and where  $S^{\sf GS}_{\rm{O}}(\q=0, \omega)$ represents the ground state and zero-point energy contribution to the structure factor, as explained below.

In order to compute eq:SO_qn0, we first note that, in our case, the excited states, for all values of $\k$
\begin{align}
\ket{\alpha_{\k}}=\balphad{\k}\ket{0_{\alpha}}\; ,\\
\ket{\beta_{\k}}=\bbetad{\k}\ket{0_{\beta}}\; ,
\end{align}
form a complete basis, where $|0_α⟩$ is the Bogoliubov ground state for the $$ bosons, i.e., $|0_α⟩=0$, and similarly for the $$ bosons. Since the Hilbert space consists of the direct product $|α⟩⊕|β⟩$,we can replace
\be
\sum_{\mu}\ket{\mu}\bra{\mu}\rightarrow\sum_{\k}\balphad{\k}\ket{0_{\alpha}}\bra{0_{\alpha}}\balpha{\k} +\sum_{\k}\bbetad{\k}\ket{0_{\beta}}\bra{0_{\beta}}\bbeta{\k}=1\; .\label{eq:ab.close}
\ee
in  \Autoref{eq:SO_qn0}.
However, by replacing \Autoref{eq:close} by \Autoref{eq:ab.close} in \Autoref{eq:SO_qn0}, we account for the 1$$ excited states and we therefore disregard the ground state and zero-point energy contribution to the structure factor, which is expressed by the term $S^GS_O(=0, ω)$ in \Autoref{eq:SO_qn0}. The ground state and zero-point energy only contribute at $=0$ and $ω=0$. We present how to calculate it below in \hyperref[sec:structure_factors_quantum_q0]{Appendix~}\ref{sec:structure_factors_quantum_q0}.

\subsection{Dipole moments: quantum structure factor at general values of $\q$}\label{sec:structure_factors_quantum_Sq}

We consider first the dynamical spin structure factor
\be
S^{\sf QM}_{\rm{S}} (\q, \omega)
= \int_{-\infty}^{\infty} \frac{\dd t}{2\pi} \exp{i\omega t}
\sum_\mu \Av{ \ops{\q}{\mu}(t) \ops{-\q}{\mu}(0) } \; .
\label{eq:SqSF_app}
\ee
Substituting \Autoref{eq:Acond2} in the expression for spin
operators, \Autoref{eq:dipole.in.terms.of.A}, and keeping terms to linear order, we find
\begin{subequations}
\begin{align}
\ops{i}{x}&\simeq i(\bbd{i}-\bb{i})\; ,\\
\ops{i}{y}&\simeq 0\; ,\\
\ops{i}{z}&\simeq-i(\bad{i}-\ba{i})\; .
\end{align}
\end{subequations}
Performing a Fourier transform and using the Bogoliubov transformation \Autoref{eq:v_e},
we can express these as
\begin{subequations}
\begin{align}
\ops{\q}{x}&
%=i\frac{\Delta_{\q}+B_{\q}}{\sqrt{\Delta_{\q}^2- B_{\q}^2}}(\bbetad{-\q}-\bbeta{\q})
\simeq i \xi_{\sf{S}}(\q) (\bbetad{-\q}-\bbeta{\q}) \label{eq:Sqm_alpha_x_app} \; , \\
\ops{\q}{y}&\simeq 0\; ,\\
\ops{\q}{z}&
%=i\frac{\Delta_{\q}+B_{\q}}{\sqrt{\Delta_{\q}^2- B_{\q}^2}}(\balphad{-\q}-\balpha{\q})
\simeq- i \xi_{\sf{S}}(\q) (\balphad{-\q}-\balpha{\q})\label{eq:Sqm_alpha_z_app} \; ,
\end{align}
\end{subequations}
where $ξ()$ is the coherence factor
\be
\xi_{\sf{S}}(\q) = \frac{\Delta_{\q}+B_{\q}}{\sqrt{\Delta_{\q}^2- B_{\q}^2}}\; .
\label{eq:coherence.factor_app}
\ee
Using \Autoref{eq:SO_qn0}, we can then calculate the structure factor for dipole moments as
\begin{align}
S^{\sf QM}_{\rm{S}} (\q, \omega)
&= \sum_{\mu,{\k}}  \norm{ \bra{n_{\k}} \ops{\q}{\mu} \ket{0} }^2
\delta(\omega-\omega_{n_{\k}})+S^{\sf GS}_{\rm{S}}(\q=0, \omega)\; ,\label{eq:SqSFtot}
\end{align}
where $|0⟩$ is the FQ ground state [\Autoref{eq:FQyGS}],
\be
\ket{n_{\k}} = \balphad{\k}\ket{0} \otimes \bbetad{\k}\ket{0} \; .\label{eq:nk}
\ee
represents the first excited states where $ ⊗$ implies a direct product, as the bosons $$ and $$ are independent. By using \Autoref{eq:nk}, we account for the 1$$ excited states and we therefore disregard the ground state and zero-point energy contribution to the structure factor which is expressed by the term $S^GS_O(=0, ω)$ in \Autoref{eq:SqSFtot}.

Finally, we find
\be
S^{\sf QM}_{\rm{S}} (\q, \omega)
= 2\frac{\sqrt{A_{\q}+B_{\q}}}{\sqrt{A_{\q}-B_{\q}}} \delta(\omega-\omega_{\q}) +S^{\sf GS}_{\rm{S}}(\q=0, \omega) \; ,
\label{eq:Sqw_qm_qn0_app}
\ee
where we used \Autoref{eq:Delta}.
Detailed calculations for $=0$ contributions to the dipole moment structure factor can be found in \hyperref[sec:structure_factors_quantum_Sq0]{Appendix~}\ref{sec:structure_factors_quantum_Sq0}.
More precisely, $S^GS_S(=0, ω)$ is given by  \Autoref{eq:SSq0_QM}, which combined with \Autoref{eq:Sqw_qm_qn0_app}  gives the total quantum structure factor for the dipole moments expressed in \Autoref{eq:Sqw_qm}.

