Purely electrical detection of the spin-splitting vector in p-wave magnets based on linear and nonlinear conductivities

A $p$-wave magnet has a momentum-dependent spin splitting and zero-net magnetization just as in the case of an altermagnet. It will be useful for high-density and ultra-fast memory, where the direction of the spin-splitting vector may be used as a bit. The spin-splitting vector corresponds to the N\'{e}el vector in altermagnets. However, it is a nontrivial problem to detect the spin-splitting vector in a $p$-wave magnet because time-reversal symmetry is preserved, while this is not a problem in an altermagnet because the anomalous Hall conductivity is present due to the breaking of time-reversal symmetry. Here, we show that it is possible to detect the in-plane component of the spin-splitting vector in the $p$-wave magnet by measuring the linear transverse and longitudinal Drude conductivity. Remarkably, this measurement is possible without using magnetization. Furthermore, we study the nonlinear Drude conductivity, the quantum-metric and the Berry curvature dipole induced nonlinear conductivity in the presence of tiny magnetization along the $z$\ axis. It is possible to detect the $z$-component of the spin-splitting vector by measuring the above nonlinear conductivities. We obtain analytic formulae for them in the first-order perturbation theory, which agree quite well with numerical results without perturbation.