\section{Supplementary}
\subsection{1.Computational details of density functional calculation}
For this prototypical case, we obtain the most relevant Hilbert space within $\pm10$ eV around the Fermi energy from the spin polarized LDA+$U$ ~\cite{Anisimov1993, Liechtenstein1995} electronic structure of the parent compound NdNiO${_2}$, using the linearized augmented plane wave~\cite{Singh} implementation~\cite{Blaha1990} of the density functional theory (DFT)~\cite{DFT1, DFT2}.
We take from Ref.~\cite{Li2019} the lattice structure with lattice constant $a=b=3.92\mathring{A},c=3.37\mathring{A}$, and the space group $P4/mmm$.
We set $U-J$ = $0.6 - 0.06$ Ry = 7.344 eV for Ni $d$-orbitals,$0.20 - 0.00$ Ry = 2.72 eV,  $0.60 - 0.00$ Ry = 8.16 eV for Nd $d$-,$f$-orbitals separately.

\vspace{-0.2cm}
\subsection{2.Extraction of many-body Hamiltonian}
We aim at obtaining a realistic SU(2)-symmetric many-body Hamiltonian $H^{\text{eff}}$ in Eq.(1).
To ensure a DFT-like level of accuracy, we further demand that when under a similar approximation, our many-body Hamiltonian needs to reproduce the self-consistent DFT Hamiltonian (equivalently the electronic band structure).
Since in LDA+$U$ the strong local electron-electron interaction is approximated in an effective Hartree-Fock manner~\cite{Anisimov1997}, a proper connection can be naturally made on the Wannier states basis by matching the self-consistent Hartree-Fock solution of our SU(2)-symmetric $H^{\text{eff}}$ with the self-consistent $H^{\text{LDA}+U}$~\cite{Wei2006}.
The requirement that the Hartree-Fock approximation of $H^{\text{eff}}$ needs to reproduce the LDA+$U$ band structure within the subspace of the active orbitals results in a rather unique set of parameters in $H^{\text{eff}}$~\cite{Wei2006}.

Specifically, since we demand that the self-consistent Hartree-Fock solution reproduce the self-consistent solution of the LDA+$U$ solution, the corresponding density matrix $\rho_{i\nu\nu^\prime}$ must be identical in both cases when represented in the same set of Wannier orbitals.
One can thus take $\rho_{i\nu\nu^\prime}$ directly from the self-consistent LDA+$U$ solution in the Wannier basis.
Furthermore, if one assumes that the structure of $U_{m m^{\prime\prime}m^{\prime\prime\prime} m^\prime}$ follows the Slater integral~\cite{Slater,Liechtenstein1995}, the entire $U_{m m^{\prime\prime}m^{\prime\prime\prime} m^\prime}$ can be fixed by just two parameters $U_{\text{eff}}$ and $J_{\text{eff}}$.
Combining $\rho_{i\nu\nu^\prime}$ and $U_{m m^{\prime\prime}m^{\prime\prime\prime} m^\prime}$, the effective Hartree-Fock potential $V^\text{HF}$ can then obtained.

In addition, orbital dependent $J_m$ in $H^{\text{eff}}$ can be straightforwardly obtained from the local site energy difference of Nd $d$-orbitals between spin-up and spin-down, when the $\frac{3}{2}$-spin $\mathbf{S}_i^f$ of the $f$-orbitals are all set to be along the spin-up direction.
Finally, the remaining intra-atomic hopping parameters $t_{ii mm^\prime}$ and site energy $\epsilon_m$ is then obtained by subtracting $V^\text{HF}$ from the intra-atomic part of $H^{\text{LDA}+U}$.

\begin{figure}[h]

    \vspace{-0.3cm}
	\begin{center}
	\includegraphics[width=0.9\columnwidth]{Reproduce.pdf}
	\end{center}
	\vspace{-0.6cm}
	\caption{
	Confirmation of the quality of $H^{^\mathbf{eff}}$ by comparing the electronic band structures with those from LDA+$U$ calculations under different magnetic configurations: (a)(b) spin majority of ferromagnetic order, (c)(d) spin minority of ferromagnetic order,  and (e)(f) antiferromagnetic order.
	The excellent agreement establishes a robust foundation for configurations with non-collinear spin correlations without long-range order.
	}
	\label{figs1}
	\vspace{-0.3cm}
\end{figure}

