Numerical test of the replica-symmetric Hamiltonian for correlations of the critical state of spin glasses in a field

A growing body of evidence indicates that the sluggish low-temperature dynamics of glass formers (e.g., supercooled liquids, colloids, or spin glasses) is due to a growing correlation length. Which is the effective field theory that describes these correlations? The natural field theory was drastically simplified by Bray and Roberts in 1980. More than 40 years later, we confirm the tenets of Bray and Roberts's theory by studying the Ising spin glass in an externally applied magnetic field, both in four spatial dimensions (data obtained from the Janus collaboration) and on the Bethe lattice.

Figure 1

Temperature dependence of the ratio of renormalized couplings $\lambda$, see Eq. (9), computed with a magnetic field $h=0.7$ on a Bethe lattice. The critical temperature is marked with a vertical line. We plot the data obtained with the three-, four-, and six-replica estimators. The black dot reports the value of $\lambda \left(T_c^{+}\right)\simeq 0.47$, see Eq. (11), that has been computed analytically in [55]. All three estimators take the same value ${\lambda }^{*}\simeq 0.55$ at the critical temperature. The continuous lines, marked with $N=\infty$, are the extrapolations of the data considering scaling corrections [56] and are compatible with the analytical computation $\lambda \left(T_c^{+}\right)$.

Figure 2

Replicon (${\chi }_R$) and longitudinal (${\chi }_L$) susceptibilities, Eqs. (4) and (5) vs temperature, as computed for the $D=4$ Edwards-Anderson model at magnetic fields $h=0.075,0.15$, and 0.3 (for each $h$, the temperature is plotted rescaled by the corresponding best estimate for $T_c$ [24]). To avoid cluttering the plot, we only show data for our largest system, $L=16$.

Figure 3

$D=4$ Edwards-Anderson model with magnetic field $h=0.15$. Left: Four-replica estimate, ${\omega }_1^{\left(4\right)}$, for the nonlinear susceptibility in Eq. (6) vs temperature. Right: Dimensionless quantity ${\mathrm{\Lambda }}_1$, recall Eq. (10), vs temperature. In both panels, $T$ is in units of the estimated critical temperature $T_c$ [24].

Figure 4

Three- and four-replica estimators for $\lambda$ as a function of the temperature in the $D=4$ Ising spin glass (the value of the magnetic field is indicated above each panel). Vertical lines report the three critical temperatures taken from [24]. The band around ${\lambda }^{*}\simeq 0.55$ is our best $L\to \infty$ extrapolation, assuming three- and four-replica estimators converge to a common value for all the three simulated values of the magnetic field (the width of the band represents the uncertainty in our extrapolation for $h=0.075$).