Structural magnetic glassiness in the spin ice ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$

The origin and nature of glassy dynamics presents one of the central enigmas of condensed-matter physics across a broad range of systems ranging from window glass to spin glasses. The spin-ice compound ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$, which is perhaps best known as hosting a three-dimensional Coulomb spin liquid with magnetically charged monopole excitations, also falls out of equilibrium at low temperature. How and why it does so remains an open question. Based on an analysis of low-temperature diffuse neutron-scattering experiments employing different cooling protocols alongside recent magnetic noise studies, combined with extensive numerical modeling, we argue that upon cooling, the spins freeze into what may be termed a “structural magnetic glass,” without an $a$ priori need for chemical or structural disorder. Specifically, our model indicates the presence of frustration on two levels, first producing a near-degenerate constrained manifold inside which phase ordering kinetics is in turn frustrated. A remarkable feature is that monopoles act as sole annealers of the spin network and their pathways and history encode the development of glass dynamics, allowing the glass formation to be visualized. Our results suggest that spin ice ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ provides one prototype of magnetic glass formation specifically and a setting for the study of kinetically constrained systems more generally.

Figure 1

Magnetic structure, monopole hopping, and relaxation time of DTO. (a) The spins in DTO are located on a pyrochlore lattice. The magnetic interaction pathways are shown in color, and dipolar coupling (not shown) is also important. These constrain the spin configurations to follow two-in/two-out ice rules. Breaking the ice rules results in the creation of a monopole antimonopole pair which can separate, creating a Dirac string as shown in (b). Monopoles are constrained by the other spins (Dirac strings) to travel over a restricted manifold. (c) Relaxation time ($\tau$) for DTO extracted from noise measurements follows a Vogel-Fulcher-Tammann (VF) behavior below 1.5 K, rather than an Arrhenius law. The color map is of ${\chi }^{\prime \prime }\left(\omega \right)$ whose maximum gives a measure of $\tau$. The inset shows the cooling protocol for one cycle of the neutron experiments: the symbols correspond to the waiting times as a function of temperature for the rapid (light blue) and slow cooling (green) protocols of the neutron experiments. These are compared with the VF law obtained from the noise and with the Arrhenius law fit to the data.

Figure 2

Glass behavior in DTO. (a) Fast (1 hour) and slow (“annealed” over 4 weeks) cooling data, taken along trajectories indicated in Fig. 1 (inset), show little discernible difference (the measurements correspond to points A2 and R2 in the sample temperature history given in the SM [37]). (b) Modeled and measured (rapid cooled) diffuse scattering agree at low temperature. (For details on the neutron-scattering measurements and quantitative line-cut comparisons, see SM [42].) (c) Simulations of the diffuse scattering show Coulombic correlations associated with the Coulomb spin liquid at 1–2 K. At lower temperatures the buildup in correlations is reflected in a sharpening of the correlations. Below 600 mK the correlations remain largely unchanged, as expected for a glass. The simulations correspond to a Hamiltonian with dipolar interactions and up to third-nearest-neighbor exchange interactions with $D=1.3224$ K, $J_1=3.41$ K, $J_2=0, J_3=-0.004$ K, and $J_3^{\prime }=0.044$ K (for details see SM [43]).

Figure 3

Spin configurations and monopole trajectories as a function of third-neighbor interaction strength. (a) Connected tetrahedra with the same net magnetic moment define ferromagnetic (FM) regions, depicted by the smooth surface wrapping around them. Ice rules allow six FM orientations. The configuration here is modeled at $T=300\mathrm{mK}<T_{\mathrm{irr}}$ using parameters from Ref. [16], see text. For clarity, only four FM orientations are shown. Monopoles are trapped in this structure, as can be seen in the animation of the time dependence presented in SI (Movie S1). (B) Calculated heat capacity over temperature ($C_v/T$) along the line $J_3-J_3\prime =0.047$ K, plotted as a function of $J_3\prime$ and $T$. Order is seen at either limit, interrupted by glass formation in a broad intermediate region. The glass formation occurs at a near-constant temperature which disrupts order at its characteristic temperature. (c) The morphology of the FM regions changes with third-nearest-neighbor interactions. A single type of FM region is shown for the parameters indicated by vertical dashed lines in (b). This shows the influence of different ordering tendencies. To the right of each subfigure, a pair of corresponding typical monopole pathways is shown.

Figure 4

Monopole pathways at different temperatures. For clarity of comparison, the same (artificially) fixed density of monopoles is shown at each temperature; in reality, the number of monopoles at lowest temperature is vanishingly small. The graph shows characteristic monopole paths taken over a finite MC time in three regions. Above the irreversibility line, $T>T_{\mathrm{irr}}$, the monopole paths are extended and mutually connected and (eventually) span the whole sample. For $T_{\mathrm{VF}}<T<T_{\mathrm{irr}}$, the monopole pathways take the form of sparse networks; as temperature is lowered their size and dimensionality is increasingly reduced and they become disconnected. For $T<T_{\mathrm{VF}}$ the monopoles are essentially frozen and the pathways become zero dimensional. The red line is the VF law that best fits the relaxation time data (shown in Fig. 1).