\subsection{Quadrupole moments: quantum structure factor at general values of $\q$}\label{sec:structure_factors_quantum_Qq}
We now consider the dynamical structure factor associated with
quadrupole moments
\be
S^{\sf QM}_{\rm{Q}} (\q, \omega)
= \int_{-\infty}^{\infty} \frac{\dd t}{2\pi} \exp{i\omega t}
\sum_{\mu\nu}  \Av{ \opq{\q}{\mu\nu}(t) \opq{-\q}{\mu\nu}(0) } \; .
\label{eq:dynamical.quadrupole.structure.factor_app}
\ee
Following the same steps as for the spin--structure factor,
we use \Autoref{eq:Acond2} to express the quadupole components up to linear order
in \Autoref{eq:quadrupole.in.terms.of.A}. We find
\begin{align}
\ophb{Q}_i & \cong
\begin{pmatrix}
\frac{2}{3} & -\bad{i}-\ba{i} & 0\\
-\bad{i}-\ba{i} & -\frac{4}{3} & -\bbd{i}-\bb{i} \\
0 &  -\bbd{i}-\bb{i}  & \frac{2}{3}
\end{pmatrix} \; ,
\label{eq:Q_bosons_app}
\end{align}
An equivalent calculation of matrix elements in the Bogoliubov basis [\Autoref{eq:v_e}]
\begin{align}
\ophb{Q}_{\q}&\cong\nonumber\\
&	\small{
\begin{pmatrix}
\frac{2}{3}\sqrt{N}\delta(\q)  &	\xi_{\sf{Q}}(\q)(\balphad{-\q}+\balpha{\q}) & 0\\
\xi_{\sf{Q}}(\q)(\balphad{-\q}+\balpha{\q})& 	-\frac{4}{3}\sqrt{N}\delta(\q)& 		\xi_{\sf{Q}}(\q)(\bbetad{-\q}+\bbeta{\q})\\
0 &  	\xi_{\sf{Q}}(\q)(\bbetad{-\q}+\bbeta{\q}) & 	\frac{2}{3}\sqrt{N}\delta(\q)
\end{pmatrix}}\; ,
\end{align}
where $N$ is the number of sites and where $ξ_Q()$ is the coherence factor for quadrupoles defined as
\be
\xi_{\sf{Q}}(\q)= \frac{B_{\q}-\Delta_{\q}}{\sqrt{\Delta_{\q}^2- B_{\q}^2}}\; .\label{eq:coherence.factor.quadrupole_app}
\ee

Using \Autoref{eq:SO_qn0}, we can then calculate the structure factor for quadrupole moments as defined in \Autoref{eq:dynamical.quadrupole.structure.factor_app}. We obtain
\begin{align}
S^{\sf QM}_{\rm{Q}}(\q, \omega)
=& 4 \frac{\sqrt{A_{\q}-B_{\q}}}{\sqrt{A_{\q}+B_{\q}}} \delta(\omega-\omega_{\q}) +S^{\sf GS}_{\rm{Q}}(\q=0, \omega) \; ,
\label{eq:Qqw_qm_qn0}
\end{align}
where we used \Autoref{eq:Delta}.
Detailed calculations for $=0$ contributions to the quadrupole moment structure factor can be found in \hyperref[sec:structure_factors_quantum_Qq0]{Appendix~}\ref{sec:structure_factors_quantum_Qq0}.
More precisely, $S^GS_Q(=0, ω)$ is given by  \Autoref{eq:SQq0_QM}, which combined with \Autoref{eq:Qqw_qm_qn0}  gives the total quantum structure factor for the quadrupole moments expressed in \Autoref{eq:Qqw_qm}.
\subsection{A-matrices : quantum structure factors at general values of $\q$}\label{sec:structure_factors_quantum_Aq}

The most fundamental objects in our theory are not dipoles or quadrupoles,
but the A--matrices which describe the quantum state of the \mbox{spin--1} moment.
It is therefore useful to introduce a dynamical structure factor
\be
S^{\sf QM}_{\rm{A}} (\q, \omega)
= \int_{-\infty}^{\infty} \frac{\dd t}{2\pi} \exp{i\omega t}
\sum_{\mu\nu} \nn{\opa{}{\mu}{\nu}(t) \opa{}{\nu}{\mu}(0)}  \; .
\label{eq:dynamical.A.matrix.structure.factor_app}
\ee
Neglecting 2\nd order and higher terms, \Autoref{eq:Acond2} becomes
\be
\ophb{A}_i\simeq
\begin{pmatrix}
0& \bad{i} & 0\\
\ba{i} & 1 & \bb{i}\\
0 & \bbd{i} & 0
\end{pmatrix}\; .
\ee
Once again we can use the Bogoliubov basis [\Autoref{eq:v_e}] to find
\be
\ophb{A}_{\q}\simeq
\begin{pmatrix}
\begin{array}{c c c c c c}
\multicolumn{2}{c}{\multirow{2}{*}[-1.5ex]{0}}& \multirow{2}{*}{{\small$\xi_{\sf{A}}^{-}(\q)\balphad{-\q}$}}& \multicolumn{2}{c}{\multirow{2}{*}[-1.5ex]{0}}\\
& & \multirow{2}{*}{{\small$-\xi_{\sf{A}}^{+}(\q)\balpha{\q} $}}& & \\
\multirow{2}{*}{{\small$-\xi_{\sf{A}}^{+}(\q)\balphad{-\q}$ }}& \multicolumn{2}{c}{\multirow{2}{*}[-1.5ex]{$\sqrt{N}\delta_{\q, 0}$ }}&	\multirow{2}{*}{{\small$-\xi_{\sf{A}}^{+}(\q)\bbetad{-\q}$}}\\
\multirow{2}{*}{{\small$+\xi_{\sf{A}}^{-}(\q)\balpha{\q}$}} & & &\multirow{2}{*}{{\small$+\xi_{\sf{A}}^{-}(\q)\bbeta{\q} $}}\\
\multicolumn{2}{c}{\multirow{2}{*}[-1.5ex]{0}}& \multirow{2}{*}{{\small$\xi_{\sf{A}}^{-}(\q)\bbetad{-\q}$}}& \multicolumn{2}{c}{\multirow{2}{*}[-1.5ex]{0}}\\
& & \multirow{2}{*}{{\small$-\xi_{\sf{A}}^{+}(\q)\bbeta{\q}$}}& &
\end{array}
\end{pmatrix}
\ee
where $N$ is the number of sites and $ξ_A^+()$ and $ξ_A^-()$ are the coherence factors for A--matrices defined as
\begin{subequations}
\begin{align}
\xi_{\sf{A}}^{+}(\q)= \frac{\xi_{\sf{S}}(\q)+\xi_{\sf{Q}}(\q)}{2}\; ,\\
\xi_{\sf{A}}^{-}(\q)=\frac{\xi_{\sf{S}}(\q)-\xi_{\sf{Q}}(\q)}{2}\; ,
\end{align}\label{eq:coherence.factor.amatrix_app}
\end{subequations}
where $ξ_S()$ and $ξ_Q()$ are defined in \Autoref{eq:coherence.factor_app} and \Autoref{eq:coherence.factor.quadrupole_app} respectively.

Using \Autoref{eq:SO_qn0}, we can then calculate the structure factor for quadrupole moments as defined in \Autoref{eq:dynamical.A.matrix.structure.factor_app}. We obtain
\begin{align}
S_{\rm{A}}^{\sf QM}(\q, \omega)
=& 2\frac{A_{\q}}{\sqrt{A_{\q}^2-B_{\q}^2}}
\delta(\omega-\omega_{\q}) +S^{\sf GS}_{\rm{A}}(\q=0, \omega)\; ,
\label{eq:Aqw_qm_qn0}
\end{align}
where we used \Autoref{eq:Delta}.
Detailed calculations for $=0$ contributions to the A--matrix structure factor can be found in \hyperref[sec:structure_factors_quantum_Aq0]{Appendix~}\ref{sec:structure_factors_quantum_Aq0}.
More precisely, $S^GS_A(=0, ω)$ is given by  \Autoref{eq:SAq0_QM}, which combined with \Autoref{eq:Aqw_qm_qn0}  gives the total quantum structure factor for the A-matrices expressed in \Autoref{eq:Aqw_qm}.
\subsection{ Quantum structure factors: contribution of the ground state at $\q=0$}\label{sec:structure_factors_quantum_q0}
We here present how to describe the zero-temperature correction of the ground state and the zero-point energy fluctuations' contribution to the quantum structure factors, which is expected to happen at $=0$ and $ω=0$.

