Spin liquids in spin-1 pyrochlore magnets: nematic Coulomb phase, intrinsic spin glass, and its realization in NaCaNi$_2$F$_7$

Abstract

The search for spin liquids, magnetic phases which lie outside the Landau paradigm, remains one of the central challenges for modern condensed matter physics. Here, motivated by NaCaNi$_2$F$_7$, we use numerical simulation to explore the phases which arise in a minimal model of a spin-1 pyrochlore magnet, once quadrupole moments are taken into account. We identify seven distinct Coulombic spin liquid phases, including one with nematic correlations. Through quantitative comparison with inelastic neutron scattering, we show that this nematic spin liquid provides a compelling scenario for both the spin liquid observed in NaCaNi$_2$F$_7$, and its spin glass transition, without the need to invoke disorder. These results highlight the great wealth of unconventional phases which arise for magnetic ions capable of supporting both dipole and quadrupole moments.

Figures (14)

• Figure 1: Finite--temperature phase diagram of the $S=1$ bilinear-biquadratic model on the pyrochlore lattice, showing an abundance of spin liquid phases. In addition to ferromagnetic (FM), and ferroquadrupolar (FQ) order, the model hosts seven distinct Coulombic spin liquids, listed in Table \ref{['table:spin.liquids']}. Solid lines indicate finite-temperature phase transitions, while dotted lines mark crossovers. Results are taken from semiclassical Monte Carlo (MC) simulations of Eq. (\ref{['eq:H.BBQ']}) with parametrization in Eq. (\ref{['eq:H.BBQ.parametrization']}). Characteristic spin configurations are illustrated in Fig. \ref{['fig:zero.T.phases']}.
• Figure 2: Ground-state phase diagram of the $S=1$ bilinear-biquadratic (BBQ) model on the pyrochlore lattice, and associated zero--temperature phases. (A) Phase diagram found in semiclassical variational calculations for $T=0$. In addition to two ordered ground states, the model supports ground-state manifolds associated with five distinct Coulombic spin liquids, as listed in Table \ref{['table:spin.liquids']}. (B) Example of spin configuration within ferromagnetic (FM) ordered state. Dipole moments are indicated with arrows. (C) Spin configuration within ferroquadrupolar (FQ) ordered state. Quadrupole moments are shown through surfaces of constant probability. (D) Spin configuration within nematic spin liquid [$\mathbb{RP}^2 \times U(1)$]. (E) Spin configuration within dipolar colour ice (dCI). (F) Spin configuration within $SU(3)$ Coulombic spin liquid (su3C). Spins have mixed dipolar and quadrupolar character. (G) Spin configuration within quadrupolar colour ice (qCI). (H) Spin configuration within ferrimagnetic spin liquid [Ferri $U(1)$]. In all cases spin configurations are shown for a single 16--site cubic unit cell, with results taken from variational simulations of Eq. (\ref{['eq:H.BBQ']}), using the parametrization Eq. (\ref{['eq:H.BBQ.parametrization']}).
• Figure 3: Correlations characteristic of the seven spin liquid phases found in the spin--1 bilinear biquadratic model on the pyrochlore lattice [Table \ref{['table:spin.liquids']}].Left column: equal--time structure factor for dipole moments, $S_{\sf S}({\bf q})$. Right column: equal--time structure factor for quadrupole moments, $S_{\sf Q}({\bf q})$. All seven spin liquids are Coulombic in character, displaying pinch points in $S_{\sf S}({\bf q})$ or $S_{\sf Q}({\bf q})$. In addition to this, the spin--nematic and ferrimagnetic spin liquids break spin--rotation symmetry, leading to Bragg peaks in the relevant structure factor. Results are taken from semiclassical Monte--Carlo (MC) simulation of Eq. (\ref{['eq:H.BBQ']}), with parametrization Eq. (\ref{['eq:H.BBQ.parametrization']}).
• Figure 4: Contrasting dynamics of nematic spin liquid and dipolar color ice phases, as revealed by the dynamical structure factors for dipole and quadrupole moments. (A) Dipolar dynamics of nematic spin liquid [$\mathbb{RP}^2 \times U(1)$] for $\phi/\pi \approx -0.1$ [K < 0], as found in $S_{\sf S}({\bf q}, \omega)$, on line ${\bf q} \in (2,2,l)$. A gap to dipolar excitations is visible at ${\bf q} = (2,2,0)$. (B) Equivalent results for quadrupolar structure factor $S_{\sf Q}({\bf q}, \omega)$, showing gapless Goldstone mode associated with spin nematic order. (C) Corresponding results for $S_{\sf S}({\bf q}, \omega)$ on line ${\bf q} \in (h,h,2)$. (D) Equivalent results for $S_{\sf Q}({\bf q}, \omega)$. (E) Dipolar dynamics of dipolar color ice (dCI) for $\phi/\pi \approx 0.1$ [K > 0], showing gapless dipolar excitations, as found in $S_{\sf S}({\bf q}, \omega)$ on the line ${\bf q} \in (2,2,l)$. (F) Equivalent results for quadrupolar channel, showing (quasi--)localization of quadrupolar excitations. (G) Corresponding results for $S_{\sf S}({\bf q}, \omega)$ on line ${\bf q} \in (h,h,2)$. (H) Equivalent results for $S_{\sf Q}({\bf q}, \omega)$. Results are taken from semiclassical molecular dynamics (MD) simulation of Eq. (\ref{['eq:H.BBQ']}) for parameters set in Eq. (\ref{['eq:RP2XU1.parameters']}, \ref{['eq:dCI.parameters']}) and structure factors defined in Eq. (\ref{['eq:Sqw']}).
• Figure 5: Comparison of dynamical structure factors found in simulation of a spin--1 model with biquadratic interactions and experiment on $\mathrm{NaCaNi}_2\mathrm{F}_7$. (A) Inelastic neutron scattering (INS) data for $\mathrm{NaCaNi}_2\mathrm{F}_7$ on the line ${\bf q} \in (2,2,l)$. (B) Dipolar structure factor $\tilde{S}_{\sf S}({\bf q}, \omega)$ found in simulations of spin--1 model, showing gapped excitations in good agreement with experiment. (C) Corresponding simulation results for the quadrupolar structure factor $S_{\sf Q}({\bf q}, \omega)$, showing dispersing mode at high energies, and strong low--energy fluctuations near ${\bf q} \in (2,2,2)$ and related zone centers. (D) INS data for $\mathrm{NaCaNi}_2\mathrm{F}_7$ on the line ${\bf q} \in (h,h,2)$. (E) Corresponding predictions for $\tilde{S}_{\sf S}({\bf q}, \omega)$. (F) Corresponding predictions for $S_{\sf Q}({\bf q}, \omega)$. Simulations were carried out for the model ${\mathscr H}_{\sf NaCaNi_2F_7}$ [Eq. (\ref{['eq:model.for.experiment']})], with parameters Eq. (\ref{['eq:parameters.for.experiment']}). For purpose of comparison with experiment, the dynamical structure factor $\tilde{S}_{\sf S}({\bf q}, \omega)$ includes a dipolar projection, and simulation results are averaged over $l \pm 0.25$ and $\omega \pm 0.2\ {\rm meV}$, as described in Supplementary Materials. INS data are taken from experiments reported in Ref. Zhang2019.