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We point out the emergence of magnetism from the interplay of electronquasi-itinerancy and quantum order by disorder in pyrochlore iridates. Likeother Mott insulating iridates, the Ir$^{4+}$ ion in pyrochlore iridatesdevelops an effective ${J=1/2}$ moment from the on-site spin-orbit coupling. Weconsider the generic symmetry-allowed exchange between these local moments on apyrochlore lattice and obtain the mean-field phase diagram. Assuming thesuperexchange is mediated by direct and/or indirect electron hopping viaintermediate oxygens, we derive the exchange interactions in the strongcoupling regime from the Hubbard model. This exchange has a degenerateclassical ground state manifold and quantum fluctuation selects a non-coplanarground state, known as quantum order by disorder. Extending to the intermediatecoupling regime, the same non-coplanar order is selected from the degeneratemanifold by the kinetic energy, which is dubbed "electron quasi-itinerancy". Wediscuss the experimental relevance of our results and electron quasi-itinerancyamong other iridates and $4d/5d$ magnets.

Electron quasi-itinerancy intertwined with quantum order by disorder in pyrochlore iridate magnetism