We therefore consider the zero--temperature quantum structure factor  at $=0$ to be given by
\be
S^{\sf GS}_{\rm{O}}(\q=0)=\sum_{\alpha\beta}\nn{\hat{O}^{\alpha}_{\q=0, \beta}\hat{O}^{\beta}_{\q=0, \alpha}}_{T=0}\; .\label{eq:SOq0_QM}
\ee
Calculating the contribution of the ground state and the zero-point energy fluctuations to the zero--temperature quantum structure factor can be achieved by adding a source term to the BBQ Hamiltonian that includes a fictive field $h$ coupled to the spin moments, similarly to what we did for the classical case [see \Autoref{sec:classical.structure.factors} and \hyperref[sec:structure_factors_classical]{Appendix~}\ref{sec:structure_factors_classical}]. The structure factors can then be calculated by taking the appropriate derivative of the free energy with respect to the fictive field $h$.

We consider the total Hamiltonian to be given by \Autoref{eq:Hfield}, and the source term to be of the form given in \Autoref{eq:DeltaH}.
We can then rewrite the operators $Ô^α_β$ of \Autoref{eq:DeltaH} in function of the fluctuations orthogonal to the FQ ground state [\Autoref{eq:GS_FQ}]. Refer to \Autoref{section:small.fluctuations} for details on the creation of orthogonal fluctuations.
Expanding the source term Hamiltonian [\Autoref{eq:DeltaH}] up to second order in bosons, Fourier transforming it and considering its contribution for $=0$, we can assume that it takes the following form
\begin{align*}
\Delta {\mathcal H} [{\bf{h}}_{\q}]=&C[{\bf{h}}_{\q=0} ] +\frac{1}{2}\sum_{\k}\left[\opdx{w}{\k}{m}_{\k}[{\bf{h}}_{\q=0}]\opx{w}{\k}\right.\\
&+\left.\left[{\V{N}}[{\bf{h}}_{\k}]^T\opx{w}{\k}+\opdx{w}{\k}{\V{N}}[{\bf{h}}_{\k}]\right]\delta_{\k,0}\right]\; ,\numberthis\label{eq:DeltaH_qm_q0}
\end{align*}
where $C[h_=0 ]$ is the coefficient for the 0$þ$ order term of the source term expended in terms of the fluctuations orthogonal to the FQ ground state,  $w$ represents these fluctuations orthogonal to the FQ ground state and is given \Autoref{eq:def_wboson},  $m_[h_=0]$ represents the interaction matrix for 2$$ order terms in fluctuations and depends on $h_=0$, and where $N[h_]^†$ and $ N[h_]$ are the coefficients for the linear terms in fluctuations. By definition of the source term Hamiltonian [\Autoref{eq:DeltaH}], all the coefficients $C[h_=0 ]$, $N_1[h_]^T$, $ N_2[h_]$, and $m_[h_=0]$  depend linearly on the fictive field $h$ and will be different whether we are considering dipole, quadrupole, or A-matrix moments for the source term [\Autoref{eq:DeltaH}].

Using \Autoref{eq:HBBQ_qm} for the BBQ Hamiltonian, we can assume that the total Hamiltonian [\Autoref{eq:Hfield}]  in terms of the bosons then take the following form
\begin{align*}
\Ham_{}=&E_0+C[{\bf{h}}_{\q=0} ] +\frac{1}{2}\sum_{\k}\left[\opdx{w}{\k}{M}_{\k}[{\bf{h}}_{\q=0}]\opx{w}{\k}\right.\\
&+\left.\left[{\V{N}}[{\bf{h}}_{\k}]^{\dagger}\opx{w}{\k}+\opdx{w}{\k}{\V{N}}[{\bf{h}}_{\k}]\right]\delta_{\k,0}\right]\; ,\numberthis\label{eq:H_qm_q0}
\end{align*}
where $E_0$ is the mean-field ground--state given in \Autoref{eq:E0}, and where $M_[h_=0]$ is the interaction matrix for the total Hamiltonian. It includes contributions from the BBQ Hamiltonian and the source term $m_[h_=0]$, and, therefore, depends on $h_=0$.

Following the method described in \hyperref[sec:bogolioubov_transfomation]{Appendix~}\ref{sec:bogolioubov_transfomation} , we perform a Bogoliubov transformation in order to diagonalize the total Hamiltonian. We assume that the new Bogoliubov bosons $v$
\be
\opx{v}{\k}=\begin{pmatrix}
\balpha{\k} \\
\balphad{-\k} \\
\bbeta{\k} \\
\bbetad{-\k} \\
\end{pmatrix} \; ,
\label{eq:def_vboson}
\ee
are given in terms of the bosons orthogonal to the FQ ground state  $w$ by \Autoref{eq:U}.
We can then assume that the total Hamiltonian in terms of  the Bogoliubov bosons $v$ becomes
\begin{align*}
\Ham_{}=&E_0+\Delta E_0[{\bf{h}}_{\q=0} ]+C[{\bf{h}}_{\q=0} ] \\
&+\left.\left[\tilde{\V{N}}[{\bf{h}}_{\k}]^{\dagger}\opx{v}{\k}+\opdx{v}{\k}\tilde{\V{N}}[{\bf{h}}_{\k}]\right]\delta_{\k,0}\right]\; ,\numberthis\label{eq:H_qm_bogo_q0}
\end{align*}
where $ϵ_,̨α[h_=0]$ and $ϵ_,̨β[h_=0]$ are the two physical eigenvalues obtained by diagonalizing $M_[h_=0]$ [see \Autoref{eq:Uev}], where $ΔE_0[h_=0 ]$ is the ground state contribution of the Bogoluibov bosons and where
\be
\begin{matrix}
\tilde{\V{N}}[{\bf{h}}_{\k}]^{\dagger}={\V{N}}[{\bf{h}}_{\k}]^{\dagger}U^{-1}\; , \\
\tilde{\V{N}}[{\bf{h}}_{\k}]={U^{\dagger}}^{-1}{\V{N}}[{\bf{h}}_{\k}]\; .\label{eq:Ntilde_q0}
\end{matrix}
\ee
Here, $U$ is the Bogoliubov matrix change defined by  \Autoref{eq:U} and remains to be determined.
We also note that $U^-1$ (and $U^†^-1$) should be calculate from \Autoref{eq:Uinv}.

