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We used resonant elastic x-ray scattering at the Cu $L_3$ and Dy $M_5$ edgesto investigate charge order in thin films of underdoped DyBa$_2$Cu$_3$O$_{6+x}$(DyBCO) epitaxially grown on NdGaO$_3$ (110) substrates. The films show anorthorhombic crystal structure with short-range ortho-II oxygen order in thecharge-reservoir layers. At the Dy $M_5$ edge we observe diffraction peaks withthe same planar wavevectors as those of the two-dimensional charge density wavein the CuO$_2$ planes and of the ortho-II oxygen order, indicating theformation of induced ordered states on the rare-earth sublattice. The intensityof the resonant diffraction peaks exhibits a non-monotonic dependence on anexternal magnetic field. Model calculations on the modulation of thecrystalline electric field at the Dy sites by charge and oxygen order capturethe salient features of the magnetic field, temperature, and photon energydependence of the scattering intensity.

Imprint of charge and oxygen orders on Dy ions in DyBa$_2$Cu$_3$O$_{6+x}$ thin films probed by resonant x-ray scattering