Quantum geometry for orbital magnetization and spintronics from parallel transport of Bloch states

Quantum geometry emerges as a unifying and quantitative guiding principle for linear and non-linear response functions of quantum matter. Going beyond current-current responses [1,2], we identify the generator of parallel transport of Bloch states, given by the commutator of the band projector and its momentum derivative, as a further essential building block for understanding responses arising from non-trivial Bloch states [3]. As a first application, I will show that orbital magnetism arises from the non-commutativity of the adiabatic transport generators in orthogonal directions. As a second application, I will discuss the quantum geometry of spin-current responses from the perspective of adiabatic Bloch state transport. We will see that the spin Berry curvature and, analogously, the spin quantum metric separate into three distinct contributions, of which the spin-mediated interband transitions and the spin-torque contribution involve quasiparticle velocities and direct band gaps. We will identify a third, entirely quantum geometric contribution arising from combined adiabatic transport and interband transitions, which is responsible for the spin quantum anomalous Hall effect. Besides making the connection to adiabatic transport theory precise, our theory enables numerical and analytically efficient evaluations for general Bloch Hamiltonians with an arbitrary number of potentially degenerate bands. I will exemplify the results in application to altermagnets [4] and p-wave magnets [5] and conclude how our framework streamlines the systematic study of non-linear spin responses, including interaction and disorder effects, via diagrammatic Green's function approaches.

[1] A. Avdoshkin*, J. Mitscherling*, J. E. Moore, PRL 135, 066901 (2025).

[2] J. Mitscherling*, A. Avdoshkin*, and J. E. Moore, PRB 112, 085104 (2025).

[3] J. Mitscherling and L. Smejkal, to be submitted.

[4] L. Smejkal, J. Sinova, and T. Jungwirth, PRX 12, 031042 (2022).

[5] A. Birk Hellenes, T. Jungwirth, R. Jaeschke-Ubiergo, A. Chakraborty, J. Sinova, and L. Smejkal, arXiv:2309.01607v3.