Charge-density-wave order with momentum (2Q,0) and (0,2Q) 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 Q(x) = (2Q,0) and Q(y) = (0,2Q) consistent with the experiments. We show that an instability towards Delta(Q)(k) = < c(k+Q)(dagger)c(k-Q)> with Q = Q(x) or Q(y) is a threshold phenomenon, but the dimensionless spin-fermion coupling is above the threshold, if the magnetic correlation length xi exceeds a certain critical value. At a critical xi, 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 Q(x) or Q(y) changes sign under k -> k + (pi, pi), but vertical bar Delta(Q)(k)vertical bar not equal vertical bar Delta(Q)(k+(pi,pi))vertical bar. In real space, such an order has both bond and site components; the bond one is larger. We further argue that Delta(Q)(k) and Delta(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 Delta(Q)(k) with Q = Q(x) or Q(y) 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 = Q(x) or Q(y), breaking Z(2) lattice rotational symmetry, and differ in phase by +/-pi/2. The selection of pi/2 or -pi/2 additionally breaks Z(2) time-reversal symmetry, such that the total order parameter manifold is U(1) x Z(2) x Z(2). In state II, density and current modulations emerge with different Q and the order parameter manifold is U(1) x U(1) x Z(2), where in the two realizations of state II, Z(2) 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 Z(2) lattice rotational symmetry (C-4 -> C-2) at T = T-n and develops a nematic order, then breaks Z(2) time-reversal symmetry at T-t < T-n 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 Delta(Q)(k) at T = T-CDW < T-t < T-n and develops a true charge order. We argue that at a mean field, T-CDW is smaller than superconducting T-SC, but preemptive composite order lifts T-CDW and reduces T-SC such that at large xi 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.