While second-order nonlinear conductivities are extensively applied to detect the Néel vector direction of antiferromagnets, they are absent in $p$-wave magnets (PWMs), an unconventional type of magnet, due to their time ($\mathcal{T}$) reversal and $p$-wave band symmetry. Here, we extend the study of nonlinear charge transports to nonlinear spin response. We first derive a quantum formula to calculate the nonlinear spin polarization, which can be applied to a system with an alternating electric field and weak disorder. In PWMs, we find that the nonlinear spin polarization is greatly enhanced near the Dirac point where the mixed Berry curvature (BC) becomes significant due to strong interband coherence. The nonlinear spin response near the Dirac point is exclusively contributed by the $\mathcal{T}$-even component induced by mixed BC, excluding the $\mathcal{T}$-odd component induced by the mixed quantum metric, which is crucial in an altermagnet. Importantly, the nonlinear spin polarization heavily depends on the Néel vector direction and spin splitting parameter in PWMs. Thus, the $\mathcal{T}$-even nonlinear spin polarization is helpful for understanding the physic of mixed BC and further provides an alternate detection scheme of the Néel vector, rather than one based on the usual nonlinear conductivity.

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• Berry curvature
• Electrical conductivity
• Spin polarization
• Spintronics