Charge-density-wave order with momentum (2 Q,0) and (0,2 Q) within the spin-fermion model: Continuous and discrete symmetry breaking, preemptive composite order, and relation to pseudogap in hole-doped cuprates

We analyze charge order in hole-doped cuprates within the the spin-fermion model. We show that a magnetically mediated interaction, which is known to give rise to d-wave superconductivity and charge order with momentum along zone diagonal, also gives rise to charge order with momenta Qx = (2Q,0) and Qy = (0,2Q) consistent with the experiments. We show that an instability towards Qk = 〈c k+Qck−Q〉 with Q = Qx or Qy is a threshold phenomenon, but the dimensionless spin-fermion coupling is above the threshold, if the magnetic correlation length ξ exceeds a certain critical value. At a critical ξ , the onset temperature for the charge order terminates at a quantum-critical point distant from the magnetic one. We argue that the charge order with Qx or Qy changes sign under k → k + (π,π ), but | Qk | = | Qk+(π,π )|. In real space, such an order has both bond and site components; the bond one is larger. We further argue that Qk and Q −k are not equivalent, and their symmetric and antisymmetric combinations describe, in real space, incommensurate density modulations and incommensurate bond current, respectively. We derive the Ginzburg-Landau functional for four-component U (1) order parameters Q±k with Q = Qx or Qy and analyze it first in mean-field theory and then beyond mean field. Within mean field we find two types of charge-density-wave (CDW) states, I and II, depending on system parameters. In state I, density and current modulations emerge with the same Q = Qx or Qy , breaking Z2 lattice rotational symmetry, and differ in phase by ±π/2. The selection of π/2 or −π/2 additionally breaks Z2 time-reversal symmetry, such that the total order parameter manifold is U (1) × Z2 × Z2. In state II, density and current modulations emerge with different Q and the order parameter manifold is U (1) × U (1) × Z2, where in the two realizations of state II, Z2 corresponds to either lattice rotational or time-reversal symmetry breaking. We extend the analysis beyond mean field and argue that discrete symmetries get broken before long-range charge order sets in. For state I, which, we argue, is related to hole-doped cuprates, we show that, upon lowering the temperature, the system first breaks Z2 lattice rotational symmetry (C4 → C2) at T = Tn and develops a nematic order, then breaks Z2 time-reversal symmetry at Tt < Tn and locks the relative phase between density and current fluctuations, and finally breaks U (1) symmetry of a common phase of even and odd components of Qk at T = TCDW < Tt < Tn and develops a true charge order. We argue that at a mean field, TCDW is smaller than superconducting TSC, but preemptive composite order lifts TCDW and reduces TSC such that at large ξ charge order develops prior to superconductivity. We obtain the full phase diagram and present quantitative comparison of our results with ARPES data for hole-doped cuprates.