  In this paper we study the transitions of entanglement complexity in an
exemplary family of states - the Rokhsar-Kivelson-sign wavefunctions - whose
degree of entanglement is controlled by a single parameter. This family of
states is known to feature a transition between a phase exhibiting volume-law
scaling of entanglement entropy and a phase with sub-extensive scaling of
entanglement, reminiscent of the many-body-localization transition of
disordered quantum Hamiltonians [Physical Review B 92, 214204 (2015)]. We study
the singularities of the Rokhsar-Kivelson-sign wavefunctions and their
entanglement complexity across the transition using several tools from quantum
information theory: fidelity metric; entanglement spectrum statistics;
entanglement entropy fluctuations; stabilizer R\'enyi Entropy; and the
performance of a disentangling algorithm. Across the whole volume-law phase the
states feature universal entanglement spectrum statistics. Yet a
"super-universal" regime appears for small values of the control parameter in
which all metrics become independent of the parameter itself; the entanglement
entropy as well as the stabilizer R\'enyi entropy appear to approach their
theoretical maximum; the entanglement fluctuations scale to zero as in output
states of random universal circuits, and the disentangling algorithm has
essentially null efficiency. All these indicators consistently reveal a complex
pattern of entanglement. In the sub-volume-law phase, on the other hand, the
entanglement spectrum statistics is no longer universal, entanglement
fluctuations are larger and exhibiting a non-universal scaling; and the
efficiency of the disentangling algorithm becomes finite. Our results, based on
model wavefunctions, suggest that a similar combination of entanglement scaling
properties and of entanglement complexity features may be found in high-energy
Hamiltonian eigenstates.
