Scaling Law Describes the Spin-Glass Response in Theory, Experiments, and Simulations

The correlation length $\xi$, a key quantity in glassy dynamics, can now be precisely measured for spin glasses both in experiments and in simulations. However, known analysis methods lead to discrepancies either for large external fields or close to the glass temperature. We solve this problem by introducing a scaling law that takes into account both the magnetic field and the time-dependent spin-glass correlation length. The scaling law is successfully tested against experimental measurements in a CuMn single crystal and against large-scale simulations on the Janus II dedicated computer.

Figure 1

Comparison of the classic experiment from Joh et al. [16] and present work. Both data sets are for ${\mathrm{Cu}}_{94}{\mathrm{Mn}}_6$. The data from Joh et al. are from a polycrystalline sample, while data from the present work come from a single crystal allowing for a much larger correlation length $\xi$ (see Table 1 for details). The figure shows the maximum of the relaxation function as a function of the squared magnetic field $H^2$. It is easy to estimate the slope at $H^2=0$ (from which $\xi$ is measured) for the data from Joh et al., which display a linear behavior for $H^2\lesssim 6\times {10}^4{\mathrm{Oe}}^2$. Instead, the large $\xi$ of the data from the present work not only causes a larger slope, but also a much larger curvature (see the enlarged region in the inset) which makes it challenging to extrapolate the slope to $H^2=0$.

Figure 2

A set of relaxation curves $S(t)=d(M/H)/d\ln t$ for CuMn at $T=29\mathrm{K}$ and $t_w={10}^4\mathrm{s}$ (top) and for the Ising-Edwards-Anderson model at $T=0.9$ and $t_w=2^{22}$ lattice sweeps (bottom). The relation between IEA and physical units is discussed in the text.

Figure 3

Experimental and numerical $\ln {t_w}^{\mathrm{eff}}$ from the maximum of the response function in Fig. 2. Top: data from the experiments in Table 1. Lines are fits to a polynomial in $H^2$, as in Eq. (9). The fit parameters are reported in Table 3. Bottom: numerical data for the runs in Table 2 (the lines are just guides for the eye).

Figure 4

The nonlinear part of the response time data: $[\ln (t_H^{\mathrm{eff}}/t_{H\to 0^{+}}^{\mathrm{eff}})-a_2(T)H^2]{\xi }^{\theta (\tilde{x})/2}$ plotted against the scaling variable $[H^2{\xi }^{D-\theta (\tilde{x})/2}{]}^2$, see Eq. (1). Left: experimental data (see Table 1). Right: numerical data (see Table 2). The main panel, in linear scale, shows a closeup for small values of $[H^2{\xi }^{D-\theta (\tilde{x})/2}{]}^2$. The inset is in log scale in order to report all our numerical data. Note that experimental and numerical data are reported in different unit systems (see main text).