Fourth- and sixth-order nonlinear spin current rectifier in three-dimensional h -wave and j -wave odd-parity magnets

Higher-order symmetric X -wave magnets consist of two groups. One includes d -wave, g -wave, and i -wave altermagnets, while the other includes p -wave and f -wave odd-parity magnets. Recently, the possibility of h -wave magnets has been discussed. Motivated by this development, we systematically construct an X -wave magnet with ( N X + 1 ) nodes in three dimensions from an X -wave magnet with N X nodes in two dimensions by means of a dimensional extension, where N X = 1 , 2 , 3 , 4 , 6 for X = p , d , f , g , i , respectively. Based on this method, we predict j -wave magnets in three dimensions. Then, we argue how to identify each of these X -wave magnets experimentally. We show that the X -wave magnet is completely identified by measuring the nonlinear spin currents. In particular, we predict that there are no spin currents other than the fourth-order ones such as σ spin x 3 y ; z in h -wave odd-parity magnets in three dimensions and the sixth-order ones such as σ spin x 5 y ; z in j -wave odd-parity magnets in three dimensions. They function as spin-current rectifiers because the spin current exhibits unidirectional flow independent of the applied electric field.