Third-order and fifth-order nonlinear spin-current generation in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>g</mml:mi></mml:math>-wave and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>i</mml:mi></mml:math>-wa

A prominent feature of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>d</a:mi></a:math>-wave altermagnets is the pure spin current generated in the absence of spin-orbit interactions. In the context of symmetry, there are <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>s</b:mi></b:math>-wave, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>p</c:mi></c:math>-wave, <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>d</d:mi></d:math>-wave, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>f</e:mi></e:math>-wave, <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>g</f:mi></f:math>-wave, and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>i</g:mi></g:math>-wave magnets. In this paper, making an analytic study of two-band Hamiltonian systems coupled with electrons, we demonstrate unexpectedly that only the <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mi>ℓ</h:mi><h:mi>th</h:mi></h:mrow></h:math> order nonlinear transverse spin current proportional to <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:msup><i:mi>E</i:mi><i:mi>ℓ</i:mi></i:msup></i:math> is generated in higher-wave symmetric magnets when the number of nodes is <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:mi>ℓ</j:mi><j:mo>+</j:mo><j:mn>1</j:mn></j:mrow></j:math>. Here <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mi>E</k:mi></k:math> 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 <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mi>g</l:mi></l:math>-wave altermagnets, while only the fifth-order spin current is generated in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>i</m:mi></m:math>-wave altermagnets. In particular, only the second-order nonlinear spin current is generated in <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mi>f</n:mi></n:math>-wave magnets, which leads to a perfect nonreciprocal spin current. On the other hand, there is no spin-current generation in <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:mi>p</o:mi></o:math>-wave magnets.