Z2 fractionalized phases of a solvable disordered t−J model

We describe the phases of a solvable $t\text{\ensuremath{-}}J$ 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 $\ensuremath{\phi}$. Both $f$ and $\ensuremath{\phi}$ carry emergent ${\mathbb{Z}}<sub>2</sub>$ gauge charges. The model has a phase in which the $\ensuremath{\phi}$ 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 $\ensuremath{\phi}$ and $f$ are gapless, and the electron operator $\ensuremath{\sim}f\ensuremath{\phi}$ has a Fermi liquid-like $1/\ensuremath{\tau}$ propagator in imaginary time, $\ensuremath{\tau}$. So while the electron spectral function has a Fermi liquid form, other properties are controlled by ${\mathbb{Z}}₂$ fractionalization and the anomalous exponents of the $f$ and $\ensuremath{\phi}$ 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.