Fractionalized Fermi liquid with bosonic chargons as a candidate for the pseudogap metal

Doping a Mott-insulating $\mathbb{Z}_2$ spin liquid can lead to a fractionalized Fermi liquid (FL*). Such a phase has several favorable features that make it a candidate for the pseudogap metal for the underdoped cuprates. We focus on a particular, simple $\mathbb{Z}_2$-FL* state which can undergo a confinement transition to a spatially uniform superconductor which is smoothly connected to the `plain vanilla' BCS superconductor with $d$-wave pairing. Such a transition occurs by the condensation of bosonic particles carrying $+e$ charge but no spin (`chargons'). We show that modifying the dispersion of the bosonic chargons can lead to confinement transitions with charge density waves and pair density waves at the same wave-vector $\mathbf{K}$, co-existing with $d$-wave superconductivity. We also compute the evolution of the Hall number in the normal state during the transition from the plain vanilla FL* state to a Fermi liquid, and argue, following Coleman, Marston, and Schofield [Phys. Rev. B 72, 245111 (2005)], that it exhibits a discontinuous jump near optimal doping. We note the distinction between these results and those obtained from models of the pseudogap with fermionic chargons.