The nature of the pseudogap phase in the cuprate superconductors has been a long-standing open question. Many theories have been proposed for capturing these physics; in this work, we model the pseudogap state as a fractionalized Fermi liquid, with the underlying topological order given by a \(\pi\)-flux spin liquid. The \(\pi\)-flux state on its own is ultimately unstable to either antiferromagnetism or valence bond solid (VBS) ordering; this is in fact desirable, as it gives a mechnaism for describing the instability of the psuedogap at low temperatures to Neel order. In this work, we study additional instabilities generated by condensation of a bosonic chargon in this \(\pi\)-flux background. A notable advantage of our theory, which is most easily seen through earlier formulations in terms of ancilla degrees of freedom, is that our theory retains explicit non-fractionalzed electron-like quasiparticles which control the doping of our system, rather than requiring the bosonic chargons to keep track of the doping.

We find that the \(\pi\)-flux spin liquid naturally leads to charge density wave (CDW) and d-wave superconductivity instabilities, which is consistent with what is observed in cuprates as well as numerical studies of the Hubbard model. The periodicity of the CDW is not fixed in our theory and in general is incommensurate, although calculations beyond mean-field may show preferences to specific commensurate wavelengths. In certain parameter regimes, one can also obtain pair density wave and d-density wave order.