Entanglement complexity of the Rokhsar-Kivelson-sign wavefunctions

Entanglement comes in different forms, some more complex than others. 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 subextensive scaling of entanglement, reminiscent of the many-body-localization transition of disordered quantum Hamiltonians. 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 ``superuniversal'' 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 nonuniversal 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---a very strong candidate being offered by the many-body localization transition of disordered lattice-spin models.