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

Sena Watanabe, Yukitoshi Motome, Haruki Watanabe

Abstract

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 the continuous $XY$ wave picture and topologically sever defect strings in the loop-gas picture, rounding the phase transitions into continuous crossovers. These theoretical findings are corroborated by classical Monte Carlo simulations.

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

Abstract

We resolve the phase diagram of the 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 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 the continuous wave picture and topologically sever defect strings in the loop-gas picture, rounding the phase transitions into continuous crossovers. These theoretical findings are corroborated by classical Monte Carlo simulations.
Paper Structure (10 sections, 31 equations, 2 figures)

This paper contains 10 sections, 31 equations, 2 figures.

Table of Contents

  1. Introduction.---
  2. Model.---
  3. Duality Mapping to the $XY$ Model.---
  4. Mapping to the Loop-Gas Model.---
  5. Topological Classification via Global Flux.---
  6. Finite-Temperature Effects.---
  7. Monte Carlo Simulations.---
  8. Conclusion and Outlook.---
  9. Dual Mapping to the $XY$ Model at Finite Temperatures
  10. Mapping to the Ising Model with a Magnetic Field

Figures (2)

  • Figure 1: (a) Geometric correspondence between the diamond and pyrochlore lattices. Black spheres represent the diamond sites, which are located at the centers of the cyan and orange tetrahedra formed by the pyrochlore sites. Arrows on the pyrochlore sites denote the spin-$1$ variables aligned with the local $z$-axis. Red and blue arrows represent $S_\ell^z=1$ and $S_\ell^z=-1$, respectively, while white spheres correspond to $S_\ell^z=0$. The illustrated spin configuration satisfies the ice rule. (b) Monopole charges and their corresponding spin configurations. Odd monopole charges can exist in the spin-1 model.
  • Figure 2: Results of MC simulations. (a) System-size dependence of the specific heat at $T=0.3$. (b) Peak positions of the specific heat for $L=4$ and the phase boundaries determined numerically in the monopole-free limit by Pandey and Damle Pandey2025. Heat maps of (c) the deviation $\Delta P$ and (d) the monopole density $\rho$ for $L=8$. Panels (c) and (d) indicate that the presence of thermal monopoles causes the macroscopic flux to deviate from integer values.