Matching Microscopic and Macroscopic Responses in Glasses

We first reproduce on the Janus and Janus II computers a milestone experiment that measures the spin-glass coherence length through the lowering of free-energy barriers induced by the Zeeman effect. Secondly, we determine the scaling behavior that allows a quantitative analysis of a new experiment reported in the companion Letter [S. Guchhait and R. Orbach, Phys. Rev. Lett. 118, 157203 (2017)]. The value of the coherence length estimated through the analysis of microscopic correlation functions turns out to be quantitatively consistent with its measurement through macroscopic response functions. Further, nonlinear susceptibilities, recently measured in glass-forming liquids, scale as powers of the same microscopic length.

Figure 1

The function $\mathcal{S}(t+t_{\mathit{w}},t_{\mathit{w}};H)$, Eq. (5), versus the time $t$ elapsed after switching on the external magnetic field $H$. In the top panel we show the $H\to 0$ extrapolation for several waiting times $t_{\mathit{w}}$ (one unit of computer time roughly corresponds to one picosecond of physical time [46]). Bottom: $\mathcal{S}(t+t_{\mathit{w}},t_{\mathit{w}};H)$ as a function of $t$ for our largest waiting time $t_{\mathit{w}}=2^{30}$ and for different values of $H$. Inset: The peak position ($H\to 0$), in units of $t_{\mathit{w}}$, depends on $t_{\mathit{w}}$ only for $t_{\mathit{w}}<10^6$.

Figure 2

The Zeeman energy follows the scaling form suggested in Eq. (6). We show a fit to $F_{\text{Zeeman}}(x)=c_{1}x+c_2{x}^2$. Inset: the data of the main panel do not collapse when plotted as a function of $H^2$.

Figure 3

The time growth of the correlation length ${\xi }_{\mathrm{mic}}$, as obtained from the microscopic correlation function $C_4(r,t_{\mathit{w}})$ [10, 11, 40], is compared to the length ${\xi }_{\mathrm{mac}}$ obtained from a fit linear in $H^2$; see Eq. (7). The microscopic time scale ${\tau }_0=1$ corresponds to a single lattice sweep in our Monte Carlo simulation (see the caption to Fig. 1). We also show the results obtained with Eq. (9), which are sensible as well. The temperature-dependent scaling variable, $T\log (t_{\mathit{w}}/{\tau }_0)$, is common in the experimental literature (e.g., see Ref. [19]).

Figure 4

The nonlinear susceptibility ${\chi }_3$ is shown as a function of $t$ for several values of $t_{\mathit{w}}$ (top), as obtained from Eq. (3). The difficulty lies in balancing systematic errors (numerical data obtained with high fields underestimate ${\chi }_3$) with statistical errors (which are larger for small values of $H$). Our compromise, shown here, tries to obtain statistical and systematic errors of comparable size. In the bottom panel we show the $G(t/t_{\mathit{w}})$ scaling function (10). Note that scaling corrections are visible only for the smallest waiting times (and, even in those cases, they only appear for small $t/t_{\mathit{w}}$).