I am currently simulating an Edwards-Anderson spin glass using standard Metropolis Monte Carlo techniques. The spins are placed on a 3D cubic lattice with periodic boundaries and take on Ising values (-1/+1), the couplings between them are drawn from a Gaussian random distribution.
What's now unclear to me is how I technically implement typical spin glass observables, such as the Edwards-Anderson order parameter: $$q_{EA} = \frac{1}{N}\sum_{i=1}^N \langle\sigma_i\rangle^2 $$
According to this definition, I first do thermal averages for each of the sites independently, and afterwards average over the system. Finally, I might do another average of independent disorder replicas.
So, in practice, how would one implement such an observable? Keeping track of the thermal "history" of every site throughout the entire simulation seems tedious and will be expensive once the lattice becomes larger. And even worse, observables such as the spin glass susceptibility $$\chi_{SG} = \frac{1}{N}\sum_{i,j}^N \langle\sigma_i\sigma_j\rangle^2 $$ require to record the history of every pair of sites...
- Literature: Parisi, Giorgio. "Order parameter for spin-glasses." Physical Review Letters 50.24 (1983): 1946.
