Coexistence of magnetic fluctuations and long-range order in the one-dimensional J1−J2 zigzag chain materials BaDy2O4 and BaHo2O4

The compounds ${\mathrm{BaDy}}₂{\mathrm{O}}₄$ and ${\mathrm{BaHo}}₂{\mathrm{O}}₄$ are part of a family of frustrated systems exhibiting interesting properties, including spin-liquid-type ground states, magnetic-field-induced phases, and the coexistence of short- and long-range magnetic order, with dominant one-dimensional correlations, which can be described as Ising ${J}₁\text{\ensuremath{-}}{J}₂$ zigzag chains along the $c$ axis. We have investigated polycrystalline samples of ${\mathrm{BaDy}}₂{\mathrm{O}}₄$ and ${\mathrm{BaHo}}₂{\mathrm{O}}₄$ with both neutron diffraction and neutron spectroscopy, coupled to detailed crystalline electric field calculations. The latter points to site-dependent anisotropic magnetism in both materials, which is corroborated by the magnetic structures we determined. The two systems show the coexistence of two different long-range orders: two double N\'eel $\ensuremath{\uparrow}\ensuremath{\uparrow}\ensuremath{\downarrow}\ensuremath{\downarrow}$ orders in the $ab$ plane with propagation vectors ${\mathbf{k}}₁=(\frac{1}{2},0,\frac{1}{2})$ and ${\mathbf{k}}₂=(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ for ${\mathrm{BaDy}}₂{\mathrm{O}}₄$, and two distinct arrangements of simple N\'eel $\ensuremath{\uparrow}\ensuremath{\downarrow}\ensuremath{\uparrow}\ensuremath{\downarrow}$ orders along the $c$ axis, both with the propagation vector ${\mathbf{k}}₀=(0,0,0)$ for ${\mathrm{BaHo}}₂{\mathrm{O}}₄$. The order for both wave vectors in ${\mathrm{BaDy}}₂{\mathrm{O}}₄$ occurs at ${T}_{\mathrm{N}}=0.48\phantom{\rule{0.28em}{0ex}}\mathrm{K}$, while in ${\mathrm{BaHo}}₂{\mathrm{O}}₄$ the first order sets in at ${T}_{\mathrm{N}}\ensuremath{\sim}\phantom{\rule{3.33333pt}{0ex}}1.3\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ and the second one has a lower ordering temperature of 0.84 K. Both compounds show extensive diffuse scattering which we successfully modeled with a one-dimensional axial next-nearest neighbor Ising (ANNNI) model. In both materials, strong diffusive scattering persists to temperatures well below where the magnetic order is fully saturated.