No Evidence for Spontaneous Orbital Currents in Numerical Studies of Three-Band Models for the CuO Planes of High Temperature Superconductors

We have numerically evaluated the current-current correlations for three-band models of the CuO planes in high-$T_c$ superconductors at hole doping $x=1/8$. The results show no evidence for the orbital current patterns proposed by Varma. If such patterns exist, the associated energy is estimated to be smaller than 5 meV per link even if ${ϵ}_{\mathrm{p}}-{ϵ}_{\mathrm{d}}=0$. Assuming that the three-band models are adequate, quantum critical fluctuations of such patterns hence cannot be responsible for phenomena occurring at significantly higher energies, such as superconductivity or the anomalous properties of the strange metal phase.

Figure 1

Orbital current pattern proposed by Varma.

Figure 2

Current-current correlations $⟨j_{k,k+\widehat{x}}{j}_{l,m}⟩$ multiplied by ${10}^2$ for the ground state of (2) with ${ϵ}_{\mathrm{p}}=3.6$ on a 24 site cluster (8 $\mathrm{Cu}=\mathrm{\text{open}}$ circles, 16 $\mathrm{O}=\mathrm{\text{filled}}$ circles) with periodic boundary conditions (PBCs). The reference link is indicated in the top and (due to the PBCs) bottom left corner.

Figure 3

As in Fig. 2, but with ${ϵ}_{\mathrm{p}}=0$. Except for the vertical links, positive correlations indicate alignment with the current pattern shown in Fig. 1.