Nonlinear thermal gradient induced magnetization in $d^{\mathrm{\prime }}$, $g^{\mathrm{\prime }}$ and $i^{\mathrm{\prime }}$ altermagnets

It is a highly nontrivial question whether a magnetization can be induced by applying a nonlinear temperature gradient in the absence of any linear component. In this work, we address this issue and provide explicit examples demonstrating that such a response can indeed arise. The spin-split band structures of $d$-wave, $g$-wave, $i$-wave altermagnets are characterized by $k^{N_X}\sin N_X\phi$, where $N_X=2,4$ and $6$, respectively. In contrast, the corresponding $d^{\mathrm{\prime }}$-wave, $g^{\mathrm{\prime }}$-wave, $i^{\mathrm{\prime }}$-wave altermagnets are described by $k^{N_X}\cos N_X\phi$. We show that a finite magnetization is induced in the $d^{\mathrm{\prime }}$-wave, $g^{\mathrm{\prime }}$-wave, $i^{\mathrm{\prime }}$-wave altermagnets under a second-order nonlinear temperature gradient, whereas no such response occurs in the $d$-wave, $g$-wave, $i$-wave altermagnets. This constitutes the leading-order contribution because the linear response is forbidden by inversion symmetry. Furthermore, we derive analytic expressions for the induced magnetization in the high-temperature regime. We also demonstrate that no analogous nonlinear thermal response appears in $p$-wave, $f$-wave, $p^{\mathrm{\prime }}$-wave and $f^{\mathrm{\prime }}$-wave odd-parity magnets.