Exact nematic and mixed magnetic phases driven by competing orders on the pyrochlore lattice

TL;DR

This work addresses how competing magnetic orders on the pyrochlore lattice can produce degenerate ground-state manifolds and novel low-temperature phases. By combining irreducible-representation analysis, a classical low-temperature expansion, Landau theory, and large-scale classical Monte Carlo, it shows that along the $A_2\oplus E\oplus T_{1-}$ line the system generically selects a $\,\mathbf{q}=0$ mixed phase $A_2\oplus\psi_2$ via order-by-disorder, while at the exact nematic point $J_{z\pm}=\pm1/\sqrt{2}$ a $\,\mathbb{Z}_2$ subsystem symmetry stabilizes a spin-nematic phase with subextensive degeneracy. The results are supported by exact symmetry arguments, duality relations, and phase diagrams revealing the interplay between $A_2$ and $E$ order parameters and the nematic sector. This establishes a concrete microscopic mechanism for a mixed irrep long-range order and a robust nematic state in a 3D frustrated magnet, with implications for experiments and the role of quantum fluctuations.

Abstract

Pyrochlore magnets are a paradigmatic example of three-dimensional frustrated systems and provide an excellent platform for studying a variety of exotic many-body phenomena, including spin liquids, nematic phases, fragmentation, and order by disorder. In recent years, increasing attention has been devoted to bilinear spin models on this lattice, where multiple magnetic phases can be degenerate in energy, often stabilizing unconventional magnetic states. In this work, we focus on one such model, parametrized by the interaction coupling $J_{z\pm}$, which defines a line in parameter space corresponding to the phase boundary between three distinct magnetic phases. Using a combination of analytical and numerical methods, we show that this model exhibits an order-by-disorder mechanism at low temperatures, giving rise to a \emph{mixed} magnetic phase. This represents the first realization of a $\mathbf{q}=0$ long-range-ordered phase in a pyrochlore magnet characterized by two distinct order parameters, which we denote as the $A_2 \oplus ψ_2$ phase. Furthermore, at $J_{z\pm} = 1/\sqrt{2}$, the model acquires a subextensive number of discrete symmetries, which preclude the stabilization of conventional long-range order and instead lead to the emergence of a novel nematic phase. We characterize this nematic phase, describe how its ground-state configurations are constructed, and analyze its stability at higher temperatures and under small deviations from $J_{z\pm} = 1/\sqrt{2}$.

Figures (23)

