Exact islands scenario for CFT systems and critical ratios in higher geometry

We study CFT d systems which are in contact with each other and symmetrically arranged. System B is treated as bath that surrounds system A in the middle. Our focus is to learn how the entanglement entropy of a bath pair system changes as a function of its size. The total size of systems A and B taken together is kept fixed in this process. It is found that for strip shaped systems the bath entropy becomes maximum when respective system sizes follow Fibonacci-type critical ratio condition. Beyond critical point when bath size increases the bath entropy starts decreasing, where island and icebergs entropies play important role. Interestingly entire effect of icebergs can be resummed giving rise to ‘exact island’ scenario for CFT d with [Formula: see text]. Post-criticality we also find important identity involving entropy differences [Formula: see text] where island contribution is exact. The mutual information of far separated bath pair follows specific law [Formula: see text]. It never vanishes for finite systems. Once system-A size approaches to Kaluza–Klein scale the bath entropy becomes discretized. In summary, knowing island corrections is vital for large bath entanglement entropy.