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 X Y deconfinement transition, whereas half-integer spins remain in a U(1) Coulomb liquid without any transition. In the large- w regime, where the local spin amplitudes are maximized ( | S z | = S ), the macroscopic flux is quantized to multiples of 2 S . By mapping the defect structure to topological loop gases, we prove that the compatibility between the physical ice rule and the modular Z 2 S flux conservation holds 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 mapping to a discrete clock model exists. Using a decomposition of the partition function, we show that the only processes sensitive to the discrete Z 2 S structure carry exponentially small statistical weight and correspond to irrelevant operators at the 3D X Y fixed point, so the deconfinement transition is of the 3D X Y 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 crossover behavior at the accessible temperatures; these temperatures lie above the estimated critical end point, so the predicted survival of the S = 3 / 2 first-order transition remains to be tested directly.