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

Sena Watanabe1,*, Yukitoshi Motome1,†, and Haruki Watanabe2,3,4,1,‡

  • *Contact author: watanabe-sena397@g.ecc.u-tokyo.ac.jp
  • †Contact author: motome@ap.t.u-tokyo.ac.jp
  • ‡Contact author: hwatanabe@ust.hk
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Phys. Rev. X 16, 041016 – Published 14 October, 2026

DOI: https://doi.org/10.1103/bkkq-q1v9

Abstract

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 the local spin amplitudes are maximized (|Sz|=S), the macroscopic flux is quantized to multiples of 2S. By mapping the defect structure to topological loop gases, we prove that the compatibility between the physical ice rule and the modular Z2S 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 Z2S structure carry exponentially small statistical 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 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.

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Popular Summary

Spin ice magnets famously host particles that behave like magnetic monopoles, but the standard picture treats each moment as a two-way switch pointing into or out of its tetrahedron, leaving open what happens when moments can take intermediate values. We settle this by mapping the problem onto simpler, well-understood statistical models, which lets us classify every phase and transition for any spin size. One value stands out: S=3/2 magnets alone switch abruptly between two distinct liquid states, while all others change smoothly. At any nonzero temperature, thermally created monopoles snip the string-like defects that keep these liquids apart, blurring the smooth transitions into gradual crossovers; the abrupt S=3/2 transition, however, survives and ends at a critical point that requires no magnetic field. Since higher-spin versions of these magnets are now being grown, and artificial arrays can imitate them, our predictions give experimenters specific signatures to hunt for.

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Phys. Rev. X 16, 041016– Published 14 October, 2026
  • Received 5 April 2026
  • Revised 15 August 2026
  • Accepted 8 September 2026
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