A simple criterion to check the quality of the resulting $H^{\text{eff}}$ (or the accuracy of the chosen $U_{\text{eff}}$ and $J_{\text{eff}}$) is the degree of the spin-independence of the intra-atomic $t_{ii mm^\prime}$.
A reasonable value of $U_{\text{eff}}$ and $J_{\text{eff}}$ should encode all the spin-dependence of the self-consistent $H^\text{HF}$.
Therefore, a non-negligible spin-dependence of the resulting $t_{ii mm^\prime}$ or site energy $\epsilon_m$ indicates clearly a need to improve the value of $U_{\text{eff}}$ and $J_{\text{eff}}$.
In practice, we find this criterion quite sufficient to pin down a rather narrow range of acceptable values of $U_{\text{eff}}$ and $J_{\text{eff}}$.
This then allows us to fix all the intra-atomic parameters in $H^{\text{eff}}$.

On the other hand, since the LDA+$U$ only included atomically local Hartree-Fock approximation, the inter-atomic hopping parameters $t_{ii^\prime mm^\prime}$ are unaffected in the approximation.
We therefore can simply take it from the spin-averaged $H^{\text{LDA}+U}$.
The next section lists some of the leading parameters.

The above procedure, if performed properly, should generate a SU(2)-symmetric many-body Hamiltonian $H^{\text{eff}}$ that can be further studied with any many-body solver one prefers, not just the Hartree-Fock approximation.
Furthermore, since the Hamiltonian does not require or guarantee a magnetic order, it can in principle be applied to metallic systems (or spin liquid insulators) that do not contain long-range order but show signs of the existence of local moments, for example Curie-Weiss susceptibility at high-temperature.

For the purpose of this work, in which we study the impact of non-collinear spin correlations, we verify that $H^{\text{eff}}$ is able to reproduce the $H^{\text{LDA}+U}$ solution under various magnetic orders.
Figure~\ref{figs1} demonstrates an excellent correspondence between our resulting band structure under the same Hartree-Fock treatment of local interactions and the LDA+$U$ band structure within the subspace of the selected orbitals.
Notice especially that the agreement occurs under both ferromagnetic and antiferromagnetic order with the \emph{same} set of parameters, despite the significantly different band structure under these two orders.
These results thus establish the high quality of these parameters and the validity of our effective Hamiltonian in describing various magnetic structures, including the paramagnetic parent compound.

\vspace{-0.2cm}
\subsection{3. Leading parameters for $H^{\text{eff}}$}
Table \ref{tabs1} lists some of the leading coefficients $\epsilon_m$ and $J_m$ of the resulting SU(2) symmetric many-body Hamiltonian.
The full interaction kernel $U_{m m^{\prime\prime}m^{\prime\prime\prime} m^\prime}$ is approximated by the Slater integral~\cite{Slater,Liechtenstein1995}, with $U_{\text{eff}} = 7.385$ eV and $J_{\text{eff}}=1.197$ eV obtained from the above procedure.
The leading terms of the resulting hopping parameters $t_{ii^\prime mm ^\prime}$ are $1.321$ eV between Ni $d_{x^2-y^2}$ and O $p_{x}$, $1.158$ eV between Nd $d_{xz}$/$d_{yz}$ and O $p_{z}$, and $1.024$ eV between Nd $d_{xy}$ and ``I'' orbitals.
The full $t_{ii^\prime mm ^\prime}$ parameters are available upon request.

\begin{table}[t]
    \caption{Leading parameters $\epsilon_m$ and $J_m$ in unit of eV.}
    \begin{ruledtabular}
        \begin{tabular}{llllll}
Nd & $d_{3z^2-r^2}$ & $d_{x^2-y^2}$ & $d_{yz}$ & $d_{xz}$ & $d_{xy}$ \\ \hline
$J_m$  & 0.272 & 0.379  & 0.268 & 0.268 & 0.217 \\
$\epsilon_m$ & 4.12 & 5.43  & 5.779 & 5.779 & 4.045 \\ \hline \hline
Ni & $d_{3z^2-r^2}$ & $d_{x^2-y^2}$ & $d_{yz}$ & $d_{xz}$ & $d_{xy}$ \\ \hline
$\epsilon_m$ & -55.448 & -55.101  & -55.616 & -55.616 & -55.284 \\\hline\hline
O & $p_{x}$ & $p_{y}$ & $p_{z}$ & ``I'' \\ \hline 
$\epsilon_m$ & -3.488 & -2.542  & -2.602 & 4.93 \\
\end{tabular}
    \end{ruledtabular}
    \label{tabs1}
    \vspace{-0.3cm}
\end{table}

