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

TL;DR

This paper investigates the competition between $ heta_{II}$-loop-current order and $d$-wave charge order in a two-dimensional model relevant to cuprate superconductors, revealing their interplay and implications for the pseudogap phase.

Contribution

It develops a mean-field theory for the competition between loop-current order and $d$-wave charge order using a three-band model and spin-fermion approach, highlighting their strong rivalry.

Findings

Strong competition between $ heta_{II}$-loop-current and $d$-wave charge order.

Constructed a mean-field framework capturing their interplay.

Relevance to pseudogap phenomena in cuprates.

Abstract

We study the fate of the so-called $\Theta_{II}$-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 $\Theta_{II}$-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. \textbf{9}, 442 (2013)] with the $\Theta_{II}$-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.