Fig. 4.

The results of the Hartree–Fock calculation. a Color plot (meV) for $E_{\mathrm{IVC}}-E_{\mathrm{VP}}$ per electron. b Color plot of self-consistency gap ${\mathrm{\Delta }}_{\mathrm{SP∕VP}}$ for the SP/VP-state in the band isolated region. (No $J$-term included) c, d Effect of the intervalley Hund’s coupling ($J$-term) on the gap for spin- and valley-polarized phases at half and quater fillings, respectively. At half-filling, $J$-term increases (decreases) ${\mathrm{\Delta }}_{\mathrm{SP}}$ (${\mathrm{\Delta }}_{\mathrm{VP}}$). At quarter-filling, $J$-term reduces the gap to the next-excited state, making the quarter-filled insulator (SP + VP) less stable than the half-filled (SP) one. e The correlated gap $\mathrm{\Delta }$ for half-filling insulators (SP, VP) as a function of in-plane $B_x$-field. $\left(\theta ,U\right)=\left(1.33^{\circ },60\mathrm{meV}\right)$. Solid lines for SP state and dotted lines for VP-state. Zeeman effect would increase (decrease) $\mathrm{\Delta }$ for the SP (VP) state with increasing $B$. The valley orbital effect $g^{x,y}\left(\mathbf{k}\right)$ leads to a linear decrease in the gap with field, thus effectively decreasing (increasing) the $g$-factor for the SPS (VP) state.