\vspace{-0.2cm}
\subsection{4. Implementation of non-collinear magnetic configuration}

In this study, the fluctuation of non-collinear spin directions is incorporated in our simulation by averaging large number of configurations with similar nearest short-range correlations $\chi = \sum_{\langle i, i^\prime\rangle}\mathbf{S}_{i} \cdot \mathbf{S}_{i^\prime}$, 
but with negligible long-range order $\mathbf{M}_{\mathbf{q}} = \sum_{i}\mathbf{S}_{i} e^{i\mathbf{q}\cdot \mathbf{x}_i}$.
Obviously, a larger supercell is necessary in the search for configurations of various short-range correlation, but without long-range order.
Furthermore, averaging over large supercells with various sizes, shapes and orientations is an efficient way to avoid fictitious gap openings and shadow band foldings related to the artificial new spatial periodicity.

The enforcement of the spin directions according to the proposed magnetic configurations can be easily achieved within the Hartree-Fock approximation by constraining the atomically local one-body density matrix $\rho_{i\nu\nu^\prime}$ to be diagonal in the spin channel $\nu=\nu^\prime$ along the assigned spin direction.
Specifically, using the Euler angle we represent (rotate) $\rho_{i\nu\nu^\prime}$ in a local spin basis whose $z$-axis is along the assigned spin direction.
We then zero out the off-diagonal elements of $\rho_{i\nu\nu^\prime}$ in this local basis, and then rotate the representation back to the global one with $z$-axis of the spin along that of the lattice.
This constrained $\rho_{i\nu\nu^\prime}$ is then combined with  $U_{m m^{\prime\prime}m^{\prime\prime\prime} m^\prime}$ in Eq.(1) to evaluate the effective orbital-dependent potential within the Hartree-Fock approximation.
Naturally, the same constraint needs to be applied in each iteration of the self-consistent cycle, until a spin-density is converged.


\section{Supplementary}
\subsection{1.Computational details of density functional calculation}
For this prototypical case, we obtain the most relevant Hilbert space within $\pm10$ eV around the Fermi energy from the spin polarized LDA+$U$ ~\cite{Anisimov1993, Liechtenstein1995} electronic structure of the parent compound NdNiO${_2}$, using the linearized augmented plane wave~\cite{Singh} implementation~\cite{Blaha1990} of the density functional theory (DFT)~\cite{DFT1, DFT2}.
We take from Ref.~\cite{Li2019} the lattice structure with lattice constant $a=b=3.92\mathring{A},c=3.37\mathring{A}$, and the space group $P4/mmm$.
We set $U-J$ = $0.6 - 0.06$ Ry = 7.344 eV for Ni $d$-orbitals,$0.20 - 0.00$ Ry = 2.72 eV,  $0.60 - 0.00$ Ry = 8.16 eV for Nd $d$-,$f$-orbitals separately.

\vspace{-0.2cm}
\subsection{2.Extraction of many-body Hamiltonian}
We aim at obtaining a realistic SU(2)-symmetric many-body Hamiltonian $H^{\text{eff}}$ in Eq.(1).
To ensure a DFT-like level of accuracy, we further demand that when under a similar approximation, our many-body Hamiltonian needs to reproduce the self-consistent DFT Hamiltonian (equivalently the electronic band structure).
Since in LDA+$U$ the strong local electron-electron interaction is approximated in an effective Hartree-Fock manner~\cite{Anisimov1997}, a proper connection can be naturally made on the Wannier states basis by matching the self-consistent Hartree-Fock solution of our SU(2)-symmetric $H^{\text{eff}}$ with the self-consistent $H^{\text{LDA}+U}$~\cite{Wei2006}.
The requirement that the Hartree-Fock approximation of $H^{\text{eff}}$ needs to reproduce the LDA+$U$ band structure within the subspace of the active orbitals results in a rather unique set of parameters in $H^{\text{eff}}$~\cite{Wei2006}.

