  The Ghatak-Sherrington (GS) spin glass model is a random probability measure
defined on the configuration space $\{0,\pm1,\pm2,\ldots, \pm \mathcal{S} \}^N$
with system size $N$ and $\mathcal{S}\ge1$ finite. This generalizes the
classical Sherrington-Kirkpatrick (SK) model on the boolean cube $\{-1,+1\}^N$
in order to capture more complex behaviors, including the spontaneous inverse
freezing phenomenon. Although many results on the physics side have been
established to understand the GS model, mathematical exploration of the model
remains scarce. Overlap, the normalized inner product of two configurations,
acts as the system order parameter to understand the phase transition of
mean-field spin glass models. In this paper, we use moment method combined with
the cavity approach to rigorously establish a quantitative joint central limit
theorem for the overlap and self-overlap array. The results hold at high
temperatures under arbitrary crystal and external fields. Compared to the SK
model, the main challenge comes from the non-trivial self-overlap terms which
also correlated with the standard overlap terms.
