  Band gap is known as an effective parameter for tuning the Lande $g$-factor
in semiconductors and can be manipulated in a wide range through the bowing
effect in ternary alloys. In this work, using the recently developed virtual
substrate technique, high-quality InAsSb alloys throughout the whole Sb
composition range are fabricated and a large $g$-factor of $g\approx -90$ at
the minimum band gap of $\sim 0.1$ eV, which is almost twice that in bulk InSb
is found. Further analysis to the zero gap limit reveals a possible gigantic
$g$-factor of $g\approx -200$ with a peculiar relativistic Zeeman effect that
disperses as the square root of magnetic field. Such a $g$-factor enhancement
toward the narrow gap limit cannot be quantitatively described by the
conventional Roth formula, as the orbital interaction effect between the nearly
triply degenerated bands becomes the dominant source for the Zeeman splitting.
These results may provide new insights into realizing large $g$-factors and
spin polarized states in semiconductors and topological materials.
