Contribution of remote bands to orbital magnetization in twisted bilayer graphene

Abstract

Motivated by recent theoretical and experimental works on orbital magnetization $M_{\mathrm{orb}}$ for the interacting system, we develop a gauge-invariant framework to compute $M_{\mathrm{orb}}$ for correlated phases of magic-angle twisted bilayer graphene within self-consistent Hartree-Fock approximation. Based on the projector formulation of the theory of orbital magnetization, we evaluate both $M_{\mathrm{orb}}$ and the self-rotation contribution $m_{\mathrm{SR}}$ directly from the Hartree-Fock Hamiltonian. We demonstrate that, in contrast to topological invariants such as the Chern number, both $M_{\mathrm{orb}}$ and $m_{\mathrm{SR}}$ obtain substantial contributions from remote bands and thus require careful convergence with respect to the number of included remote bands. Applying this approach to correlated phases at integer fillings, we obtain converged $M_{\mathrm{orb}}$ and $m_{\mathrm{SR}}$ for time reversal symmetry broken Chern insulating states at $ν=\pm3$ and for competing correlated phases at other integer fillings. Our results establish a systematic and controlled approach for evaluating orbital magnetization in correlated moiré systems and clarify the crucial role of remote bands in determining their magnetic response.

Figures (4)

• Figure 1: (a) Low-energy band structure of non-interacting TBG calculated using the BM model. The black solid and red dashed curves correspond to the $K’$ and $K$ valleys, respectively. (b) Zoomed-in view of the isolated flat bands near charge neutrality, highlighting their separation from the higher-energy remote bands. (c) Schematic illustration of Brillouin zone folding in TBG. The small hexagon denotes the moiré Brillouin zone (mBZ), reciprocal to the moiré superlattice. The cyan arrows indicate the momentum path in the enlarged moiré Brillouin zone along which the band dispersions in panels (a) and (b) are plotted.
• Figure 2: (a,b) Band dispersions of the Chern insulating states at fillings $\nu = 3$ (a) and $\nu = -3$ (b), obtained within the HF approximation. The solid curves represent the eight interaction-renormalized HF bands, while the dashed curves denote the remote bands that are excluded from the self-consistent HF calculation. Bands carrying nontrivial Chern numbers are highlighted in red ($C = 1$) and blue ($C = -1$), corresponding to the experimentally observed Chern insulating states with $C = \pm 1$ at $\nu = \pm 3$. (c,d) Calculated total orbital magnetization $M_{\mathrm{orb}}$ and self-rotation contribution $m_{\mathrm{SR}}$ per moiré cell for the states shown in (a,b), plotted as functions of the number of remote-band pairs ($n_{\mathrm{cut}}$) included in the projection-matrix formalism. Both quantities exhibit clear convergence for $n_{\mathrm{cut}} \approx 20$. When evaluating $M_{\mathrm{orb}}$, the chemical potential $\mu$ is fixed at the top of the valence band for $\nu = 3$ and at the bottom of the conduction band for $\nu = -3$, as indicated by the horizontal dashed lines in panels (a) and (b).
• Figure 3: Calculated total orbital magnetization $M_{\mathrm{orb}}$ (a) and self-rotation contribution $m_{\mathrm{SR}}$ (b) per moiré cell as functions of the chemical potential for the Chern insulating state at $\nu = 3$ inside the insulating gap, whose band dispersion is shown in Fig. \ref{['fig:fig2']}(a). $M_{\mathrm{orb}}$ shows a linear dependence with $\mu$ and changes sign within the gap while $m_{\mathrm{SR}}$ remains constant. The number of remote band pairs $n_{cut}$ is set to 30 in the calculation.
• Figure 4: (a,b) Band dispersions of the correlated Chern insulating state (CCI) (a) and the intervalley-coherent (IVC) state (b) at filling $\nu = 1$, obtained within the HF approximation. The solid curves represent the eight interaction-renormalized HF bands, while the dashed curves denote the remote bands excluded from the self-consistent HF calculation. Bands carrying nontrivial Chern numbers are highlighted in red ($C = 1$) and blue ($C = -1$). For the CCI, the Chern bands are triply degenerate, yielding a total Chern number $C = 3$. (c,d) Calculated total orbital magnetization $M_{\mathrm{orb}}$ (c) and self-rotation contribution $m_{\mathrm{SR}}$ (d) per moiré cell as functions of the chemical potential $\mu$ inside the insulating gap for the two states shown in (a) and (b). Solid circles correspond to the CCI, and solid squares to the IVC state. $M_{\mathrm{orb}}$ varies linearly with $\mu$ and changes sign across the gap, while $m_{\mathrm{SR}}$ remains constant. The number of remote band pairs $n_{cut}$ is set to 30 in the calculation.