Abstract
Quantum geometry is a differential geometry based on quantum mechanics. It is related to various transport and optical properties in condensed matter physics. The Zeeman quantum geometry is a generalization of quantum geometry including the spin degrees of freedom. It is related to electromagnetic cross responses. Quantum geometry is generalized to non-Hermitian systems and density matrices. Especially, the latter is quantum information geometry, where the quantum Fisher information naturally arises as quantum metric. We apply these results to the X-wave magnets, which include d% -wave, g-wave and i-wave altermagnets as well as p-wave and f-wave magnets. They have universal physics for anomalous Hall conductivity, tunneling magneto-resistance and planar Hall effect. We also study magneto-optical conductivity, magnetic circular dichroism and Friedel oscillations in the X-wave magnets. Various analytic formulas are derived in the case of two-band Hamiltonians. This paper presents a review of recent progress together with some original results.