Dualities and Topological Classification of the $S=1$ Pyrochlore Spin Ice

We resolve the phase diagram of the $S=1$ pyrochlore spin ice, which exhibits trivial paramagnetic, U(1) Coulomb, and spin nematic phases. In the monopole-free limit, the system can be effectively mapped onto 3D $XY$ and Ising loop-gas models depending on the spin anisotropy, which provides theoretical estimates for the phase boundaries, while a macroscopic flux vector classifies the topological sectors via geometric parity rules. At finite temperatures, thermal monopoles act as a symmetry-breaking field in both 3D $XY$ and Ising loop-gas pictures, rounding the phase transitions into continuous crossovers. These theoretical findings are corroborated by classical Monte Carlo simulations.