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The complex magnetism and transport properties of tetragonalCe$_3$Cu$_4$As$_4$O$_2$ were examined through neutron scattering and physicalproperties measurements on polycrystalline samples. The lamellar structureconsists of alternating layers of $\rm CeCu_4As_4$ with a single square Celattice and oxygen-linked Ce bi-layer $\rm Ce_2O_2$. Extending along $\bf c$, atube-like Fermi surface from DFT calculations points to a quasi-two-dimensionalelectronic system. Peaks in the specific heat at the Ne\'{e}l temperature$T_{N}=24$ $\rm K$, $T_{2}~=~16 $ $\rm K$ and $T_{3}~=~1.9$ $ \rm K$ indicatethree magnetic phase transitions or distinct cross-over phenomena. For$T<T_{N}$ neutron diffraction indicates the development of ferromagnetic absheets for both Ce sites, with alternating polarization along $\bf{c}$, a wavevector ${\bf k}_{1}={\bf c}^*$. For $T<T_{2}$, quasi-two-dimensional low-energyspin fluctuations with ${\bf k}_{2}=\frac{1}{2}{\bf a}^*$ and polarizedperpendicular to ${\bf k}_{2}$ are suppressed. The data are consistent withquasi-two-dimensional antiferromagnetic order in the $\rm CeCu_4As_4$ planespolarized along the ${\bf k}_{2}$ wave vector. $T_{3}$ marks a spin-floptransition where the ${\bf k}_{1}$ staggered magnetization switches to in-planepolarization. While the narrow 4f bands lie deep below the Fermi surface, thereare significant transport anomalies associated with the transitions; inparticular a substantial reduction in resistivity for $T<T_{N}$. At $T=100$ $\rm mK$ the ${\bf k}_1$ modulated staggered moment is $0.85~\mu_B$, whichmatches the $0.8~\mu_B$ saturation magnetization achieved for H $~=~7$ $ \rm T$at $T~=~2$ $ \rm K$. From low T Lorentzian fits the correlation length is inexcess of 75 \AA. We argue the unusual sequence of magnetic transitions resultsfrom competing interactions and anisotropies for the two Ce sites.

Modulated magnetism and anomalous electronic transport in $\rm Ce_3Cu_4As_4O_2$