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arXiv:2604.04346 · Physical Review X

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

Sena Watanabe, Yukitoshi Motome, Haruki Watanabe

Published Sep 8, 2026• 0 citations
arXivHTMLPDF

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 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.

DOI
10.1103/bkkq-q1v9

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Three dimensional frustrated magnets realized in the rare-earth pyrochlores R_2B_2O_7, R=rare-earth ion, B transition metal ion) have attracted much interest since the discovery of spin ice in Ho_2Ti_2O_7 by Harris et al. Since then, a number of interesting phenomena in pyrochlore magnets have been observed and the interest in these systems, from an experimental and theoretical point of view, has nothing but increased. Perhaps the most appealing and unique possibility is that these systems may realize a quantum disordered phase; a so-called quantum spin ice. Theoretically, it has been predicted by Hermele et al. that a U(1) quantum spin liquid is realized in the XXZ quantum spin ice (with global U(1)xZ_2 symmetry) in the perturbative regime and more recently Savary and Balents have introduced a quantum rotor model with bosonic spinons coupled to U(1)U(1)U(1) gauge fields to describe non-peturbatively the deconfined phase predicted. They studied an anisotropic quantum spin ice (QSI) model which corresponds to the XXZ QSI with an additional anisotropic coupling. They solved the quantum rotor model using a mean-field approximation combined with a large-N approximation and predicted the possibility of a U(1) liquid in addition to an exotic ferromagnetic phase. Further work by Lee et al. has addressed the more complicated fully frustrated XXZ quantum spin ice regime, where all components of the model are frustrated, by using some insights provided by the previous perturbative results. \n \nIn this thesis, we will address the non-perturbative regime of the fully frustrated XXZ QSI beyond the simplest perturbative treatment. We will begin by mapping the quantum spin ice model to a compact U(1) gauge theory with deconfined electric and magnetic monopoles. Neglecting the interaction between the gapless photon excitation and the electric charges, we will study the matter sector with only electric charges, the so-called spinons. Building up on insights provided by high order degenerate perturbation theory calculations, we will construct some symmetric spin liquid ansatz for the static part of the gauge fields. The two novel spin liquids that we construct break time reversal symmetry and thus have a non-trivial chirality. In order to solve the matter sector of the theory and determine the stability of the deconfined phases, we will introduce a novel representation of a XY quantum rotor in terms of ``exclusive" bosons, a representation which is dynamically implemented in the dilute limit. We will benchmark this mapping in different ways by comparing our results with Quantum Monte Carlo simulations and exact diagonalization in some limiting cases. Using this mapping we will compute the region of stability of the U(1) liquid phase of XXZ QSI. Within this mapping we will reproduce qualitatively the results of previous work by finding only the so-called zero and \\pi-flux U(1) liquids in the vicinity of the classical spin ice point, while improving greatly on the estimation of the exchange coupling at which the bosonic spinons condense to form a superfluid phase. In the fully frustrated XXZ QSI, we will provide evidence, using exact diagonalization of the quantum rotor model, that one of the chiral spin liquid may be realized in the strongly correlated and frustrated QSI regime, where all previous approximations break down.
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We calculate the magnetic phase diagram of the spin-1/21/21/2 nearest neighbor XXZ pyrochlore model using the pseudo-Majorana functional renormalization group in the temperature flow formalism. Our phase diagram as a function of temperature and coupling ratio, allowing both longitudinal and transverse couplings to be ferromagnetic and antiferromagnetic, reveals a large non-magnetic regime at low temperatures, which includes the quantum spin ice phase near the antiferromagnetic Ising model, as well as the antiferromagnetic Heisenberg and XY models. We are able to detect magnetic phase transitions via critical finite size scaling down to temperatures two orders of magnitude smaller than the spin interactions, demonstrating the remarkably good performance of our method upon approaching the ground state. Specifically, the low temperature transition from the zero-flux quantum spin ice phase into the transverse ferromagnetic phase shows very good agreement with previous quantum Monte Carlo results. Comparing our findings with classical results, we identify a quantum order-by-disorder effect near the antiferromagnetic XY model. In magnetically disordered regimes, we find characteristic patterns of broadened pinch points in the spin structure factor and investigate their evolution when approaching magnetically ordered phases. We also compute linear responses to lattice symmetry breaking perturbations and identify a possible lattice nematic ground state of the antiferromagnetic XY model.
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For nearly three decades, research on frustrated magnetism in three dimensions (3D) has centered on the pyrochlore lattice of corner-sharing tetrahedra and the classical spin liquid (CSL) known as spin ice. We propose that a lattice of corner-sharing octahedra -- the octochlore lattice -- may provide a next-generation platform for 3D frustrated magnetism, with realizations in anti-perovskite and alkali-rare-earth fluoride compounds. We study the phase diagram of Ising moments on the octochlore lattice, finding a variety of frustrated phases including CSLs and phases with subextensive ground state degeneracy intermediate between spin liquids and long-range order. Utilizing a cluster multipole framework, we present a unified treatment of this variety of frustrated behaviors. In addition to a spin ice CSL, we identify a fracton CSL with excitations restricted to move along one-dimensional (1D) lines, a classical U(1) equivalent of the paradigmatic X-cube model harboring fracton topological order. These "lineon" quasiparticles carry magnetic quadrupole moments, contrasting the famous magnetic monopoles of spin ice. These two CSLs lie at the boundaries of a parent "frustrated chains" phase with subextensive degeneracy. Each CSL corresponds to a condensate of different bound states of 1D ferro-spinons, giving rise to quasi-critical dimensional crossovers near the ends of the frustrated chains phase associated to avoided Kasteleyn-like transitions. We also find a spin nematic phase whose ground states may be viewed as fracton crystals, exhibiting both uniaxial and biaxial orders. The latter is caused by spontaneous dimensional reduction owing to accidental symmetries of the subextensive ground state manifold. This work paves the way for the realization of fracton CSLs and the exploration of other exotic states in underexplored octochlore magnetic materials.
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