# Spin glasses overlap

Suppose you have a spin glass simulation in which the standard Metropolis MC algorithm is used to sample phase space. The we calculate the equivalent for the lattice system of the self intermediate scattering function, namely: $$C(\tau) = \frac{1}{N}\sum_i^N\left<\sigma_i(t)\sigma_i(t+\tau)\right>$$ in which $\sigma_I$ is the i-th spin, and the sum is average over all $t$, steps of the MC simulation.

What information can be extracted from this correlation function? Is this the same information we can obtain from the self intermediate scattering function for "normal" (= NON spin) glasses?

Is it correct to say that the long asymptotic long time value is the $q_{EA}$ (Edward-Anderson)?

I don't know how this function is called in the spin glass framework. Do you know how can I find it in the literature and/or have you some paper to suggest about it?

• Can I ask you if you were looking for something different from my answer? Commented Sep 19, 2018 at 20:12

This is called the Temporal Autocorrelation Function. A bit like spatial correlations, it is related to critical dynamics. For example, in a simple, non frustrated 2d Ising model, time autocorrelation is low in the high temperature phase (which is noisy and therefore forgets quickly), and also in the low temperature phase (which remembers for a long time, but has no variability), but is high in the vicinity of the critical point. In other words, it tells us how much a spin at a given time is related to a spin at a later time. $$C$$ is obviously a monotonically decreasing function.
I don't know much about actual spin glasses, but your intuition that $$C = q$$ at infinity seems correct to me, and is reported in these notes. These could also be a starting point to find further literature.