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

$g$-factor engineering with InAsSb alloys toward zero band gap limit