  To investigate the cluster glass phase of ultrasoft particles, we examine an
annealed two-replica system endowed with an attractive inter-replica field
similar to that of a binary symmetric electrolyte. Leveraging this analogy, we
conduct pole analysis on the total correlation functions in the two-replica
system where the inter-replica field will eventually be switched off. By
synthesizing discussions grounded in the pole analysis with a hierarchical view
of the free-energy landscape, we derive an analytical form of the mean overlap
between two replicas within the mean field approximation of the Gaussian core
model. This formula elucidates novel numerical findings observed in the cluster
glass phase.
