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

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. In the small-$w$ regime, where the single-ion term favors low spin amplitudes, exact 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 ($|S^z| = S$), 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 emergent $\mathbb{Z}_{2S}$ flux conservation holds if and only if $S \le 3/2$. For $S=3/2$, this maps the system to the 3-state Potts model, whose symmetry-allowed cubic invariant drives a first-order transition. For $S \ge 2$, monopole contamination breaks the discrete clock mapping. Using an exact decomposition of the partition function, we show that the hierarchical string fusion cascade exponentially suppresses the discrete perturbations, which act as a dangerously irrelevant operator at the 3D $XY$ fixed point, protecting 3D $XY$ criticality. 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 endpoint. Classical Monte Carlo simulations for $S$ up to $7/2$ corroborate these analytical predictions.

Figures (6)

• Figure 1: Schematic phase diagram in the $(\mu, T)$ plane for (a) integer $S \ge 1$, (b) $S = 3/2$, and (c) half-integer $S \ge 5/2$. The single-ion fugacity is $w \coloneqq e^{-\mu/T}$, so the monopole-free transition lines $\mu_c = -T \ln w_c$ are straight lines emanating from the origin with slopes determined by $w_c$ (Tables \ref{['tab:integer_spins']} and \ref{['tab:halfinteger_spins']}). For integer spins (a), the three phases are the trivial paramagnet ($\mu > 0$, small $w$), the $U(1)$ Coulomb liquid, and the $\mathbb{Z}_{2S}$-confined Coulomb phase ($\mu < 0$, large $w$). For half-integer spins (b, c), the non-magnetic $S^z = 0$ state is absent, so the small-$w$ phase is the $U(1)$ Coulomb liquid rather than the trivial paramagnet. Dashed lines denote continuous transitions (crossovers at $T > 0$); the solid line in (b) denotes the first-order coexistence line that survives at finite temperature and terminates at a critical endpoint (open circle). The $S = 1/2$ case (not shown) has no transition and resides in the $U(1)$ Coulomb phase for all $\mu$.
• Figure 2: Ice rule compatibility at a single $z=4$ diamond-lattice vertex. Arrows indicate the bipartite A$\to$B link orientation along which $\phi_\ell$ is defined. (a) For $S=3/2$, three fundamental defects ($\phi=1$) converge and annihilate simultaneously: the defect sum is $1+1+1 = 3 = 2S \equiv 0\pmod{2S}$, and the remaining vacuum link carries $\phi = 2S$, so the exact ice rule $\sum_{\ell \in \bm{r}} \phi_\ell = 1+1+1+2S = 4S$ is preserved. (b) For $S=2$, four fundamental defects ($\phi=1$) converge: $\sum_{\ell \in \bm{r}} \phi_\ell = 1+1+1+1 = 4 = 2S$. (c) The spin configuration $S^z_\ell$ corresponding to (a); the vertex satisfies $\sum_{\ell \in \bm{r}} S^z_\ell = 0$, confirming that the ice rule is preserved. (d) The spin configuration corresponding to (b); the vertex gives $\sum_{\ell \in \bm{r}} S^z_\ell = -4 \neq 0$, violating the ice rule despite the $\mathbb{Z}_{2S}$ modular condition being satisfied. The four bonds are drawn in a planar cross for clarity; on the diamond lattice they point toward the corners of a regular tetrahedron.
• Figure 3: Hierarchical fusion cascade for $S \ge 2$. Arrows indicate the bipartite A$\to$B link orientation along which $\phi_\ell$ is defined. Because the exact ice rule on the $z=4$ diamond lattice forbids simultaneous annihilation of $2S \ge 4$ fundamental defects at a single vertex [Fig. \ref{['fig:annihilation']}(b,d)], the defects must fuse sequentially through $2S-2$ intermediate vertices. At each vertex, an incoming fundamental defect ($\phi=1$) merges with the accumulated intermediate defect, incrementing the effective charge by one. The displayed link labels $\phi_\ell$ alternate between $\phi$ and $2S - \phi$ along the chain because vertices alternate between A and B sublattices; on an A$\to$B link where the effective defect charge $\phi$ propagates from B to A, one has $\phi_\ell = 2S - \phi$. The figure illustrates the half-odd-integer case; for integer $S$ the same cascade structure applies with uniform link orientations. The resulting bridge penalty $\mathcal{P}_{\text{bridge}} = \prod_{\phi=2}^{2S-2} x_\phi$ [Eq. \ref{['eq:cascade_suppression']}] is exponentially suppressed in $S$. Open and filled circles denote A- and B-sublattice vertices, respectively. The cascade is drawn schematically as a linear chain; on the actual diamond lattice the backbone follows a zigzag path with tetrahedral bond angles.
• Figure 4: Defect string topology on the diamond lattice. Arrows indicate the bipartite A$\to$B link orientation along which $\phi_\ell$ is defined. Solid lines denote defect links carrying $\phi_\ell=1$ or $\phi_\ell=2S-1$ (alternating due to the A$\to$B sublattice convention); dashed lines denote vacuum links ($\phi_\ell=0$ or $\phi_\ell=2S$). (a) At $v=0$ (monopole-free ice rule), defect strings form closed directed loops with $Q_{\bm{r}}=0$ at every vertex. (b) At finite $v>0$, thermal monopoles ($|Q_{\bm{r}}|=1$) act as string endpoints, severing closed loops into open segments. This maps to an effective magnetic field $h_{\text{eff}} \approx \sqrt{3}\,v$ in the emergent $O(2)$ (i.e., 3D $XY$) spin model [Eq. \ref{['eq:h_eff']}]. Open and filled circles denote A- and B-sublattice vertices, respectively. The hexagonal loop shown is the shortest closed loop on the diamond lattice, drawn schematically in two dimensions; on the actual lattice it forms a chair-shaped six-membered ring.
• Figure 5: Monte Carlo results for integer spins at $T = 0.2$ with $L = 2$, 4, 8. (a,b) Specific heat per spin $c$ for $S = 2$ and $S = 3$. Vertical dashed lines indicate the analytically predicted crossover scales $-T \ln w_{c1}$ (purple), $-T \ln w_{c2}$ (dark yellow), and the Schottky peak position $\mu_{\text{Sch}} = 2.4T/(2S-1)$ (cyan). Insets zoom in on the $\mu < 0$ peak region. The peak heights saturate with $L$, confirming crossover rather than a true phase transition. (c,d) Thermal occupancy fractions $n_s = \langle \delta_{|S^z_\ell|, s} \rangle$ for $S = 2$ and $S = 3$ ($L = 8$), showing the sequential depopulation of higher spin amplitudes as $\mu$ increases. (e) Comparison of the specific heat across $S = 1$, 2, and 3 ($L = 8$). The Schottky shoulder (visible for $S \ge 2$ near $\mu \approx 0.1$) shifts toward $\mu = 0$ with increasing $S$, consistent with the $1/(2S-1)$ scaling predicted analytically.