Magnetization induced by a nonlinear response to temperature gradient in d, g and i altermagnets

It is a highly nontrivial question whether magnetization is induced by a nonlinear response to a temperature gradient in systems where the linear response is forbidden by inversion symmetry. In this work, we address this issue and provide explicit examples demonstrating that such a response can indeed arise. The spin-split electron band structures induced by d-wave, g-wave, i-wave altermagnets are characterized by k N X N Xϕ around the band bottom, where N X=2,4 and 6, respectively. In contrast, those of the corresponding d-wave, g-wave, i-wave altermagnets are described by k N X N Xϕ. We show that a finite magnetization is induced in the d-wave, g-wave, i-wave altermagnets under a second-order nonlinear response to a temperature gradient (d xT) squared when the temperature gradient is linear and along either the x or y direction, whereas no such response occurs in the d-wave, g-wave and i-wave altermagnets. We also study the case where the direction of the temperature gradient is arbitrary. In this case, a finite magnetization is induced also in d-wave altermagnets but not in g-wave and i-wave altermagnets. We discuss the difference in physics between the X-wave and X-wave altermagnets with X=d,g,i based on the tight-binding model. Furthermore, we show the emergence of the magnetization as a response when the temperature profile is not linear, d x squaredT not equal to 0.