Dynamical Scaling as a Signature of Multiple Phase Competition in ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$

${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ is a celebrated example of a pyrochlore magnet with highly frustrated, anisotropic exchange interactions. To date, attention has largely focused on its unusual, static properties, many of which can be understood as coming from the competition between different types of magnetic order. Here we use inelastic neutron scattering with exceptionally high energy resolution to explore the dynamical properties of ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. We find that spin correlations exhibit dynamical scaling, analogous to behavior found near to a quantum critical point. We show that the observed scaling collapse can be explained within a phenomenological theory of multiple-phase competition, and confirm that a scaling collapse is also seen in semiclassical simulations of a microscopic model of ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. These results suggest that dynamical scaling may be general to systems with competing ground states.

Figure 1

Low-energy neutron scattering from the short-range correlated phase of ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. Panels (a)–(i) show color plots of neutron scattered intensity, with the horizontal rows showing three different temperatures and the vertical columns show different constant energy slices in the $hh\ell$ scattering planes. All temperatures and energies show diffuse scattering rods along $\{111\}$ directions along with crosses of scattering centered at (220). Panel (j) shows the data integrated over the $\{111\}$ scattering rods [indicated by the red box in panel (a)] scaled by the temperature. Up to 2 K, the data collapse onto themselves and follow a scaling relation of type Eq. (1). The specific form of scaling predicted by our phenomenological theory of multiple-phase competition, Eq. (3), is shown with a solid line. Error bars indicate 1 standard deviation.

Figure 2

Dynamical scaling collapse of $S_{\mathrm{rod}}^{\mathrm{cl}}(\omega )$ calculated using molecular dynamics simulations compared with the theoretical scaling relation Eq. (6). (a) Calculations with exchange parameters set to the values estimated for ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ in [32] (point $A$ in Fig. 3). A near, but imperfect, collapse is observed. (b) Calculations with a modified value of $J_1$ such that the exchange parameters lie on the FM-AFM boundary (point $B$ in Fig. 3). A much closer data collapse is observed compared to (a). (c) Calculations at the spin liquid point $J_1=J_2=J_4=0$, $J_3<0$ (point $C$ in Fig. 3). The collapse is observed with a vanishing value of the rod criticality temperature $T_{\mathrm{rod}}$.

Figure 3

Finite temperature phase diagram of the pyrochlore $\{J_1,J_2,J_3,J_4\}$ exchange model [17, 48, 50], determined from classical Monte Carlo simulations. The horizontal axis is $J_1$, the vertical axis is temperature, and the out-of-the-page axis is $J_2$, with $J_3=-0.322\mathrm{meV}$ and $J_4=-0.091J_2$. The solid lines show $T_{\text{order}}$ as a function of $J_1$, for a series of values of $J_2$. Point $A$ shows the ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ exchange parameters [32]. Point $B$ has the same values of $J_{2,3,4}$ as $A$, but $J_1$ is adjusted so as to lie exactly on the phase boundary. Point $C$ corresponds to the spin liquid at $J_1=J_2=0$ [36], where FM and ${\mathrm{\Gamma }}_5$ orders meet Palmer-Chalker antiferromagnetic order. The green line shows the finite temperature boundary between the ferromagnet (FM) and antiferromagnetic ${\mathrm{\Gamma }}_5$ (AFM) states, which goes to zero at the classical spin liquid point. Thus in the finite temperature regime, ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ is continuously connected to a zero temperature spin liquid phase.

Figure 4

Variation of thermodynamic transition temperature $T_{\text{order}}$, and dynamic criticality temperature $T_{\mathrm{rod}}$, found in simulation. Results are shown for a path in parameter space that connects ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ (A) to the spin-liquid point $J_2=0$ (C), shown by a white line in the inset. Both $T_{\text{order}}$ and $T_{\mathrm{rod}}$ tend to zero approaching the spin liquid. $T_{\mathrm{rod}}<T_{\text{ordering}}$ for all parameters, meaning that the approach to criticality on the rods is cutoff by the ordering transition as temperature is lowered. The effects of this hidden critical point are nevertheless seen in the paramagnetic phase.