Topological Phase Transitions and Their Thermodynamic Fate in Arbitrary- S Classical Pyrochlore Spin Ice

We develop a self-contained theoretical framework that classifies the topological phases and critical phenomena of classical pyrochlore magnets with arbitrary spin S, subject to competing exchange and single-ion anisotropies. We work first in the monopole-free (ice-rule) limit. In the small- w regime, where the single-ion term favors low spin amplitudes, dualities reveal a dichotomy: Integer spins exhibit a continuous 3D XY deconfinement transition, whereas half-integer spins remain in a U(1) Coulomb liquid without any transition. In the large- w regime, where S z = S, the macroscopic flux is quantized to multiples of 2S. Mapping the defect structure to loop gases, we prove that the ice rule is compatible with the modular Z 2S flux conservation at the vertex level if and only if S 3/2. For S=3/2, this maps the system to the three-state Potts model, whose symmetry-allowed cubic invariant drives a first-order transition. For S 2, no discrete clock mapping exists; a decomposition of the partition function shows that the only processes sensitive to the discrete Z 2S structure carry exponentially small weight and correspond to irrelevant operators at the 3D XY fixed point, so the deconfinement transition is of the 3D XY type. Finally, incorporating thermal monopoles, we show that they act as a symmetry-breaking effective magnetic field that severs defect strings. Consequently, the continuous transitions are rounded into crossovers, whereas the first-order S=3/2 transition is predicted to survive at finite temperatures, terminating at a critical end point. Classical Monte Carlo simulations for S up to 7/2 corroborate the predicted crossovers at the accessible temperatures, which lie above the estimated critical end point, so the survival of the S=3/2 first-order transition remains to be tested directly.