Two observable features of the staggered-flux phase at nonzero doping

Abstract

We investigate whether the staggered-flux phase (SFP) is realized in slightly doped phases of the Cu-O high-${\mathit{T}}_{\mathit{c}}$ superconductors. Using a mean-field solution of the t-J model, we calculate the size of circulating currents in the ${\mathrm{CuO}}_{2}$ planes. For realistic parameters we find nonzero currents when the doping \ensuremath{\delta}0.12. Taking into account structural details, we calculate the physical magnetic-field strength and the neutron-scattering cross section. The static field at the muon site varies between 0 and 100 G depending mainly on doping but with additional complications being the size of the Wannier functions, temperature, screening, localization, and the mean-field-approximation itself. These fields are not detected in muon-spin-relaxation experiments but cannot be ruled out both because of the aforementioned complications and because at low doping the muon is also affected by residual quasistatic spin moments. Neutrons scattering off orbital moments of the SFP exhibit a Bragg peak at wave vector (\ensuremath{\pi}/a,\ensuremath{\pi}/a) even at nonzero doping; however, this peak is perhaps 70 times weaker than that produced by static spin moments in a fully N\'eel ordered phase and is therefore difficult to observe. The absence of quasistatic spin moments in our description conflicts with neutron experiments on lightly doped samples. The inelastic spin structure does, however, exhibit a split peak at wave vector (\ensuremath{\pi}/a,\ensuremath{\pi}/a) in qualitative agreement with neutron experiments on superconducting ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{CuO}}_{4}$ samples but additional structure along the (${\mathit{Q}}_{\mathit{x}}$,0) and (0,${\mathit{Q}}_{\mathit{y}}$) directions has not been seen. The absence of magnetic fields when \ensuremath{\delta}>0.12 is consistent with the limits set by the muon experiments on superconducting samples. We show that similar results are obtained using the Gutzwiller-projected SFP.