Ferroically Ordered Magnetic Octupoles in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>d</mml:mi></mml:math>-Wave Altermagnets
Sayantika Bhowal, Nicola A. Spaldin
Abstract
We show that time-reversal symmetry-broken, centrosymmetric antiferromagnets with nonrelativistic spin splitting of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>d</a:mi></a:math>-wave symmetry—the so-called <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>d</c:mi></c:math>-wave —are conveniently described in terms of the ferroic ordering of magnetic octupoles. The magnetic octupoles are the lowest-order ferroically ordered magnetic quantity in this case and so are the natural order parameter for the transition into the magnetically ordered state. They provide a unified description of the broken time-reversal symmetry and the nonrelativistic spin splitting, as well as a platform for manipulating the latter, and account for other phenomena, such as piezomagnetism, characteristic of this class of antiferromagnets. Unusually for antiferromagnets, we show that the magnetic octupoles cause a nonzero magnetic Compton scattering, providing a route for their direct experimental detection. We illustrate these concepts using density-functional and model calculations for the prototypical nonrelativistic spin-split antiferromagnet, rutile-structure manganese difluoride <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:msub><e:mrow><e:mi>MnF</e:mi></e:mrow><e:mrow><e:mn>2</e:mn></e:mrow></e:msub></e:mrow></e:math>. Published by the American Physical Society 2024