  Recent work found an enhanced correction to the entanglement entropy of a
subsystem in a chaotic energy eigenstate. The enhanced correction appears near
a phase transition in the entanglement entropy that happens when the subsystem
size is half of the entire system size. Here we study the appearance of such
enhanced corrections holographically. We show explicitly how to find these
corrections in the example of chaotic eigenstates by summing over contributions
of all bulk saddle point solutions, including those that break the replica
symmetry. With the help of an emergent rotational symmetry, the sum over all
saddle points is written in terms of an effective action for cosmic branes. The
resulting Renyi and entanglement entropies are then naturally organized in a
basis of fixed-area states and can be evaluated directly, showing an enhanced
correction near holographic entanglement transitions. We comment on several
intriguing features of our tractable example and discuss the implications for
finding a convincing derivation of the enhanced corrections in other, more
general holographic examples.