Specifically, since we demand that the self-consistent Hartree-Fock solution reproduce the self-consistent solution of the LDA+$U$ solution, the corresponding density matrix $\rho_{i\nu\nu^\prime}$ must be identical in both cases when represented in the same set of Wannier orbitals.
One can thus take $\rho_{i\nu\nu^\prime}$ directly from the self-consistent LDA+$U$ solution in the Wannier basis.
Furthermore, if one assumes that the structure of $U_{m m^{\prime\prime}m^{\prime\prime\prime} m^\prime}$ follows the Slater integral~\cite{Slater,Liechtenstein1995}, the entire $U_{m m^{\prime\prime}m^{\prime\prime\prime} m^\prime}$ can be fixed by just two parameters $U_{\text{eff}}$ and $J_{\text{eff}}$.
Combining $\rho_{i\nu\nu^\prime}$ and $U_{m m^{\prime\prime}m^{\prime\prime\prime} m^\prime}$, the effective Hartree-Fock potential $V^\text{HF}$ can then obtained.

In addition, orbital dependent $J_m$ in $H^{\text{eff}}$ can be straightforwardly obtained from the local site energy difference of Nd $d$-orbitals between spin-up and spin-down, when the $\frac{3}{2}$-spin $\mathbf{S}_i^f$ of the $f$-orbitals are all set to be along the spin-up direction.
Finally, the remaining intra-atomic hopping parameters $t_{ii mm^\prime}$ and site energy $\epsilon_m$ is then obtained by subtracting $V^\text{HF}$ from the intra-atomic part of $H^{\text{LDA}+U}$.

\begin{figure}[h]

    \vspace{-0.3cm}
	\begin{center}
	\includegraphics[width=0.9\columnwidth]{Reproduce.pdf}
	\end{center}
	\vspace{-0.6cm}
	\caption{
	Confirmation of the quality of $H^{^\mathbf{eff}}$ by comparing the electronic band structures with those from LDA+$U$ calculations under different magnetic configurations: (a)(b) spin majority of ferromagnetic order, (c)(d) spin minority of ferromagnetic order,  and (e)(f) antiferromagnetic order.
	The excellent agreement establishes a robust foundation for configurations with non-collinear spin correlations without long-range order.
	}
	\label{figs1}
	\vspace{-0.3cm}
\end{figure}

A simple criterion to check the quality of the resulting $H^{\text{eff}}$ (or the accuracy of the chosen $U_{\text{eff}}$ and $J_{\text{eff}}$) is the degree of the spin-independence of the intra-atomic $t_{ii mm^\prime}$.
A reasonable value of $U_{\text{eff}}$ and $J_{\text{eff}}$ should encode all the spin-dependence of the self-consistent $H^\text{HF}$.
Therefore, a non-negligible spin-dependence of the resulting $t_{ii mm^\prime}$ or site energy $\epsilon_m$ indicates clearly a need to improve the value of $U_{\text{eff}}$ and $J_{\text{eff}}$.
In practice, we find this criterion quite sufficient to pin down a rather narrow range of acceptable values of $U_{\text{eff}}$ and $J_{\text{eff}}$.
This then allows us to fix all the intra-atomic parameters in $H^{\text{eff}}$.

On the other hand, since the LDA+$U$ only included atomically local Hartree-Fock approximation, the inter-atomic hopping parameters $t_{ii^\prime mm^\prime}$ are unaffected in the approximation.
We therefore can simply take it from the spin-averaged $H^{\text{LDA}+U}$.
The next section lists some of the leading parameters.

The above procedure, if performed properly, should generate a SU(2)-symmetric many-body Hamiltonian $H^{\text{eff}}$ that can be further studied with any many-body solver one prefers, not just the Hartree-Fock approximation.
Furthermore, since the Hamiltonian does not require or guarantee a magnetic order, it can in principle be applied to metallic systems (or spin liquid insulators) that do not contain long-range order but show signs of the existence of local moments, for example Curie-Weiss susceptibility at high-temperature.

For the purpose of this work, in which we study the impact of non-collinear spin correlations, we verify that $H^{\text{eff}}$ is able to reproduce the $H^{\text{LDA}+U}$ solution under various magnetic orders.
Figure~\ref{figs1} demonstrates an excellent correspondence between our resulting band structure under the same Hartree-Fock treatment of local interactions and the LDA+$U$ band structure within the subspace of the selected orbitals.
Notice especially that the agreement occurs under both ferromagnetic and antiferromagnetic order with the \emph{same} set of parameters, despite the significantly different band structure under these two orders.
These results thus establish the high quality of these parameters and the validity of our effective Hamiltonian in describing various magnetic structures, including the paramagnetic parent compound.

