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 $\Delta_k^Q = \langle c^\dagger_{{\bf k}+{\bf Q}} c_{{\bf k}-{\bf Q}}\rangle$withmomenta${\bf Q}=Q_x =(2Q,0)$and${\bf Q}=Q_y =(0,2Q)$, if the magnetic correlation length $\xi$ exceeds some critical value. We argue that ${\mathrm{\Delta }}_k^Q$ and ${\mathrm{\Delta }}_{-k}^Q$ 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 ${\mathrm{\Delta }}_{\pm k}^Q$ with $\mathbf{Q}=Q_x$ 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 $\mathbf{Q}=Q_x$ or $Q_y$, breaking $Z_2$ lattice rotational symmetry, and differ in phase by $\pm \pi \mathrm{/}2$. The selection of $\pi \mathrm{/}2$ or $-\pi \mathrm{/}2$ 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 $\mathbf{Q}$ 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 ${\mathrm{\Delta }}_k^Q$ at $T=T_{\mathrm{c}\mathrm{d}\mathrm{w}}<T_t<T_n$ and develops a true charge order. We argue that the preemptive orders lift $T_{\mathrm{c}\mathrm{d}\mathrm{w}}$ and reduces $T_{\mathrm{s}\mathrm{c}}$ such that at large $\xi$ charge order may develop prior to superconductivity. We obtain the phase diagram and present quantitative comparison with ARPES data for hole-doped cuprates.