 Spin overlap

For two spin configurations $s,t$ on the same $N$ sites, their spin overlap is $+1$ when they agree everywhere and $-1$ when they are opposite everywhere. Two independent canonical ensemble draws must use the same quenched disorder. Otherwise their comparison mixes changes of thermal state with changes of material sample.

An original finite example uses the six equally weighted ground states of the frustrated Ising triangle. For each first configuration, the second has one identical choice, one fully reversed choice, two choices differing at one site and two differing at two sites. Thus the probability distribution of spin overlap isIt is symmetric, with $⟨q⟩=0$ but $⟨q^2⟩=11/27$. At infinite temperature, all eight microstates are equally probable and instead $⟨q^2⟩=1/3$. This computable change in similarity statistics is a finite-model fact, not evidence for replica symmetry breaking.