Quantum geometry and <i>X</i> -wave magnets with <i>X = p, d, f, g, i</i>

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. In particular, the latter is quantum information geometry, where the quantum Fisher information naturally arises as a 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 effects. 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 the recent progress together with some original results.