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Description
In this work, We study the interplay among charge density wave (CDW), pair density wave (PDW), and d-wave superconducting (SC) orders in high-temperature cuprate superconductors. Using the renormalized mean-field theory based on the t-J model formulated in real space, we derive self-consistent equations incorporating for CDW, PDW and d-wave SC order parameters with different doping. We focus on wavevectors Q=(Qx,Qy) and their multiples (nQx,mQy) with Qx=Qy=pi/4 and m, n=0,+/-1, +/-2…+/-7 . The results show that density waves are dominated by wavevectors being pi/4 and its multiples in the underdoped regime, and they are dominated by wavevectors being pi/4 and its multiples around the optimum doping regime. Furthermore, density waves continue to the overly doped regime and vanish at doping being around 0.19, displaying the quantum critical point at doping level 0.19.