Topological metamagnetism: Thermodynamics and dynamics of the transition in spin ice under uniaxial compression

Metamagnetic transitions are analogs of a pressure-driven gas-liquid transition in water. In insulators, they are marked by a superlinear increase in the magnetization that occurs at a field strength set by the spin exchange interactions. Here we study topological metamagnets, in which the magnetization is itself a topological quantity and for which we find a single transition line for two materials with substantially different magnetic interactions: the spin ices ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. We study single crystals under magnetic field and stress applied along the [001] direction and show that this transition, of the Kasteleyn type, has a magnetization versus field curve with upward convexity and a distinctive asymmetric peak in the susceptibility. We also show that the dynamical response of ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ is sensitive to changes in the ${\mathrm{Ho}}^{3+}$ environment induced by compression along [001]. Uniaxial compression may open up experimental access to equilibrium properties of spin ice at lower temperatures.

Figure 1

(a) Magnetization $M$ and static susceptibility (${\chi }_{\text{DC}}\equiv dM/d{B}_{\mathrm{local}}$, inset) for DTO as a function of internal magnetic field along [001] $B_{\mathrm{local}}$ at different temperatures. (b) ${\chi }_{\text{DC}}$ for HTO as a function of internal magnetic field. The metamagneticlike features are sharper for HTO than DTO, in agreement with its smaller density of local excitations (monopoles) at a given temperature. Inset: Ideal paramagnetic behavior (dotted lines) and Husimi tree results for the NN model without monopole excitations. The shape and behavior of the ${\chi }_{\text{DC}}$ peaks support a Kasteleyn transition broadened by thermal effects.

Figure 2

(a) Real (${\chi }^{\prime }$) and (b) imaginary (${\chi }^{\prime \prime }$) parts of the ac susceptibility in unstrained HTO as a function of $\mathbf{B}\left|\right|$[001] for different temperatures. The monotonic ${\chi }^{\prime }\left(B\right)$ contrasts with the peaks in ${\chi }^{\prime \prime }$ at fields near those observed in ${\chi }_{\text{DC}}$. The inset in (a) shows the pyrochlore spin lattice. A strong $\mathbf{B}\left|\right|$[001] polarizes all spins (black arrows); in red, a string of reversed spins is excited at a finite temperature. Tetrahedra with two spins in and two out are magnetically neutral; an imbalance of spins results in a positive or negative magnetic monopole (blue and red spheres). The green arrows mark the direction of compression. The insets in (b) show the HTO crystal, cut in an hourglass shape along [001] (bottom), and the susceptometer and strain device (top). The superconducting primary (gray) winds around the two mutually opposing secondary coils (orange). The ends of the sample were epoxied to mobile piezoelectric anvils and covered by metallic plates.

Figure 3

The experimental points, obtained for materials with different $J_{\mathrm{eff}}$ and diverse measurement techniques, determine a single line that separates two regions: a high $B/T$ region where the magnetization is close to saturation and a low one where fluctuations proliferate. The coincidence (within experimental error) of theses curves for HTO and DTO reflects the topological nature of the transition. The lines indicate the Kasteleyn transition curve for the NN and dumbbell models (dashed line, taken from Ref. [10]) and extrapolated from high $T$ for the dipolar model (solid line, taken from Ref. [22]). The open pentagons are the results for ${\chi }^{\prime \prime }$ ($f\to 0$) under strain.

Figure 4

Imaginary part of the ac susceptibility ${\chi }^{\prime \prime }$ for HTO at zero field, measured as a function of temperature at fixed frequency $f$ for different values of strain along [001]. The peak moves toward lower temperatures as compression increases, indicating a faster dynamics.

Figure 5

Magnetization $M$ and static susceptibility ${\chi }_{\text{DC}}$ (inset) at $0.9\mathrm{K}$ for (a) DTO and (b) HTO as a function of internal magnetic field along [001] $B_{\mathrm{local}}$. We have used two field sweep rates $R\equiv dB/dt$ differing by a factor of $\approx 3$, obtaining the same results. This suggests that the measurements approach the static limit. It is easy to see that the superlinear increase in the magnetization indicating metamagnetism (correlated to a susceptibility peak) is much better defined and more asymmetric for HTO, where the spin-ice condition two spins in/two out at a given temperature is fulfilled for a much bigger fraction of tetrahedra. The positions of the susceptibility peak are very similar in both cases, in spite of the marked differences in their effective nearest-neighbor magnetic interactions $J_{\mathrm{eff}}$.

Figure 6

Effects of compression. Imaginary part of the susceptibility ${\chi }_{\mathrm{AC}}^{\prime \prime }$ vs field along [001] for different values of applied stress parallel to $\mathbf{B}$. In all cases the temperature is fixed at 1.4 K, and the frequency is fixed at $f=17.7\mathrm{Hz}$. The inset shows the phase diagram under compressing strain for $f\to 0$. The solid line indicates the predicted Kasteleyn transition in the presence of dipolar interactions under no applied stress.

Figure 7

Extrapolation to $f\to 0$. Each point represents the field maximum $B_{\max }$ in the imaginary part of the ac susceptibility ${\chi }^{\prime \prime }(T,B)$ at a given temperature $T$ and frequency $f$. Solid circles: ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ crystals at a fixed strain of $-0.3\%$. $B_{\max }$ increases with decreasing frequency, a trend that is more pronounced at low temperature. The points in the phase diagram corresponding to $f\to 0$ were taken after a linear extrapolation, as illustrated here. Open symbols: ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ (circles) and ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ crystals (stars) at nominally zero compression. The curves have a bigger negative slope for DTO at fixed temperature, reflecting its slower dynamics in our working temperature range.