Interplay of charge density wave, pairing density wave and superconductivity in high Tc cuprate superconductors

We investigate how exactly the charge density wave (CDW), pair density wave (PDW) and d-wave uniform superconducting order (d-wave SC) interplay in high Tc cuprate superconductors. By applying the renormalized mean-field theory in real space for the t-J model, we derive self-consistent renormalized mean field equations for CDW, PDW and d-wave SC order parameters (n(Q), △(Q), △₀). Here we focus on wavevectors  Q = (Q_x, Q_y) and their multiples (nQ_x, mQ_y)  with Q_x= Q_y = π/4  and n, m = 0, ±1, ±2, ±3, ...,±7.  Solving the obtained mean-field equations, phase boundaries of for different s. In particular, we find that density waves in the underdoped regime are dominated by wavevectors being  π/4 and its multiples, while density waves are dominated by wavevectors being π/2 and its multiples around the optimum doping regime and continue to the overly doped regime and vanish at doping being around 0.19, resulting in a quantum critical point at doping level 0.19.