Nonlinear optical (NLO) effects have drawn considerable attention in condensed matter physics. In magnetically ordered materials, NLO responses are widely believed to be time-reversal odd ($T$-odd). In particular, the widely studied $PT$-symmetric collinear antiferromagnetic (AFM) materials have shown that their NLO responses depend critically on spin-orbit coupling (SOC). In this work, we demonstrate that $E$-type AFM materials, as a class of type-II multiferroic materials independent of SOC, exhibit NLO responses that break the traditional conception. Using both magnetic group and spin group methods, we compare the symmetry-breaking mechanisms between $PT$-symmetric AFM systems and $E$-type AFM systems. Through first-principles calculations on representative $E$-type AFM materials (${\mathrm{MnS}}_2, {\mathrm{LuMnO}}_3$, and ${\mathrm{NdNiO}}_3$), we find that the NLO coefficients of the two spin sublattices are identical in both magnitude and sign. Therefore, these NLO responses persist in the absence of SOC effect, and are time-reversal even ($T$-even). Additionally, we discover that the NLO responses can be effectively tuned across distinct magnetic polymorphs. The above results provide new insights into the symmetry origins and microscopic mechanisms of NLO effects in SOC-free magnetically ordered materials.

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• Antiferroelectricity
• Magneto-optics
• Multiferroics
• Nonlinear optical materials
• Density functional calculations