Z2 fractionalized phases of a solvable disordered t-J model

We describe the phases of a solvable t-J model of electrons with infinite-range, and random, hopping, and exchange interactions, similar to those in the Sachdev-Ye-Kitaev models. The electron fractionalizes, as in an "orthogonal metal," into a fermion f, which carries both the electron spin and charge, and a boson phi. Both f and phi carry emergent Z(2) gauge charges. The model has a phase in which the phi bosons are gapped, and the f fermions are gapless and critical, and so the electron spectral function is gapped. This phase can be considered as a toy model for the underdoped cuprates, without spatial structure. The model also has an extended, critical, "quasi-Higgs" phase where both phi and f are gapless, and the electron operator similar to f phi has a Fermi liquid-like 1/tau propagator in imaginary time, tau. So while the electron spectral function has a Fermi liquid form, other properties are controlled by Z(2) fractionalization and the anomalous exponents of the f and phi excitations. This quasi-Higgs phase is proposed as a toy model of the overdoped cuprates. We also describe the critical state separating these two phases.