Satisfaction and Violation of the Fluctuation-Dissipation Relation in spin ice materials

TL;DR

This work tests the fluctuation-dissipation relation in spin ice materials Dy$_2$Ti$_2$O$_7$ and Ho$_2$Ti$_2$O$_7$ by simultaneously measuring magnetic noise $S(f)$ and the dissipative susceptibility $\chi''(f)$. They find FDR holds over a broad frequency range in the high-temperature and thermally localized regimes, and persists in a local-equilibrium nonergodic regime down to about 300–400 mK, but exhibits a low-frequency violation below ~0.1 Hz with an excess of noise and aging phenomena, indicating multiple slow relaxation processes. Ho$_2$Ti$_2$O$_7$ shows two distinct relaxation times and a stronger FDR violation (S/D up to ~6 at 163 mK) than Dy$_2$Ti$_2$O$_7$, highlighting material-specific aging dynamics linked to monopole kinetics and Pauling-state bandwidth effects. The results motivate further theoretical and experimental exploration of aging, nonergodicity, and relaxation pathways in spin ice, including controlled quench protocols to probe the out-of-equilibrium dynamics more deeply.

Abstract

We test the fluctuation-dissipation relation (FDR) in spin ice materials Dy$_2$Ti$_2$O$_7$ and Ho$_2$Ti$_2$O$_7$ by measuring both the magnetic noise and the out-of-phase part of the susceptibility and comparing their ratio. We show that it is satisfied at temperatures well into the non-ergodic region below 600 mK, indicating local equilibrium. In both materials, below 400 mK, low frequency violations develop, showing an excess of noise as in spin glasses, with a frequency threshold of 0.1 Hz. New relaxation pathways and aging properties are unveiled in this frequency range in the ac susceptibility. The FDR remains valid at higher frequencies down to 150 mK.

Figures (19)

• Figure 1: FDR plot for Dy$_2$Ti$_2$O$_7$ on a logarithmic scale: $S(f)$ (small dots) and $D(f)$ (big dots) measured between 4.2 K and 150 mK. Lines are guides to the eye. Inset: ac susceptibility $\chi'$ and $\chi"$ vs $f$ measured at 700 mK. The solid lines show the fit to the Cole-Davidson equation (\ref{['Davidson_Cole']}) with $\tau=0.429$ s, $\beta=0.67$, $\chi_S=0$ and $\chi_T=0.213$ emu.cm$^{-3}$.
• Figure 2: FDR plot for Ho$_2$Ti$_2$O$_7$ on a logarithmic scale: $S(f)$ (small dots) and $D(f)$ (big dots) measured between 4.2 K and 163 mK. Lines are guides to the eye. Inset: ac susceptibility $\chi'$ and $\chi"$ vs $f$ measured at 1 K. The solid lines show the fit to a sum of two Cole-Davidson equations (\ref{['Davidson_Cole']}) with $\tau_1=0.098$ s, $\beta_1=0.31$, $\chi_{T1}=0.116$ emu.cm$^{-3}$, $\tau_2=0.002$ s, $\beta_2=0.41$, $\chi_{T2}=0.13$ emu.cm$^{-3}$ and $\chi_{S1}=\chi_{S2}=0$.
• Figure 3: FDR plot for (a) Dy$_2$Ti$_2$O$_7$ and (b) Ho$_2$Ti$_2$O$_7$ plotted on a logarithmic scale at low temperature and low frequency: $S(f)$ (small dots) and $D(f)$ (big dots). (c) Fluctuation dissipation ratio $S(f)/D(f)$ calculated below 0.016 Hz for Dy$_2$Ti$_2$O$_7$ (in blue) and Ho$_2$Ti$_2$O$_7$ (in gold) from base temperature to 800 mK. The solid black line corresponds to a ratio equal to 1 which is expected when there is no violation of the FDR.
• Figure 4: (a) Relaxation time $\tau$ (top panel) and exponent $\alpha$ (bottom panel) in Dy$_2$Ti$_2$O$_7$ obtained from the fits of the noise data using Equation (\ref{['fit_S']}), together with the $\tau$ obtained from the fits of $\chi(f)$ with Equation (\ref{['Davidson_Cole']}) and the results from Refs. Samarakoon22Matsuhira11Dusad19. Below 600 mK, it is not possible to fit the noise and ac susceptibility due to the absence of shoulder in the noise and of maxima in $\chi"$. The $\alpha$ parameter can nevertheless be obtained from the slope of the noise. (b) Relaxation times $\tau_{\rm slow}$ and $\tau_{\rm fast}$ (top panel) and exponents $\alpha_{\rm fast}$ and $\alpha_{\rm slow}$ (bottom panel) in Ho$_2$Ti$_2$O$_7$ obtained from noise measurements. Above 2 K, $\alpha_{\rm fast}$ is obtained with a high uncertainty because the power law slope of the mode is at the limit of the frequency window (diamond symbols). The relaxation times obtained from ac susceptibility are also shown. The shaded backgrounds indicate the three temperature regimes: (i) equilibrium (blue), (ii) local equilibrium (white) and (iii) out of equilibrium (purple).
• Figure 5: ac susceptibility $\chi'$ (top panel) and $\chi"$ (bottom panel) vs $f$ for different temperatures below 300 mK in Ho$_2$Ti$_2$O$_7$. Inserts: Susceptibility measured at 163 mK after different waiting times.