Magnetic excitations in quantum rare earth pyrochlores

Rare-earth pyrochlores are materials with chemical formula A₂B₂O₇, where A is the rare-earth ion and B is a transition metal. At low temperature, these systems host various magnetic states such as spin ice, spin liquid state, ferromagnetic ordering, all in-all out, and anti-ferromagnetic ordering. For each rare-earth ion with total angular momentum J, the 2J + 1 fold degeneracy splits into singlets and doublets due to the crystal electric field. However, the crystal electric field ground state for most of the magnetic ions is a doublet that comes into three different varieties, labeled as ┌₃, ┌₄, ┌₅,₆. This work focuses only on systems in which the ground state doublet is well-separated from the first excited state so that we end up with effective two=state systems, referred to as quantum rare-earth pyrochlores. The low temperature excitations of interacting spins have a wave nature and are referred to as spin waves or magnons, where the energy of these waves is quantized. To study these magnons, we apply the Holstein-Primakoff transformation on the effective spin Hamiltonian to construct a bosonic Hamiltonian that describes magnons. In this study, we limit ourselves to the linear spin-wave approximation in which we diagonalize the magnonic Hamiltonian analytically and numerically for various systems of interest. In particular, we study magnons in Nd₂Zr₂O₇ which orders in an all in-all out state near 0.285 K, in Er₂Ti₂O₇ with a antiferromagnetic state below 1.2 K, and finally the Yb₂Ti₂O₇ which orders ferromagnetically near 0.2 K