For arbitrary S, integer spins show a continuous 3D XY deconfinement transition in the small-anisotropy regime while half-integer spins stay in a U(1) Coulomb liquid; only S=3/2 retains a first-order transition at finite temperature because thermal monopoles round all continuous transitions into a 3
Since at most two fundamental strings can be fused per vertex (using the two outward background links), the process requires a chain of 2S−2 diamond- lattice verticesr 1,r 2,
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Topological Phase Transitions and Their Thermodynamic Fate in Arbitrary-$S$ Pyrochlore Spin Ice
For arbitrary S, integer spins show a continuous 3D XY deconfinement transition in the small-anisotropy regime while half-integer spins stay in a U(1) Coulomb liquid; only S=3/2 retains a first-order transition at finite temperature because thermal monopoles round all continuous transitions into a 3