
\subsection{I. Unit cell}

\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.8\hsize]{sfig1.eps} 
\end{center}
\caption{\label{fig1} Low temperature structure of U$_3$O$_8$. 
(a) Top view of the $Amm2$ orthorhombic cell. The U atom labels are for reference in the text.
(b) Perspective view of the same cell, showing the layered character of the structure.
(c) The oxygen pentagonal bipyramids coordinating the U atoms.}
\end{figure}

\clearpage

\subsection{II. Derivation of the magnetic subgroups of the U$_3$O$_8$ $Amm2$ structure}

Magnetic phase transitions involving ordering of localized magnetic moments break the symmetry of the non-magnetic parent phase. The magnetic ordering is then described using an eigenvector basis because, for any ordered property expressed as propagation vectors of the Brillouin zone of the crystal, Wigner’s theorem warrants that eigenvectors transform as irreducible representations. In the current case, the simplest propagation vector compatible with the observed reflections is the
$Z$ point $(\frac{1}{2} 0 0)$ of the centred orthorhombic Brillouin zone of the $Amm2$ parent structure. The magnetic representation of the magnetic moments of the U atoms decomposes onto the following unidimensional irreducible representations of the parent group:
\begin{equation}
mZ_1\oplus 3\,mZ_2\oplus 3\,mZ_3\oplus 2\,mZ_4 \nonumber
\end{equation}

\noindent
Each one of these four irreducible representations gives a compatible magnetic space group describing the magnetic order in the daughter phase:

\begin{table}[!h]
\begin{ruledtabular}
\begin{tabular}{lll}
 Magnetic Irrep & Magnetic Space Group & Comments \\
\hline
 $mZ_1$    & $A_{2a}mm2$ OG \#38.6.270 & AFM, collinear, U1 non-magnetic  \\ [2pt]
 $mZ_2$    & $A_{2a}mm'2$ OG \#38.9.273 & Planar, non-collinear  \\ [2pt]
 $mZ_3$    & $A_{2a}mm2$ OG \#38.6.270 & Planar, non-collinear  \\ [2pt]
 $mZ_4$    & $A_{2a}mm'2$ OG \#38.9.273 & FM intralayer, AFM interlayer, collinear  \\ [2pt]
\end{tabular}
\end{ruledtabular}
\end{table}

\begin{figure}[!h]
\begin{center}
\includegraphics[width=\hsize]{sfig2.eps} 
\end{center}
\caption{\label{fig2} The schematic figures below display the characteristic ordering of the magnetic moments of U atoms within each one of the four possible structures. Only one of the two layers composing the magnetic structure is represented. In the next layer magnetic moments are always flipped.}  
\end{figure}

\clearpage

List of the low {\bf q} magnetic reflections predicted by the $mZ_1$ AFM model. The reflections indexes refer to the magnetic lattice with the a lattice parameter doubled with symmetry
$A_{2a}mm2$ (38.6.270) in the OG symbol convention \cite{litvin2008}.

\begin{table}[!h]
\begin{ruledtabular}
\begin{tabular}{cccc}
 HKL & $q$ (\AA$^{-1}$) & Model prediction & Miskowiec \cite{miskowiec2021} \\
\hline
(100)    & 0.76 & extinct  & \\ [2pt]
(110)    & 0.92 & extinct  & \\ [2pt]
(101)    & 1.20 & extinct  & \\ [2pt]
(120)    & 1.29 & extinct  & \\ [2pt]
(111)    & 1.31 & ``strong'' & observed \\ [2pt]
(121)    & 1.60 & extinct &  \\ [2pt]
(130)    & 1.75 & extinct &  \\ [2pt]
(131)    & 1.98 & very weak & not observed \\ [2pt]
(102)    & 2.02 & extinct &  \\ [2pt]
(112)    & 2.09 & extinct &  \\ [2pt]
(140)    & 2.23 & extinct &  \\ [2pt]
(300)    & 2.27 & extinct &  \\ [2pt]
(122)    & 2.28 & very weak & observed \\ [2pt]
(310)    & 2.33 & extinct  &
\end{tabular}
\end{ruledtabular}
\end{table}

