Strong competition between ${\mathrm{\Theta }}_{II}$-loop-current order and $d$-wave charge order along the diagonal direction in a two-dimensional hot spot model

We study the fate of the so-called ${\mathrm{\Theta }}_{II}$-loop-current order that breaks both time-reversal and parity symmetries in a two-dimensional hot spot model with antiferromagnetically mediated interactions, using Fermi surfaces relevant to the phenomenology of the cuprate superconductors. We start from a three-band Emery model describing the hopping of holes in the CuO${}_2$ plane that includes two hopping parameters $t_{pp}$ and $t_{pd}$, local on-site Coulomb interactions $U_d$ and $U_p$ and nearest-neighbor $V_{pd}$ couplings between the fermions in the copper [Cu$(3d_{x^2-y^2})$] and oxygen [O$(2p_x)$ and O$(2p_y)$] orbitals. By focusing on the lowest-energy band, we proceed to decouple the local interaction $U_d$ of the Cu orbital in the spin channel using a Hubbard-Stratonovich transformation to arrive at the interacting part of the so-called spin-fermion model. We also decouple the nearest-neighbor interaction $V_{pd}$ to introduce the order parameter of the ${\mathrm{\Theta }}_{II}$-loop-current order. In this way, we are able to construct a consistent mean-field theory that describes the strong competition between the composite order parameter made of a quadrupole-density-wave and $d$-wave pairing fluctuations proposed in Efetov \emph{et al.} [Nat. Phys. $\mathbf{\text{9}}$, 442 (2013)] with the ${\mathrm{\Theta }}_{II}$-loop-current order parameter that is argued to be relevant for explaining important aspects of the physics of the pseudogap phase displayed in the underdoped cuprates.