\vspace{-0.2cm}
\subsection{3. Leading parameters for $H^{\text{eff}}$}
Table \ref{tabs1} lists some of the leading coefficients $\epsilon_m$ and $J_m$ of the resulting SU(2) symmetric many-body Hamiltonian.
The full interaction kernel $U_{m m^{\prime\prime}m^{\prime\prime\prime} m^\prime}$ is approximated by the Slater integral~\cite{Slater,Liechtenstein1995}, with $U_{\text{eff}} = 7.385$ eV and $J_{\text{eff}}=1.197$ eV obtained from the above procedure.
The leading terms of the resulting hopping parameters $t_{ii^\prime mm ^\prime}$ are $1.321$ eV between Ni $d_{x^2-y^2}$ and O $p_{x}$, $1.158$ eV between Nd $d_{xz}$/$d_{yz}$ and O $p_{z}$, and $1.024$ eV between Nd $d_{xy}$ and ``I'' orbitals.
The full $t_{ii^\prime mm ^\prime}$ parameters are available upon request.

\begin{table}[t]
    \caption{Leading parameters $\epsilon_m$ and $J_m$ in unit of eV.}
    \begin{ruledtabular}
        \begin{tabular}{llllll}
Nd & $d_{3z^2-r^2}$ & $d_{x^2-y^2}$ & $d_{yz}$ & $d_{xz}$ & $d_{xy}$ \\ \hline
$J_m$  & 0.272 & 0.379  & 0.268 & 0.268 & 0.217 \\
$\epsilon_m$ & 4.12 & 5.43  & 5.779 & 5.779 & 4.045 \\ \hline \hline
Ni & $d_{3z^2-r^2}$ & $d_{x^2-y^2}$ & $d_{yz}$ & $d_{xz}$ & $d_{xy}$ \\ \hline
$\epsilon_m$ & -55.448 & -55.101  & -55.616 & -55.616 & -55.284 \\\hline\hline
O & $p_{x}$ & $p_{y}$ & $p_{z}$ & ``I'' \\ \hline 
$\epsilon_m$ & -3.488 & -2.542  & -2.602 & 4.93 \\
\end{tabular}
    \end{ruledtabular}
    \label{tabs1}
    \vspace{-0.3cm}
\end{table}

\vspace{-0.2cm}
\subsection{4. Implementation of non-collinear magnetic configuration}

In this study, the fluctuation of non-collinear spin directions is incorporated in our simulation by averaging large number of configurations with similar nearest short-range correlations $\chi = \sum_{\langle i, i^\prime\rangle}\mathbf{S}_{i} \cdot \mathbf{S}_{i^\prime}$, 
but with negligible long-range order $\mathbf{M}_{\mathbf{q}} = \sum_{i}\mathbf{S}_{i} e^{i\mathbf{q}\cdot \mathbf{x}_i}$.
Obviously, a larger supercell is necessary in the search for configurations of various short-range correlation, but without long-range order.
Furthermore, averaging over large supercells with various sizes, shapes and orientations is an efficient way to avoid fictitious gap openings and shadow band foldings related to the artificial new spatial periodicity.

The enforcement of the spin directions according to the proposed magnetic configurations can be easily achieved within the Hartree-Fock approximation by constraining the atomically local one-body density matrix $\rho_{i\nu\nu^\prime}$ to be diagonal in the spin channel $\nu=\nu^\prime$ along the assigned spin direction.
Specifically, using the Euler angle we represent (rotate) $\rho_{i\nu\nu^\prime}$ in a local spin basis whose $z$-axis is along the assigned spin direction.
We then zero out the off-diagonal elements of $\rho_{i\nu\nu^\prime}$ in this local basis, and then rotate the representation back to the global one with $z$-axis of the spin along that of the lattice.
This constrained $\rho_{i\nu\nu^\prime}$ is then combined with  $U_{m m^{\prime\prime}m^{\prime\prime\prime} m^\prime}$ in Eq.(1) to evaluate the effective orbital-dependent potential within the Hartree-Fock approximation.
Naturally, the same constraint needs to be applied in each iteration of the self-consistent cycle, until a spin-density is converged.

