Overlap locking and non-perturbative effects in spin glasses

Abstract

We study the phenomenon of the locking of the order parameter (or synchronization) in spin glasses at low temperatures. When two systems with independent disorders are coupled, their overlaps become similar. A crucial question is how this effect depends on the strength of the coupling between the two systems. Non-perturbative phenomena are present when $1 \ll ΔH \ll N$, being $ΔH$ the coupling Hamiltonian and $N$ the size of the system. In this intermediate-coupling region, the effect is related to finite-size free-energy corrections and to the correlations in the Dyson hierarchical spin glass, a model that mimics the physics of finite-dimensional systems. We study this phenomenon in the mean-field approach, both analytically and numerically, and we finally compute the critical exponents for finite-volume corrections in mean-field theory and for the decay of correlations in the Dyson hierarchical model.

Figures (9)

• Figure 1: The normalized $\chi^2$ as function of the exponent $\omega$ using the data with $N\ge 512$ and $N\ge 1024$
• Figure 2: Our prediction for the exponent $\alpha(q,\rho)$ in the one dimensional long range model as function of $\rho$
• Figure 3: The semi-analytic predictions for $z^{1/2}{\cal D}(z)$ plotted versus $z^{1/2}$ for $M=10^4,10^5,10^6,10^7$. A constant behaviour at large $z$ of this quantity implies that ${\cal D}(z)\propto z^{-1/2}$.
• Figure 4: Data for $D(t=z/N,N)$ as a function of $z$ for several system sizes. The dashed line is the predicted power behavior for ${\cal D}(z)=\lim_{N \to \infty} D(z/N,N)$. In the insert, we show the same data zoomed on the small $z$ region, where the size dependence is almost absent, and the function ${\cal D}(z)$ can be measured directly.
• Figure 5: Scaling of the function $D(t=z/N,N)$ according to the theory that predicts the decays $D(t,N) \propto t^{-3/2}$ at fixed $N$ and ${\cal D}(z) \propto z^{-1/2}$.