Coexistence of charge-density-wave and pair-density-wave orders in underdoped cuprates

We analyze incommensurate charge-density-wave (CDW) and pair-density-wave (PDW) orders with transferred momenta $(\pm Q,0)$/$(0,\pm Q)$ in underdoped cuprates within the spin-fermion model. Both orders appear due to exchange of spin fluctuations before magnetic order develops.We argue that the ordered state with the lowest energy has non-zero CDW and PDW components with the same momentum. Such a state breaks $C_4$ lattice rotational symmetry, time-reversal symmetry, and mirror symmetries. We argue that the feedback from CDW/PDW order on fermionic dispersion is consistent with ARPES data. We discuss the interplay between the CDW/PDW order and $d_{x^2-y^2}$ superconductivity and make specific predictions for experiments.