Instability towards staggered loop currents in the three-orbital model for cuprate superconductors

We present evidence for the existence of a spontaneous instability towards an orbital loop-current phase in a multiorbital Hubbard model for the ${\mathrm{CuO}}_2$ planes in cuprates. Contrary to the previously proposed ${\theta }_{II}$ phase with intra-unit cell currents, the identified instability is towards a staggered pattern of intertwined current loops. The orbitally resolved current pattern thereby shares its staggered character with the proposal of $d$-density wave order. The current pattern will cause a Fermi surface reconstruction and the opening of a pseudogap. We argue that the pseudogap phase with time-reversal symmetry breaking currents is susceptible to further phase transitions and therefore offers a route to account for axial incommensurate charge order and a polar Kerr effect in underdoped cuprates.

Figure 1

Staggered pattern of spontaneous loop currents. Open circle,“x” and “y” symbols denote $\mathrm{Cu}{d}_{x^2-y^2}, \mathrm{O}{p}_x$, and $\mathrm{O}{p}_y$ orbitals, respectively. Currents along $p-d$ bonds (black arrows) are about three times stronger than those in $p-p$ bonds (green arrows).

Figure 2

Unit cell of a ${\mathrm{CuO}}_2$ plane (dashed box). The orbital phase convention is indicated by the sign of the hopping matrix elements. Numbers in green enumerate the inequivalent bonds, and the direction of the blue arrows indicates a positive sign of the current flow for the current operator definitions given in Appendix pp1.

Figure 3

Diagrammatic structure of gRPA. (a) Interacting susceptibility, (b) vertex function, (c) effective interaction. Reprinted with permission from [34]. ©American Physical Society.

Figure 4

Temperature evolution of ${\chi }_{11}^J(\mathbf{Q},\omega =0)$ for $\mathbf{Q}=(\pi ,\pi )$. (Inset) ${\chi }_{11}^J(\mathbf{q},\omega =0)$ at $T=0.011$. The parameter values are $V_{pd}=2.2, V_{pp}=1$, and the hole density is $p=0.13$.

Figure 5

(Left) Normal (red) and reconstructed (blue) Fermi surfaces. The loop current strengths are set to $z_{pd}=0.04$ and $z_{pp}=z_{pd}/3$. The hole density here is $p=0.10$. (Right) Density of states near the Fermi energy.