Effects of Electron-Phonon Coupling on the Magnetic Order in Rare-Earth Nickelates

It is no coincidence that rare-earth nickel perovskites (RNiO$_3$) have recently been gathering a lot of attention. Their exciting and extensive array of novel properties, such as the metal-to-insulator transition (MIT), bond disproportionation and an unusual antiferromagnetic ground state, with the four-site spin order, characterized by the wavevector $\mathbf{q} = (\frac{1}{4},\frac{1}{4},\frac{1}{4})$ (with many of these properties tunable in thin film RNiO$_3$ and heterostructures), makes them highly desirable as a new generation ``quantum material'', for possible applications in electronics, energy storage, and even quantum computing. Given that a breathing-mode lattice distortion mirrors the MIT, its impact on bond disproportionation has been well-studied. However, fewer studies have considered the influence of distortion on the magnetic order. We construct an effective two-orbital Hubbard Hamiltonian and couple it to the distortion (which we treat semiclassically), then find the phase diagram using the Hartree-Fock approximation. We discuss how the different terms in the Hamiltonian affect the phase diagram, specifically the magnetic order stabilized in the various regions of parameter space.