The canonical partition function is defined by
\be
Z=Tr(\exp{-\beta \hat{\Ham}})\; ,
\ee
where $β$ is defined by \Autoref{eq:def_beta}, and $$ is the operator Hamiltonian.
However in order to compute the partition function and the free energy, we want to get rid of the linear terms $Ñ[h_]^†$ and $Ñ[h_]$, which only contribute for $=̨0$.
The partition function is then the one of a set of  independent harmonic oscillators for the $≠̨0$ terms, but still contains linear terms with respect to the bosons for $=̨0$:
\begin{align*}
Z=&Tr\left(\exp{-\beta E_0-\beta\Delta E_0[{\bf{h}}_{\q=0} ]-\beta C[{\bf{h}}_{\q=0} ] }\right. \\
&\times\prod_{\k\neq 0}(\exp{-\beta \epsilon_{\k,\alpha}[{\bf{h}}_{\q=0}]\balphad{\k}\balpha{\k}})(\exp{\beta\epsilon_{\k,\beta}[{\bf{h}}_{\q=0}]\bbetad{\k}\bbeta{\k}})\\
&\times(\exp{-\frac{1}{2}\beta\epsilon_{\k=0,\alpha}[{\bf{h}}_{\q=0}]\balphad{\k=0}\balpha{\k=0}-\frac{1}{2}\beta n_1[{\bf{h}}_{\k=0}]\balphad{\k=0}-\frac{1}{2}\beta n_2[{\bf{h}}_{\k=0}]\balpha{\k=0}})\\
&\left.\times(\exp{-\frac{1}{2}\beta\epsilon_{\k=0,\beta}[{\bf{h}}_{\q=0}]\bbetad{\k=0}\bbeta{\k=0}-\frac{1}{2}\beta n_3[{\bf{h}}_{\k=0}]\bbetad{\k=0}-\frac{1}{2}\beta n_4[{\bf{h}}_{\k=0}]\bbeta{\k=0}})\right)\; .\numberthis
\end{align*}}
\be
\begin{matrix}
n_1[{\bf{h}}_{\k=0}]=\tilde{N}[{\bf{h}}_{-\k=0}]^{\dagger,2}+	\tilde{N}[{\bf{h}}_{\k=0}]^1 \; , \\
n_2[{\bf{h}}_{\k=0}]= \tilde{N}[{\bf{h}}_{\k=0}]^{\dagger,1}+	\tilde{N}[{\bf{h}}_{-\k=0}]^2\; ,\\
n_3[{\bf{h}}_{\k=0}]=\tilde{N}[{\bf{h}}_{-\k=0}]^{\dagger,4}+	\tilde{N}[{\bf{h}}_{\k=0}]^3 \; , \\
n_4[{\bf{h}}_{\k=0}]= \tilde{N}[{\bf{h}}_{\k=0}]^{\dagger,3}+	\tilde{N}[{\bf{h}}_{-\k=0}]^4\; ,
\end{matrix}
\ee
with $Ñ[h_]^†,1$ denoting the first component of $Ñ[h_]^†$.
To do this, we note that we can perform a change a variables by completing the square. For the $=̨0$ term, for the the $, $ bosons for instance, we have
\begin{align*}
%&\textrm{for~} \k=0\; :\\
-\frac{1}{2}&\beta\epsilon_{0,\alpha}[{\bf{h}}_{\q=0}]\balphad{0}\balpha{0}-\frac{1}{2}\beta n_1[{\bf{h}}_{\k=0}]\balphad{0}-\frac{1}{2}\beta n_2[{\bf{h}}_{\k=0}]\balpha{0}=\\
&-\frac{1}{2}\beta\epsilon_{0,\alpha}[{\bf{h}}_{\q=0}](\balphad{0}+\frac{n_1[{\bf{h}}_{\k=0}]}{\epsilon_{0,\alpha}[{\bf{h}}_{\q=0}]}) (\balpha{0}+\frac{n_2[{\bf{h}}_{\k=0}]}{\epsilon_{0,\alpha}[{\bf{h}}_{\q=0}]})\\
& +\beta\frac{n_1[{\bf{h}}_{\k=0}]n_2[{\bf{h}}_{\k=0}]}{\epsilon_{0,\alpha}[{\bf{h}}_{\q=0}]}  \; ,    \numberthis
\end{align*}
We note that we have
\be
n_1[{\bf{h}}_{\k=0}]=n_2[{\bf{h}}_{\k=0}]^{\dagger}\; ,
\ee
so that we can define the change of variable
\be
\begin{matrix}
\brhod{\k=0}=\balphad{\k=0}+\frac{n_1[{\bf{h}}_{\k=0}]}{\epsilon_{\k=0,\alpha}[{\bf{h}}_{\q=0}]}	 \; , \\
\brho{\k=0}=\balpha{\k=0}+\frac{n_2[{\bf{h}}_{\k=0}]}{\epsilon_{\k=0,\alpha}[{\bf{h}}_{\q=0}]}\; ,
\end{matrix}
\ee
which ensures that $=̨0$ and $=̨0$ have bosonic commutation relations and are associated with the eigenmode $ϵ_=̨0,α[h_=0]$.
We follow the same argument for the  $=̨0, =̨0$ bosons, and get new bosons $=̨0$ and $=̨0$:
\be
\begin{matrix}
\bsigmad{\k=0}=\bbetad{\k=0}+\frac{n_3[{\bf{h}}_{\k=0}]}{\epsilon_{\k=0,\beta}[{\bf{h}}_{\q=0}]}	 \; , \\
\bsigma{\k=0}=\bbeta{\k=0}+\frac{n_4[{\bf{h}}_{\k=0}]}{\epsilon_{\k=0,\beta}[{\bf{h}}_{\q=0}]}\; ,
\end{matrix}
\ee
associated with the eigenmode $ϵ_=̨0,β[h_=0]$
The partition function is then the one of a set of  independent harmonic oscillators. We obtain
\begin{align*}
Z=&Tr\left(\exp{-\beta E_0-\beta\Delta E_0[{\bf{h}}_{\q=0} ]-\beta C[{\bf{h}}_{\q=0} ] }\right. \\
&\times\prod_{\k\neq 0}(\exp{-\beta \epsilon_{\k,\alpha}[{\bf{h}}_{\q=0}]\balphad{\k}\balpha{\k}})(\exp{-\beta\epsilon_{\k,\beta}[{\bf{h}}_{\q=0}]\bbetad{\k}\bbeta{\k}})\\
&\times(\exp{-\beta \epsilon_{\k=0,\alpha}[{\bf{h}}_{\q=0}]\brhod{\k}\brho{\k}})(\exp{-\beta\epsilon_{\k=0,\beta}[{\bf{h}}_{\q=0}]\bsigmad{\k}\bsigma{\k}})\\
&\left.\times\exp{\beta\frac{n_1[{\bf{h}}_{\k=0}]n_2[{\bf{h}}_{\k=0}]}{\epsilon_{\k=0,\alpha}[{\bf{h}}_{\q=0}]}  } \exp{\beta\frac{n_3[{\bf{h}}_{\k=0}]n_4[{\bf{h}}_{\k=0}]}{\epsilon_{\k=0,\alpha}[{\bf{h}}_{\q=0}]}}\right)\; .\numberthis
\end{align*}
We then perform the trace on the Fock space, and use the fact that the trace is independent of the choice of the basis. This means we can compute it separately for the $$ bosons on their respective Fock basis $|n^α_⟩$ and for the $=̨0$ bosons on its respective Fock basis $|n^ρ_=̨0⟩$, and similarly for $$ and $=̨0$.
Taking the trace over the Fock space as explained above, we obtain
\begin{align*}
Z=&\exp{-\beta E_0-\beta\Delta E_0[{\bf{h}}_{\q=0} ]-\beta C[{\bf{h}}_{\q=0} ] } \\
&\times\prod_{\k}\frac{1}{1-\exp{-\beta\epsilon_{\k,\alpha}[{\bf{h}}_{\q=0}]}} \frac{1}{1-\exp{-\beta\epsilon_{\k,\alpha}[{\bf{h}}_{\q=0}]}}\\
&\times\exp{\beta\frac{n_1[{\bf{h}}_{\k=0}]n_2[{\bf{h}}_{\k=0}]}{\epsilon_{\k,\alpha}[{\bf{h}}_{\q=0}]}  } \exp{\beta\frac{n_3[{\bf{h}}_{\k=0}]n_4[{\bf{h}}_{\k=0}]}{\epsilon_{\k,\alpha}[{\bf{h}}_{\q=0}]}  }\; .\numberthis
\end{align*}
We note here that instead of doing the change of variable for $=̨0$, we could also
The free energy is given by
\begin{align*}
F &=-\frac{\log(Z)}{\beta}\\
&=E_0+\Delta E_0[{\bf{h}}_{\q=0} ]+C[{\bf{h}}_{\q=0} ]\\
&+\frac{1}{ \beta}\sum_{\k}\log(1-\exp{-\beta\epsilon_{\k,\alpha}[{\bf{h}}_{\q=0}]})\\
&+\frac{1}{ \beta}\sum_{\k}\log(1-\exp{-\beta\epsilon_{\k,\beta}[{\bf{h}}_{\q=0}]})\\
& - 2\frac{n_1[{\bf{h}}_{\k=0}]n_2[{\bf{h}}_{\k=0}]}{\epsilon_{\k,\alpha}[{\bf{h}}_{\q=0}]} -2\frac{n_3[{\bf{h}}_{\k=0}]n_4[{\bf{h}}_{\k=0}]}{\epsilon_{\k,\alpha}[{\bf{h}}_{\q=0}]} +{\mathcal O}(T^2) \; . \numberthis
\label{eq:Fq0_QM}
\end{align*}

