Gap opening mechanism for correlated Dirac electrons in organic compounds $\alpha$-(BEDT-TTF)${}_2$I${}_3$ and $\alpha$-(BEDT-TSeF)${}_2$I${}_3$

To determine how electron correlations open a gap in two-dimensional massless Dirac electrons in the organic compounds $\alpha$-(BEDT-TTF)${}_2$I${}_3$ [$\alpha$-(ET)${}_2$I${}_3$] and $\alpha$-(BEDT-TSeF)${}_2$I${}_3$ [$\alpha$-(BETS)${}_2$I${}_3$], we derive and analyze $ab$ $initio$ low-energy effective Hamiltonians for these two compounds. We find that the horizontal stripe charge ordering opens a gap in the massless Dirac electrons in $\alpha$-(ET)${}_2$I${}_3$, while an insulating phase without explicit symmetry breaking appears in $\alpha$-(BETS)${}_2$I${}_3$. We clarify that the combination of the anisotropic transfer integrals and the electron correlations induces a dimensional reduction in the spin correlations, i.e., one-dimensional spin correlations develop in $\alpha$-(BETS)${}_2$I${}_3$. We show that the one-dimensional spin correlations open a gap in the massless Dirac electrons. Our finding paves the way for opening gaps for massless Dirac electrons using strong electronic correlations.