\clearpage

\subsection{III. Electronic structure}

\begin{figure}[!h]
\includegraphics[width=0.7\hsize]{sfig3a.eps}
\includegraphics[width=0.7\hsize]{sfig3b.eps}
\caption{\label{fig3} (a)
Band structure of U$_3$O$_8$, with the colors highlighting
the contributions of the U $5f$, O $2p$, and U $6d$ states, which are the main contributors in the energy range shown [the valence band maximum (VBM) is at 0]. Close inspection, using
a $10\times10\times14$ {\bf k}-point mesh, shows that the conduction band minimum falls
within the triangle formed by the $\Gamma$, X, and Y (see Fig.~\ref{fig4}).
(b) Atom-type projected density of states. The density of states calculations indicate that the band gap is
close to 1.27 eV. The U1 and U2 $5f$ labels are for reference in the main text.}
\end{figure}

\begin{figure}[!h]
\includegraphics[width=0.6\hsize]{sfig4.eps}
\caption{\label{fig4} 
{\bf k}-point path for the band plot in Fig.~\ref{fig2}(a). Note here we
use the $C2mm$ setting of the orthorhombic
cell (see Fig.~12 in Ref.~\citenum{setyawan2010}).}
\end{figure}


\clearpage

\begin{center}
\subsection{IV. Projected densities of states}
\end{center}

\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.36\hsize]{sfig5a.eps}
\includegraphics[width=0.36\hsize]{sfig5b.eps}
\includegraphics[width=0.36\hsize]{sfig5c.eps}
\includegraphics[width=0.36\hsize]{sfig5d.eps}
\includegraphics[width=0.36\hsize]{sfig5e.eps}
\includegraphics[width=0.36\hsize]{sfig5f.eps}
\includegraphics[width=0.36\hsize]{sfig5g.eps}
\includegraphics[width=0.36\hsize]{sfig5h.eps}
\end{center}
\caption{\label{fig5} (a) Comparison of DOS with and without SOC over a wider range of
energies than in the main text (VBM is at 0). SOC increases the band widths, as can be seen in
the conduction band, and especially in the lower valence band, where there is a split not
present in the calculations neglecting SOC. Which states make the difference at which energies
 is made clear in (b) and (c). More specifically, figures (d) and (e) compare the contributions
 of uranium atom U1, which carry no net magnetic moment. Clearly, SOC lowers the energy of
 the $5f$ states with respect to the VBM, while increasing the band width. Further, with SOC the
 $6p$ states split, with a narrow, strong peak at $\sim-25$ eV. (The DOS of U1 for down spin is
 indistinguishable from the up spin DOS, as the magnetic moment is 0). Figures (f), (g), and (h)
 compare the contribution of the uranium U2 atoms, which carry a net magnetic moment. Figures
 (f) and (h) show that the magnetization is essentially due to the $5f$ states just below the VBM.
 Figure (g) again shows that SOC causes a splitting of the $6p$ states, with a narrow, strong
 peak below a shallower extended continuum. These $6p$ states (from U1 and U2) contribute to
 lower considerably the total energy of the system.}
\end{figure}


\clearpage

\begin{center}
\subsection{V. Dielectric function}
\end{center}


\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.49\hsize]{sfig6a.eps}
\includegraphics[width=0.49\hsize]{sfig6b.eps}  
\end{center}
\caption{\label{fig6} Imaginary part (left panel) and real part (right panel) of the
dielectric function. Convergence of the real part is more difficult because it is calculated
based on the imaginary part, using the Kramers-Kronig relations. For the real part to be converged up to a certain energy, sufficient bands should be included. Here convergence is reasonable up to
5--6 eV with 448 bands.}
\end{figure}