The moments are given by taking the appropriate derivative of the free energy. They are given by the same expression that we obtained for the classical case expressed in \Autoref{eq:Ocorq0} and \Autoref{eq:OOcorq0}.
We now note that we are interested in the zero temperature $T=0$ structure factor, and we can disregard the terms with $1/ β$ in the free energy.
\Autoref{eq:Ocorq0} then becomes
\begin{align*}
\nn{\hat{O}^{\alpha}_{\q=0, \beta}}_{T=0}
&=	-\left. \frac{\ddp\left[ \Delta E_0[{\bf{h}}_{\q=0} ] +C[{\bf{h}}_{\q=0} ]\right]}{\ddp h^{\alpha}_{\q=0,\beta}} \right|_{\V{h}=0}\\
&+2\left. \frac{\ddp\left[ \frac{n_1[{\bf{h}}_{\k=0}]n_2[{\bf{h}}_{\k=0}]}{\epsilon_{\k,\alpha}[{\bf{h}}_{\q=0}]} +\frac{n_3[{\bf{h}}_{\k=0}]n_4[{\bf{h}}_{\k=0}]}{\epsilon_{\k,\alpha}[{\bf{h}}_{\q=0}]}\right]}{\ddp h^{\alpha}_{\q=0,\beta}} \right|_{\V{h}=0}\numberthis \; .
\label{eq:Ocorq01q0}
\end{align*}
We also note that the terms with $n_1[h_=̨0]n_2[h_=̨0]$ and $n_3[h_=̨0]n_4[h_=̨0]$ are at least quadratic (if not of higher order, depending on $ϵ_[h_=0]$)  in the field components $h_=0$, as any of the $n_i=1,2,3,4$ is  independently  linear in $h_-=̨0$ by definition. Therefore taking the first derivative of the terms with $n_1[h_=̨0]n_2[h_=̨0]$ and $n_3[h_=̨0]n_4[h_=̨0]$  and evaluating them at zero field will inevitably lead to a null contribution. We then are simply left with
\begin{align*}
\nn{\hat{O}^{\alpha}_{\q=0, \beta}}_{T=0}
&=	-\left. \frac{\ddp \Delta E_0[{\bf{h}}_{\q=0} ] +C[{\bf{h}}_{\q=0} ]}{\ddp h^{\alpha}_{\q=0,\beta}} \right|_{\V{h}=0}\numberthis \; .
\label{eq:Ocorq0_QM}
\end{align*}
The second moments are given by \Autoref{eq:Ocorq0} and disregarding again the term with $1/ β$, it simply becomes
\begin{eqnarray}
\nn{\hat{O}^{\alpha}_{\q=0, \beta}\hat{O}^{\mu}_{\q=0, \nu}}_{T=0}
&=&	\nn{\hat{O}^{\alpha}_{\q=0, \beta}}_{T=0}	\nn{\hat{O}^{\mu}_{\q=0, \nu}}_{T=0}\; .	\label{eq:OOcorq0_QM}
\nonumber\\
\end{eqnarray}
We can insert \Autoref{eq:OOcorq0_QM} into \Autoref{eq:SOq0_QM} to calculate the ground state contribution to the quantum structure factor.
Therefore all we need to do is find the 0$þ$ order contribution of the source term, i.e., find $C[h_=0 ]$, and compute the zero--point energy of the Bogoliubov transformation $ΔE_0[h_=0 ] $.

