Altermagnetic instabilities from quantum geometry

Altermagnets are a newly identified type of collinear antiferromagnetism with vanishing net magnetic moment, characterized by lifted Kramers' degeneracy in parts of the Brillouin zone. Their time-reversal symmetry-broken band structure has been observed experimentally and is theoretically well understood. On the contrary, altermagnetic fluctuations and the formation of the corresponding instabilities remains largely unexplored. We establish a correspondence between the quantum metric of normal and the altermagnetic spin-splitting of ordered phases. We analytically derive a criterion for the formation of instabilities and show that the quantum metric favors altermagnetism. We recover the expression for conventional <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>q</a:mi><a:mo>=</a:mo><a:mn>0</a:mn></a:mrow></a:math> instabilities where the spin-splitting terms of the normal-state model are locally absent. As an example, we construct an effective model of MnTe and illustrate the relationship between quantum geometry and altermagnetic fluctuations by explicitly computing the quantum metric and the generalized magnetic susceptibility.