Quantum Geometry Induced Nonlinear Transport in Altermagnets

Quantum geometry plays a pivotal role in the second-order response of PT-symmetric antiferromagnets. Here we study the nonlinear response of 2D altermagnets protected by C_{n}T symmetry and show that their leading nonlinear response is third order. Furthermore, we show that the contributions from the quantum metric and Berry curvature enter separately: the longitudinal response for all planar altermagnets only has a contribution from the quantum metric quadrupole (QMQ), while transverse responses in general have contributions from both the Berry curvature quadrupole (BCQ) and QMQ. We show that for the well-known example of d-wave altermagnets the Hall response is dominated by the BCQ. Both longitudinal and transverse responses are strongly dependent on the crystalline anisotropy. While altermagnets are strictly defined in the limit of vanishing spin orbit coupling (SOC), real altermagnets exhibit weak SOC, which is essential to observe this response. Specifically, SOC gaps the spin-group protected nodal line, generating a response peak that is sharpest when SOC is weak. Two Dirac nodes also contribute a divergence to the nonlinear response, whose scaling changes as a function of SOC. Finally, we apply our results to thin films of the 3D altermagnet RuO_{2}. Our work uncovers distinct features of altermagnets in nonlinear transport, providing experimental signatures as well as a guide to disentangling the different components of their quantum geometry.