\subsection{Dipole moments: contribution of the ground state to the quantum structure factor at $\q=0$}\label{sec:structure_factors_quantum_Sq0}

First, we consider the structure factor for dipole moments of spin
\begin{eqnarray}
S^{\sf GS}_{\rm{S}}(\q=0)
&=& \sum_{\alpha} \nn{\ops{\q=0}{\alpha} \ops{-\q=0}{\alpha}}_{T=0}  \; .\label{eq:Sq0_QM}
\end{eqnarray}
The relevant source term is given by \Autoref{eq:HamLowTS}
According to \Autoref{eq:dipole.in.terms.of.A} and using \Autoref{eq:Acond2},  we can express \Autoref{eq:HamLowTS} in function of fluctuations orthogonal to the FQ ground state [\Autoref{eq:GS_FQ}].
Considering fluctuation terms up to 2$$ order, we have
\begin{align*}
\ops{i}{x}&=-i(\bb{i}-\bbd{i})\; ,\\
\ops{i}{y}&=i(\bad{i}\bb{i}-\ba{i}\bbd{i})\; , \numberthis\\
\ops{i}{z}&=-i(\bad{i}-\ba{i})\; .
\end{align*}
After performing a Fourier transform, and considering the source term Hamiltonian  $Δ[ h^α_i,β ] $ [\Autoref{eq:HamLowTS}] at $=0$, we have
\begin{align*}
\Delta\Ham [{\bf{h}}_{\q}] =-\sum_{\q}&\left[ih^{x}_{-\q}(\bbd{\q}-\bb{\q})-ih^{z}_{-\q}(\bad{\q}-\ba{\q})\right]\delta_{\q,0} \\
-\sum_{\k}&\frac{i}{\sqrt{N}}h^{y}_{\q=0}(\bad{\k}\bb{\k}-\ba{\k}\bbd{\k})\; .\numberthis \label{eq:Ham_S_q0}
\end{align*}
And using \Autoref{eq:HBBQ_qm} for the BBQ Hamiltonian, the total Hamiltonian [\Autoref{eq:Hfield}] in terms of the bosons takes the same form as in \Autoref{eq:H_qm_q0},
where $M_[h_=0]$ is given by
\begin{subequations}
\begin{align*}
A_{\k}& -B_{\k} &  -\frac{i}{\sqrt{N}}h^{y}_{\q=0} & 0 \\
-B_{\k} &A_{\k} &  0& \frac{i}{\sqrt{N}}h^{y}_{\q=0} \\
\frac{i}{\sqrt{N}}h^{y}_{\q=0} &  0 & A_{\k} &- B_{\k}  \\
0 & -\frac{i}{\sqrt{N}}h^{y}_{\q=0} & -B_{\k} & A_{\k}\\
\end{pmatrix}\; ,\label{eq:def_phi_vec_M_k_Sq0}\numberthis
\end{align*}
where $A_{\k}$ and $B_{\k}$ are given in \Autoref{eq:DefAB}
and where ${\V{N}}[{\bf{h}}_{\k}]$ is given by
\be
\begin{matrix}
-h^{x}_{-\k}	\\
\end{pmatrix}\; ,\label{eq:N_S_q0}
\end{matrix}
\ee
and where $ C[{\bf{h}}_{\q=0} ] $ holds
\be
C[{\bf{h}}_{\q=0} ] =0\; . \label{eq:C_S_q0}
\ee
\end{subequations}

Following the procedure depicted in \Autoref{section:quantum.theory} and detailed in \hyperref[sec:bogolioubov_transfomation]{Appendix~}\ref{sec:bogolioubov_transfomation}, we perform a Bogoliubov transformation and
the eigenvalues $ϵ_,̨λ$ are given by
\begin{subequations}
\begin{align}
\epsilon_{\k,1}[{\bf{h}}_{\q=0}]&=-\epsilon_{\k,2}=+\sqrt{A_{\k}^2-B_{\k}^2} +\frac{1}{\sqrt{N}}h^{y}_{\q=0}\; ,\\
\epsilon_{\k,3}[{\bf{h}}_{\q=0}]&=-\epsilon_{\k,4}=+\sqrt{A_{\k}^2-B_{\k}^2}-\frac{1}{\sqrt{N}}h^{y}_{\q=0}\; .
\end{align}\label{eq:omega_qm_S_q0}
\end{subequations}
After performing the Bogoliubov transformation, the Hamiltonian can be rewritten as follows:
\begin{align*}
\Ham=&E_0+\Delta E_0[{\bf{h}}_{\q=0}]\\
&+\left[\sum_{\k}\epsilon_{\k,1}[{\bf{h}}_{\q=0}]\balphad{\k}\balpha{\k}+\epsilon_{\k,3}[{\bf{h}}_{\q=0}]\bbetad{\k}\bbeta{\k}\right]\; ,\numberthis\label{eq:HamBogo2_q0}
\end{align*}
where $C[h_=0 ]$ is disregarded since it is null [\Autoref{eq:C_S_q0}], and where $ΔE_0[h_=0]$ is the zero--point energy
\begin{eqnarray}
\Delta E_0 [{\bf{h}}_{\q=0}]&=& \frac{1}{2}\sum_\k \left[\epsilon_{\k,1}[{\bf{h}}_{\q=0}]+\epsilon_{\k,3}[{\bf{h}}_{\q=0}]\right]\; .
\end{eqnarray}
According to \Autoref{eq:Ocorq0_QM}, the ground state contribution to the first moments yield
\begin{subequations}
\begin{align}
\nn{S^{x}_{\q}}_{T=0}&=\nn{S^{z}_{\q}}_{T=0}=0\; ,\\
\nn{S^{y}_{\q}}_{T=0}&=\frac{1}{2}\sum_\k \left[\frac{1}{\sqrt{N}}-\frac{1}{\sqrt{N}}\right]=0 \; .
\end{align}\label{eq:SxSzSyq0}
\end{subequations}
And according to \Autoref{eq:OOcorq0_QM} and \Autoref{eq:Sq0_QM}, the spin dipole structure factor at $=0$ yields
\begin{eqnarray}
S^{\sf GS}_{\rm{S}}(\q=0)
&=& 0 \; . \label{eq:SSq0_QM}
\end{eqnarray}
Indeed, the ground state is quadrupolar and does not break time--reversal symmetry. Therefore, at zero temperature, the contribution of quantum fluctuations from the zero--point energy should average to zero for the spin dipole moments.
The spectral representation of \Autoref{eq:SSq0_QM} is then also trivially null.
Combining \Autoref{eq:Sqw_qm_qn0_app} and \Autoref{eq:SSq0_QM}, we obtain \Autoref{eq:Sqw_qm}.

