Mapping the Phase Diagram of a Frustrated Magnet: Degeneracies, Flat Bands, and Canting Cycles on the Pyrochlore Lattice

We map the complete classical phase diagram of the spin Hamiltonian describing pyrochlore rare-earth magnets with all symmetry-allowed nearest-neighbor bond-dependent anisotropic two-spin interactions. We provide a simple derivation of the organization of spins into tensor degrees of freedom describing the multipole moments of a tetrahedron, whose components correspond to irreducible representations (irreps) of the tetrahedral symmetry group $T_d$. By parameterizing the Hamiltonian directly in terms of the energies of the individual irreps, we perform an exhaustive search of all possible irrep degeneracies which may host stable classical spin liquids. Doing so reveals four one-parameter families of models along which three phases are degenerate, all four of which merge at the Heisenberg antiferromagnet and its dual, and we give a complete three-dimensional picture of the phase diagram showing all of the phases and their intersections. The appearance of two copies of a single irrep implies an extra degenerate locus which pierces the phase boundaries at special points, yielding two additional isolated triple points. We demonstrate that one-parameter families of Hamiltonians are characterized by a topological invariant describing how the ground state spin configuration winds when adiabatically transported around this degenerate locus, analogous to a "diabolical locus". We provide a comprehensive catalog of all flat band degeneracies in the phase diagram and discuss the mechanisms that may allow for or impede the realization of a variety of classical spin liquids described by tensor gauge fields exhibiting pinch line singularities and concomitant four-fold pinch points. Lastly, we provide a list of all cases where three irreps are degenerate above the ground state, which may lead to interesting features in the spin wave spectrum within each ordered phase.