Plaquette chirality patterns for robust zero-gap states in $\alpha$-type organic conductor

Dirac electrons with a zero-gap state (ZGS) in organic conductor $\alpha$-(BEDT-TTF)$_2$I$_3$ result from a fine tuning of the seven nearest neighbors transfer integrals ($a_1, a_2, a_3, b_1, b_2, b_3, b_4$) between the four molecules of the unit cell. In this work we show that for given moduli $|a_1|,...|b_4|$, the possibility of having Dirac electron with a ZGS at $3/4$ (or $1/4$) filling strongly depends on the specific configurations of signs of the seven transfer integral. More precisely it is possible to classify the sign configurations into essentially four classes determined by $\chi_{a}={\rm sign} (a_2a_3)$ and $\chi_b={\rm sign} (b_1b_2b_3b_4)$. Using extended numerics, we show that for both weak and large inhomogeneity in the moduli, the class $(\chi_a,\chi_b)=(-,-)$ is the most favorable to find Dirac electrons with ZGS at $3/4$ (or $1/4$) filling. For the class $(\chi_a,\chi_b)=(+,+)$ in the opposite case, we never found any ZGS at either $1/4$ or $3/4$ filling. The last two classes \suzum{given by} $(\chi_a,\chi_b)=(+,-)$ and $(\chi_a,\chi_b)=(-,+)$ corresponding to an intermediate situation; they allow for ZGS at $3/4$ (resp. $1/4$) filling but are much less favorable than class $(\chi_a,\chi_b)=(-,-)$. As a matter of fact, all previous numerical studies of Dirac electrons and ZGS in $\alpha$-(BEDT-TTF)$_2$I$_3$ correspond to class $(\chi_a,\chi_b)=(-,+)$.