\subsection{Quadrupole moments: contribution of the ground state to the quantum structure factor at $\q=0$}\label{sec:structure_factors_quantum_Qq0}
We now consider the quadrupole structure factor at the $Γ$--point, which is defined as
\begin{eqnarray}
S^{\sf GS}_{\rm{Q}}(\q=0)
&=& \sum_{\alpha\beta}
\nn{\opq{\q=0}{\alpha\beta} \opq{\q=0}{\beta\alpha}} _{T=0}\; . \label{eq:Qq0_QM}
\end{eqnarray}
We follow the same procedure as depicted in \hyperref[sec:structure_factors_quantum_q0]{Appendix~}\ref{sec:structure_factors_quantum_q0}. The relevant source term for quadrupole moments is given by \Autoref{eq:HamLowTQ}. We can express \Autoref{eq:HamLowTQ} up to second order in terms of the bosons by using \Autoref{eq:Acond2} and \Autoref{eq:quadrupole.in.terms.of.A}. We use \Autoref{eq:HBBQ_qm} for the BBQ Hamiltonian. We then obtain for the total Hamiltonian given by \Autoref{eq:Hfield}, written in the form of \Autoref{eq:H_qm_q0}, where
$M_[h_=0]$ is given by
\begin{subequations}
\be
A_{\k}+\alpha_1 & -B_{\k} &   \beta_1&0\\
-B_{\k} &A_{\k}+\alpha_1  &   0& \beta_1 \\
\beta_1 &  0& A_{\k}+\alpha_2 &- B_{\k}  \\
\beta_1& 0 & -B_{\k} & A_{\k}+\alpha_2  \\
\end{pmatrix} \; ,\label{eq:M_Qq0_QM}
\ee
where ${\V{N}}[{\bf{h}}_{\k}]$ is given by
\be
\begin{matrix}
%	{\V{N}}_1[{\bf{h}}_{\k}]^{\dagger}=i\begin{pmatrix}
%		-h^{z}_{-\k}, &
%		h^{z}_{\k},&
%		h^{x}_{-\k} &
%		-h^{x}_{\k}
%	\end{pmatrix}\; , \\
\xi^1_{-\k}\\
\xi^1_{\k}\\
\xi^2_{-\k}\\
\xi^2_{-\k}\label{eq:N_Q_q0}
\end{pmatrix}
\end{matrix}\; ,
\ee
and where $ C[{\bf{h}}_{\q=0} ] $ holds
\be
C[{\bf{h}}_{\q=0} ] =\sqrt{N}\left(\frac{4}{3}h_{\q=0,y}^{y}-\frac{2}{3}(h_{\q=0,x}^{x}+h_{\q=0,z}^{z})\right)\; ,\label{eq:C_Qq0_QM}
\ee
\end{subequations}
with $A_$ and $B_$ being given in \Autoref{eq:DefAB} and with the following definitions
\be
\begin{matrix}
\begin{matrix}
\alpha_1=\frac{2}{\sqrt{N}}(h_{\q=0,x}^{x}-h_{\q=0,y}^{y})\; ,&\alpha_2=\frac{2}{\sqrt{N}}(h_{\q=0,z}^{z}-h_{\q=0,y}^{y})\; ,
\end{matrix}\\
\begin{matrix}
\beta_1=\frac{1}{\sqrt{N}}(h_{\q=0,z}^{x}+h_{\q=0,x}^{z})\;,
\end{matrix}\\
\begin{matrix}
\xi^1_{\k}=(h^{x}_{\k,y}+h^{y}_{\k,x}) & \; ,& \xi^2_{\k}=(h^{y}_{\k,z}+h^{z}_{\k,y})
\end{matrix}\; .
\end{matrix}
\ee
Following the procedure depicted in \Autoref{section:quantum.theory} and detailed in \hyperref[sec:bogolioubov_transfomation]{Appendix~}\ref{sec:bogolioubov_transfomation}, we perform a Bogoliubov transformation and
the eigenvalues $ϵ_,̨λ$ are given by
\begin{subequations}
\begin{align*}
&\epsilon_{\k,1}[{\bf{h}}_{\q=0}]= -\epsilon_{\k,2}[{\bf{h}}_{\q=0}]=\\
&\sqrt{A_{\k}^2-B_{\k}^2+\beta_1^2+\frac{1}{2} \left(\alpha_{1}^2+\alpha_{2}^2\right)+A_{\k} \alpha^{+}-\Delta  \left(A_{\k}+\frac{\alpha^{+}}{2}\right)}\; ,\numberthis\\
&\epsilon_{\k,3}[{\bf{h}}_{\q=0}]= -\epsilon_{\k,4}[{\bf{h}}_{\q=0}]=\\
&\sqrt{A_{\k}^2-B_{\k}^2+\beta_1^2+\frac{1}{2} \left(\alpha_{1}^2+\alpha_{2}^2\right)+A_{\k} \alpha^{+}+\Delta  \left(A_{\k}+\frac{\alpha^{+}}{2}\right)}\; ,\numberthis\\
\end{align*} \label{eq:omega_qm_Q_q0}
\end{subequations}
where $α^+$ and $Δ$ are defined in \Autoref{eq:a+_Delta}.
After performing the Bogoliubov transformation, the Hamiltonian can be rewritten as follows:
\begin{align*}
\Ham=&E_0+\Delta E_0[{\bf{h}}_{\q=0}]+C[{\bf{h}}_{\q=0} ]\\
&+\left[\sum_{\k}\epsilon_{\k,1}[{\bf{h}}_{\q=0}]\balphad{\k}\balpha{\k}+\epsilon_{\k,3}[{\bf{h}}_{\q=0}]\bbetad{\k}\bbeta{\k}\right]\; ,\numberthis\label{eq:HamBogo2_Qq0}
\end{align*}
where  $C[h_=0 ]$ is given in \Autoref{eq:C_Qq0_QM}, and where$ΔE_0[h_=0]$ is the zero--point energy and yields
\begin{eqnarray}
\Delta E_0[{\bf{h}}_{\q=0}] &=& \frac{1}{2}\sum_\k \left[\epsilon_{\k,1}[{\bf{h}}_{\q=0}]+\epsilon_{\k,3}[{\bf{h}}_{\q=0}]\right]\; .
\end{eqnarray}
According to \Autoref{eq:Ocorq0_QM}, the ground state contribution to the first moments yield
\begin{subequations}
\begin{align}
\nn{Q^{xx}_{\q=0}}_{T=0}&=-\frac{2}{3}\sqrt{N}+ \frac{1}{\sqrt{N} }\sum_\k \left[\frac{A_{\k}}{\sqrt{A_{\k}^2-B_{\k}^2}} \right]\; ,\\
\nn{Q^{xy}_{\q=0,y}}_{T=0}&=\nn{Q^{yx}_{\q=0}}_{T=0}=0\; ,\\
\nn{Q^{xz}_{\q=0,z}}_{T=0}&=\nn{Q^{zx}_{\q=0}}_{T=0}=0\; ,\\
\nn{Q^{yy}_{\q=0,y}}_{T=0}&=\frac{4}{3}\sqrt{N}- \frac{1}{\sqrt{N} }\sum_\k \left[\frac{2A_{\k}}{\sqrt{A_{\k}^2-B_{\k}^2}} \right]\; ,\\
\nn{Q^{yz}_{\q=0,z}}_{T=0}&=\nn{Q^{zy}_{\q=0}}_{T=0}=0\; ,\\
\nn{Q^{zz}_{\q=0}}_{T=0}&=-\frac{2}{3}\sqrt{N}+ \frac{1}{\sqrt{N}} \sum_\k \left[\frac{A_{\k}}{\sqrt{A_{\k}^2-B_{\k}^2}} \right]\; .
\end{align}\label{eq:nQq0_QM}
\end{subequations}
