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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 within the the spin-fermion model. We show that magnetically-mediated interaction gives rise to charge order $Δ_k^Q = \langle c^\dagger_{{\bf k}+{\bf Q}} c_{{\bf k}-{\bf Q}}\rangle$ with momenta ${\bf Q}=Q_x =(2Q,0)$ and ${\bf Q}=Q_y =(0,2Q)$, if the magnetic correlation length $ξ$ exceeds some critical value. We argue that $Δ_k^Q$ and $Δ_{-k}^Q$ are not equivalent, and their symmetric and antisymmetric combinations describe density modulations and bond current. We derive GL functional for four-component $U(1)$ order parameters $Δ^Q_{\pm k}$ with ${\bf Q} = Q_x$ or $Q_y$. Within mean-field we find two types of CDW states, I and II, depending on system parameters. In state I density and current modulations emerge with the same ${\bf Q} = Q_x$ or $Q_y$, breaking $Z_2$ lattice rotational symmetry, and differ in phase by $\pmπ/2$. The selection of $π/2$ or $-π/2$ additionally breaks $Z_2$ time-reversal symmetry, such that the total order parameter manifold is $U(1) \times Z_2 \times Z_2$. In state II density and current modulations emerge with different $\bf Q$ and the order parameter manifold is $U(1) \times U (1) \times Z_2$. We go beyond mean-field and show that discrete symmetries get broken before long-range charge order sets in. For state I, the system first breaks $Z_2$ lattice rotational symmetry ($C_4 \to C_2$) at $T= T_n$ and develops a nematic order, then breaks $Z_2$ time-reversal symmetry at $T_t < T_n$, and finally breaks $U(1)$ symmetry of a common phase of even and odd components of $Δ^Q_{k}$ at $T= T_{\rm cdw} < T_t < T_n$ and develops a true charge order. We argue that the preemptive orders lift $T_{\rm cdw}$ and reduces $T_{\rm sc}$ such that at large $ξ$ charge order may develop prior to superconductivity. We obtain the phase diagram and present quantitative comparison with ARPES data for hole-doped cuprates.