  We study the effects of marginally spinful electron-electron interactions on
the low-energy instabilities and favorable phase transitions in a
two-dimensional (2D) spin-$1/2$ semimetal that owns a quadratic band crossing
point (QBCP) parabolically touched by the upper and lower bands. In the
framework of a renormalization group procedure, all sorts of interactions are
treated on the equal footing to derive the coupled energy-dependent evolutions
of all interaction couplings that govern the low-energy properties. Deciphering
the essential physical information of such flows, we at first find that the
tendencies of interaction parameters fall into three categories including Limit
case, Special case, and General case based on the initial conditions. In
addition, the 2D QBCP system is attracted to several distinct kinds of fixed
points (FPs) in the interaction-parameter space, namely
$\mathrm{FP}_1^{+}$/$\mathrm{FP}_2^{-}$, $\mathrm{FP}_1^{\pm}$/
$\mathrm{FP}_2^{\pm}$/$\mathrm{FP}_3^{\pm}$, and
$\mathrm{FP}_1^{\pm}/\mathrm{FP}_3^{\pm}/\mathrm{FP}_{41,42,43}^{\pm}$ with the
subscripts characterizing the features of FPs for the Limit, Special, and
General cases, respectively. Furthermore, as approaching these FPs, we
demonstrate that the spinful fermion-fermion interactions can induce a number
of favorable instabilities accompanied by certain phase transitions.
Specifically, the quantum anomalous Hall (QAH), quantum spin Hall (QSH), and
nematic (Nem.) site(bond) states are dominant for $\mathrm{FP}_{1}^{\pm}$,
$\mathrm{FP}_{2}^{\pm}$, and $\mathrm{FP}_{3}^{\pm}$, respectively. Rather, QSH
becomes anisotropic nearby $\mathrm{FP}_{41,42,43}^{\pm}$ with one component
leading and the others subleading. Besides, Nem.site(bond), chiral
superconductivity, and nematic-spin-nematic (NSN.) site(bond) are subleading
candidates around these FPs.
