Spatial modulations of mid-gap states in (001)La$_{1.88}$Sr$_{0.12}$CuO$_4$ films: Indications for anti-phase ordering of the d-wave order parameter

Using scanning tunneling spectroscopy we have investigated the spatial evolution of the anomalous c-axis zero bias conductance peak, discovered in a previous study by our group, in epitaxial La$_{1.88}$Sr$_{0.12}$CuO$_4$ thin films. We found an anisotropic spatial dependence of the corresponding low-energy density of states which complies with the predicted spectral features of an anti-phase ordering of the d-wave order parameter within the ab-plane. Such an ordering was recently suggested to account for the 1/8 anomaly in the high temperature superconductors and the dynamical layer decoupling recently reported to occur in the transport studies of La$_{15/8}$Ba$_{1/8}$CuO$_4$.