= Spin overlap
{title2=$q(s,t)=N^{-1}\sum_i s_it_i$}
= Overlap of spin configurations
{synonym}
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> is
$$
P(q=1)=P(q=-1)=\frac16,\qquad
P(q=1/3)=P(q=-1/3)=\frac13.
$$
It is symmetric, with $\langle q\rangle=0$ but $\langle q^2\rangle=11/27$. At infinite <temperature>, all eight <microstates> are equally probable and instead $\langle q^2\rangle=1/3$. This computable change in similarity statistics is a finite-model fact, not evidence for <replica symmetry breaking>.
Back to article page
