Dynamics of competing orders in two-dimensional metals with antiferromagnetic exchange interactions

We study the dynamics of bond order parameters after a quantum quench in a two-dimensional square lattice model with nearest-neighbor exchange and repulsion, using an unrestricted time-dependent Hartree-Fock computation. The mean-field model can be constructed by a set of operators, including $d$-wave Cooper pair and particle-hole pair, which form a SU(4) algebra, and thus their equations of motion are closed. After the quench, we find enhanced oscillation amplitude of the $d$-wave charge order below superconducting critical temperature ($T_c$) as observed in recent experiments in $YBa_2Cu_3O_{6+x}$. We also observe a phase shift when crossing $T_c$ and temperature-dependent frequencies.