Third-order and fifth-order nonlinear spin-current generation in $g$-wave and $i$-wave altermagnets, and perfectly nonreciprocal spin-current in $f$-wave magnets

Abstract: A prominent feature of $d$-wave altermagnets is the pure spin current generated in the absence of spin-orbit interactions. In the context of symmetry, there are the $s$-wave, the $p$-wave, the $d$-wave, the $f$-wave, the $g$-wave and the $i$-wave magnets. In this paper, making an analytic study of two-band Hamiltonian systems coupled with electrons, we demonstrate unexpectedly that only the $\ell$-th order nonlinear transverse spin current proportional to $E^{\ell }$ is generated in higher-wave symmetric magnets when the number of the nodes is $\ell +1$. Here $E$ is applied electric field. The nonlinear spin current is essential provided the linear spin current is absent. Indeed, only the third-order nonlinear spin current is generated in $g$-wave altermagnets, while only the fifth-order spin current is generated in $i$-wave altermagnets. In particular, only the second-order nonlinear spin current is generated in $f$-wave magnets, which leads to a perfect nonreciprocal spin current. On the other hand, there is no spin-current generation in $p$-wave magnets.