Charge-Lattice Coupling in Hole-Doped LuFe_{2}O_{4+δ}: The Origin of Second-Order Modulation

Understanding singularities in ordered structures, such as dislocations in lattice modulation and solitons in charge ordering, offers great opportunities to disentangle the interactions between the electronic degrees of freedom and the lattice. Specifically, a modulated structure has traditionally been expressed in the form of a discrete Fourier series with a constant phase and amplitude for each component. Here, we report atomic scale observation and analysis of a new modulation wave in hole-doped LuFe_{2}O_{4+δ} that requires significant modifications to the conventional modeling of ordered structures. This new modulation with an unusual quasiperiodic singularity can be accurately described only by introducing a well-defined secondary modulation vector in both the phase and amplitude parameter spaces. Correlated with density-functional-theory (DFT) calculations, our results reveal that those singularities originate from the discontinuity of lattice displacement induced by interstitial oxygen in the system. The approach of our work is applicable to a wide range of ordered systems, advancing our understanding of the nature of singularity and modulation.