Purely electrical detection of the spin-splitting vector in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-wave magnets based on linear and nonlinear conductivities

A <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>p</a:mi></a:math>-wave magnet has zero-net magnetization and induces a momentum-dependent spin-splitted band structure just as in the case of an altermagnet. It will be useful for high-density and ultrafast memory, where the direction of the spin-splitting vector may be used as a bit. However, it is a nontrivial problem to detect the spin-splitting vector in a <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>p</b:mi></b:math>-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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>p</c:mi></c:math>-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 <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>z</d:mi></d:math> axis. It is possible to detect the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>z</e:mi></e:math>-component of the spin-splitting vector by measuring the above nonlinear conductivities. We obtain analytic formulas for them in the first-order perturbation theory, which agree quite well with numerical results without perturbation.