represents the intervalley coherence $\mathcal{C}$. (b), (c) The displacement field dependence of physical quantities $\mathcal{C},N_v,\mathrm{Ch}$, and $S^{\pm }$ on an $8\times 8$ lattice at fixed $w_0/w_1=0.2$ (b) and $w_0/w_1=0.8$ (c).

Figure 6

The HF band structure at $w_0/w_1=0.8$ for $U=0$ in region I (a), $w_0/w_1=0.8$ for $U=240\mathrm{meV}$ in region II (b), $w_0/w_1=0.2$ for $U=180\mathrm{meV}$ in region III (c), and $w_0/w_1=0.2$ for $U=280\mathrm{meV}$ in region IV (d) on a $10\times 10$ lattice at filling factor $\nu =-1$. The color represents the valley polarization $v_i\left(\mathbf{k}\right)$ of each single-body state.

Figure 7

Phase diagrams at filling factor $\nu =0$. (a) The two-dimensional phase diagram on an $8\times 8$ momentum lattice in $(w_0,U)$ parameter space. It can be seen that in the weak $U$ phase, the intervalley coherence $\mathcal{C}\approx 1$ shows that there are four occupied intervalley coherent bands. (b) The energy gap along the high-symmetry lines as a function of $w_0/w_1$ and $U$. Here we use the method discussed in the Supplemental Material [131] to obtain the Hartree-Fock Hamiltonian along the high-symmetry lines, therefore, we are able to estimate the energy gap from the $8\times 8$ lattice. (c), (d) The displacement field dependence of several quantities $\mathcal{C},N_v,S^{\pm }$, and $\mathrm{Ch}$ on an $8\times 8$ lattice with $w_0/w_1=0.2$ (c) and $w_0/w_1=0.8$ (d).

Figure 8

(a)–(c) The HF band structure on a $10\times 10$ lattice at filling factor $\nu =0$ at $w_0/w_1=0.8$ for $U=50\mathrm{meV}$ in region I (a), at $w_0/w_1=0.8$ for $U=200\mathrm{meV}$ in region II (b), and at $w_0/w_1=0.2$ for $U=250\mathrm{meV}$ in region III (c), respectively. The color stands for the valley polarization $v_i\left(\mathbf{k}\right)$ of each single-body state. The zoom-in band structures around $K_M, K_M^{\prime }$, and ${\mathrm{\Gamma }}_M$ points in the dashed boxes in (c) are also shown. It is visible that the HF band structure is discontinuous at these points, and it is also gapless at $K_M$ and $K_M^{\prime }$ points.