• Figure 1: Phase diagram as a function of temperature $T$ and coupling $J_{z\pm}$ along the $A_2\oplus E\oplus T_{1-}$ line from classical Monte Carlo simulations, with the color scale indicating (a) specific heat, (b) $A_2$ order parameter $|m_{A_2}|$, and (c) $E$ order parameter $|\mathbf{m}_E|$. The green arrow marks the position of the exact nematic point at $J_{z\pm}=1/\sqrt{2}$. The black arrow highlights the position of the jump of the $A_2\oplus E$ mixing angle. Lastly, the red arrow signals the exact Heisenberg AFM point at $J_{z\pm}=\sqrt{2}$ where a spin-liquid phase is stabilized down to zero temperature. The specific heat peaks and the order parameters define several regions in the $(J_{z\pm},T)$ phase diagram. The high-temperature paramagnetic phase freezes into different phases depending on $J_{z\pm}$. For $J_{z\pm}\lesssim 1/\sqrt{2}$, there is an intermediate $\mathbf{q}=0$$A_2$ phase which is driven into a $A_2\oplus \psi_2$ long-range order at lower temperatures, while for $J_{z\pm}\gtrsim1/\sqrt{2}$, this mixed phase is realized directly from the paramagnetic phase, with an intermediate region characterized by $|m_E| \gg |m_{A_2}| > 0$, separated from the low-temperature region by a crossover. From the exact spin-nematic point at $(J_{z\pm},T)=(1/\sqrt{2},0)$ stems a finite-temperature nematic phase fan (SN). Lines are guides to the eye, with solid lines indicating true phase transitions and dashed lines indicating crossovers.
• Figure 2: $A_2\oplus E$ sphere, defined by $\theta = \pi/2$ in Eq. \ref{['eq:angle-representation']}, for (a) $J_{z\pm} =0$ and (b) $J_{z\pm} \approx0.14$. The white dot at the north pole corresponds to a spin configuration that is uniquely described by the $A_2$ irrep. The white circle along the equator marks the $\Gamma_5$ manifold, corresponding to spin configurations that are composed of the $E$ irrep mode only. The orange circles correspond to spin configurations in the U(1) manifolds parametrized by Eqs. \ref{['eq:A2E-U1-manifolds']}, which can fluctuate away from the $A_2\oplus E$ sphere with no associated energy cost.
• Figure 3: Entropy $\mathcal{S}$ (modulo a $T$ and $J_{z\pm}$ dependent constant) on the $A_2\oplus E$ sphere for different values of $J_{z\pm}$ from classical low-temperature expansion. Panels (a), (c), (f), (h), and (d) show the stereographic projection of the entropy, where we identify the north pole as the point where $A_2$ is maximal [cf. Fig. \ref{['fig:U1-manifolds']}]. The azimuthal angle $\eta$ parametrizes the $\Gamma_5$ manifold and the polar angle $\phi$ parametrizes the mixing with the $A_2$ configuration. The white full (dashed) lines denote spin configurations that are linear combinations of the $A_2$ and $\psi_2$ ($-\psi_2$) states. Panels (b), (d), (g), and (i) illustrate the evolution of the entropy $\mathcal{S}$ for $\eta = 0$. Panel (j) illustrates the stereographic projection of the high-entropy paths defined by Eq. \ref{['eq:A2E-U1-manifolds']} and shown in Fig. \ref{['fig:U1-manifolds']} in orange, taking $J_{z\pm}=0.4$ as a representative example. The red dots (light blue squares) in panel (j) indicate the locations in the $A_2\oplus E$ manifold where the entropy is maximal for $J_{z\pm} <1.26$ ($J_{z\pm} > 1.26$). Panel (k) illustrates the evolution of the entropy along the $A_2\oplus \psi_2$ line, defined by $\eta=0$, as a function of the coupling $J_{z\pm}$. In panels (b), (d), (g), and (i) the vertial red lines mark the location of the two high-entropy branches.
• Figure 4: Mean-field phase diagram as a function of $r_1$ and $r_0$ for fixed $(r_2,r_{xyz},r_3,r_4,\omega,f_6)=(1,1,1,1,2,0.5)$ from Landau theory, with the color scale indicating (a) the $E$ order parameter $|\mathbf{m}_E|$ and (b) the $A_2$ order parameter $|m_{A_2}|$. Panels (c)-(e) illustrate the evolution of the order parameters $|\mathbf{m}_E|$ (blue curve) and $|m_{A_2}|$ (red curve) along the different one-dimensional cuts marked in green in panels (a) and (b): Panel (c) shows the order parameters along the path $1\rightarrow2$, panel (d) along the path $1\rightarrow 3$, and panel (e) along the path $2\rightarrow 4$. Panel (f) shows the order parameters also along the path $2 \rightarrow 4$, but for a smaller value $\omega = 1$ of the coupling between $m_{A_2}$ and $m_E$.
• Figure 5: Monte Carlo results in the non-Kramers case $J_{z\pm} = 0$. (a) Specific heat, (b) ${A_2}$ order parameter, and (c) $E$ order parameter as functions of the temperature $T$ for different lattice sizes $L$. The continuous transition at $T_{\mathrm{c}}$, indicated by the vertical dashed line, separates the paramagnetic phase at high temperatures from the $A_2$ long-range-ordered phase at low temperatures. In (a), the horizontal dashed line at $C = 7/8$ highlights the low-temperature limit of the specific heat, as expected from the equipartition theorem with $n_2=6$ quadratic modes and $n_4=2$ quartic modes.