Numerical analysis of three-band models for CuO planes as candidates for a spontaneous T-violating orbital current phase

Abstract

Recently, we have numerically evaluated the current-current correlation function for the ground states of three-band models for the CuO planes of high-${T}_{\mathrm{c}}$ superconductors at hole doping $x=1∕8$ using systems with 24 sites and periodic boundary conditions. In this paper, the numerical analysis is extended to a wider range of parameters. Our results show no evidence for the time-reversal symmetry violating current patterns recently proposed by Varma [C. M. Varma, Phys. Rev. B 73, 155113 (2006)]. If such current patterns exist, our results indicate that the energy associated with the loop currents must be smaller than $5\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$ per link even if the on-site chemical potential on the oxygen sites, which is generally assumed to be of the order of ${ϵ}_{\mathrm{p}}\ensuremath{-}{ϵ}_{\mathrm{d}}=3.6\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, is taken to 1.8, 0.9, 0.4, and finally $0\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$. The current-current correlations remain virtually unaffected as we increase the interatomic Coulomb repulsion ${V}_{\mathrm{pd}}$, the term driving the system into the current carrying phase in Varma's analysis, from $1.2\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}2.4\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$. 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 pseudogap phase. We further derive an upper bound for the magnetic moment per unit cell from the upper bound we obtain for spontaneous currents, and find it smaller than the magnetic moment measured in a recent neutron scattering experiment. In this context, we observe that if the observed magnetic moments were due to a current pattern, the magnitude of these currents would be insufficient to determine the phase diagram. Finally, we discuss the role of finite size effects in our numerical experiments. In particular, we show that the net spin $1∕2$ of our finite size ground states does not infringe on the validity of our conclusions.