Pair density wave solution for self-consistent model

In the t-U-V Hubbard model on the square lattice we found self-consistent analytic solution for the ground state with coexisting d-wave symmetric bond ordered pair density wave (PDW) and spin (SDW) or charge (CDW) density waves, as observed in some high-temperature superconductors. In particular, the solution gives the same periodicity for CDW and PDW, and a pseudogap in the fermi-excitation spectrum.