Strong competition between ΘII -loop-current order and d -wave charge order along the diagonal direction in a two-dimensional hot spot model

Abstract

We study the fate of the so-called ΘII-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 CuO2 plane that includes two hopping parameters tpp and tpd, local onsite Coulomb interactions Ud and Up, and nearest-neighbor Vpd couplings between the fermions in the copper [Cu(3dx2-y2)] and oxygen [O(2px) and O(2py)] orbitals. By focusing on the lowest-energy band, we proceed to decouple the local interaction Ud 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 Vpd to introduce the order parameter of the ΘII-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 et al. [Nat. Phys. 9, 442 (2013)1745-247310.1038/nphys2641] with the ΘII-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.