Before calculating the structure factor, we note that, as given in \Autoref{eq:nQq0_QM}, the first quadrupole moments consist of two terms with different scaling behaviour with respect to the parameter we expand fluctuations about, which is the length of the spin $s$ . Indeed, similarly to multi-boson expansion, or its linear spin-wave version with Holstein--Primakoff bosons or Schwinger bosons in the case of a $su(2)$ representation of the spin, we assume the fluctuation to be sufficiently small compared to the spin length $s$. In other words, $C[h_=0 ]$  from \Autoref{eq:C_Qq0_QM} and the eigenvalues in \Autoref{eq:omega_qm_Q_q0} scale with $s$ as
\begin{subequations}
\be
C[{\bf{h}}_{\q=0} ]\sim s h^{\mu}_{\q=0\mu} \; ,
\ee
\be
\epsilon_{\k,\lambda}[{\bf{h}}_{\q=0}]\sim s \sqrt{ Const. +\frac{h^{\mu}_{\q=0\mu}}{s^2}+\frac{\O({\bf{h}}_{\q=0}^2)}{s^2} }\; .
\ee
\end{subequations}
Their  derivatives with respect to $h^μ_=0,μ$ that enters the quadrupole moments [\Autoref{eq:Ocorq0_QM}] yield
\begin{subequations}
\be
\left.\frac{C[{\bf{h}}_{\q=0} ] }{\ddp h^{\mu}_{\q=0,\mu}}\right|_{\bf{h}=0}\sim s \; ,
\ee
\begin{align*}
\left.\frac{\ddp\epsilon_{\k,\lambda}[{\bf{h}}_{\q=0}] }{\ddp h^{\mu}_{\q=0,\mu}}\right|_{\bf{h}=0}\sim  \left.\frac{s}{\sqrt{ Const.}}\frac{\ddp \frac{h^{\mu}_{\q=0\mu}}{s^2}}{\ddp h^{\mu}_{\q=0,\mu}}\right|_{\bf{h}=0} \sim  \frac{1}{s} \; .
\end{align*}
\end{subequations}
This implies that the scaling behaviour of the first quadrupole moments goes as
\be
\nn{Q^{\mu\mu}_{\q=0}}_{T=0}\sim s+ \frac{1}{s}\; ,
\ee
where $s$ is the length of the spin.
We now argue that because our approximation is valid up to linear order in $1/s$, i.e., second order in fluctuations,  we can disregard  $1/s^2$ terms.  $1/s^2$ terms are physical but should not enter into our level of approximation. Indeed, one would expect additional contribution to the $1/s^2$ term coming from higher orders in perturbation theory. However, we do not take these into account here and simply consider terms up to $1/s$.
According to \Autoref{eq:OOcorq0_QM} and \Autoref{eq:Qq0_QM}, the spin quadrupole structure factor at $=0$ yields
%	\begin{eqnarray}
%		S^{\sf GS}_{\rm{Q}}(\q=0)&=& \frac{8}{3}N-8 \sum_\k \left[\frac{A_{\k}}{\sqrt{A_{\k}^2-B_{\k}^2}} \right]\nonumber\\
%		&&+\frac{6}{N}\left( \sum_\k \left[\frac{A_{\k}}{\sqrt{A_{\k}^2-B_{\k}^2}} \right]\right)^2  \label{eq:SQq0_QM}\; .
%	\end{eqnarray}
\begin{eqnarray}
S^{\sf GS}_{\rm{Q}}(\q=0)&=& \frac{8}{3}N-8 \sum_\k \left[\frac{A_{\k}}{\sqrt{A_{\k}^2-B_{\k}^2}} \right]\nonumber\\
&&+\O(\frac{1}{s^2})  \label{eq:SQq0_QM}\; .
\end{eqnarray}
Its spectral representation is given by
%		\begin{eqnarray}
%		S^{\sf GS}_{\rm{Q}}(\q=0,\omega)&=& \left(\frac{8}{3}N-8 \sum_\k \left[\frac{A_{\k}}{\sqrt{A_{\k}^2-B_{\k}^2}} \right]\right)\delta(\omega)\nonumber\\
%		&&+\frac{6}{N}\left( \sum_\k \left[\frac{A_{\k}}{\sqrt{A_{\k}^2-B_{\k}^2}} \right]\right)^2 \delta(\omega) \label{eq:SQq0_QM_w}\; .
%	\end{eqnarray}
\begin{eqnarray}
S^{\sf GS}_{\rm{Q}}(\q=0,\omega)&=& \left(\frac{8}{3}N-8 \sum_\k \left[\frac{A_{\k}}{\sqrt{A_{\k}^2-B_{\k}^2}} \right]\right)\delta(\omega)\nonumber\\
&&+\O(\frac{1}{s^2}) \label{eq:SQq0_QM_w}\; .
\end{eqnarray}
Combining \Autoref{eq:Qqw_qm_qn0} and \Autoref{eq:SQq0_QM_w}, we obtain \Autoref{eq:Qqw_qm}.
\subsection{A-matrices: contribution of the ground state to the quantum structure factor at $\q=0$}\label{sec:structure_factors_quantum_Aq0}
For the quantum zero temperature structure factor for the A--matrices at $=0$, we make use of the sum rule given in \Autoref{eq:sum.rule.AQS}.
This leads to
\begin{eqnarray}
S^{\sf GS}_{\rm{A}}(\q=0)&=&  \frac{1}{4} S^{\sf GS}_{\rm{Q}}(\q=0)
+  \frac{1}{2} S^{\sf GS}_{\rm{S}}(\q=0)
+  \frac{1}{3}N \delta_{\q,0}\nonumber \\
&=& N-2 \sum_\k \left[\frac{A_{\k}}{\sqrt{A_{\k}^2-B_{\k}^2}} \right] +\O(\frac{1}{s^2})\; ,\nonumber\\
\label{eq:SAq0_QM}
\end{eqnarray}
%	\begin{eqnarray}
%	S^{\sf GS}_{\rm{A}}(\q=0)&=&  \frac{1}{4} S^{\sf GS}_{\rm{Q}}(\q=0)
%	+  \frac{1}{2} S^{\sf GS}_{\rm{S}}(\q=0)
%	+  \frac{1}{3}N \delta_{\q,0}\nonumber \\
%	&=& N-2 \sum_\k \left[\frac{A_{\k}}{\sqrt{A_{\k}^2-B_{\k}^2}} \right]\nonumber\\
%	&&+\frac{3}{2N}\left( \sum_\k \left[\frac{A_{\k}}{\sqrt{A_{\k}^2-B_{\k}^2}} \right]\right)^2 \; ,\label{eq:SAq0_QM}
where we used \Autoref{eq:SSq0_QM} and \Autoref{eq:SQq0_QM}.
Its spectral representation yields
\begin{eqnarray}
S^{\sf GS}_{\rm{A}}(\q=0, \omega)&=& \left(N-2 \sum_\k \left[\frac{A_{\k}}{\sqrt{A_{\k}^2-B_{\k}^2}} \right]\right)\delta(\omega) \nonumber\\
&&+\O(\frac{1}{s^2}) \; .\nonumber\\
\label{eq:SAq0_QM_w}
\end{eqnarray}
Combining \Autoref{eq:Aqw_qm_qn0} with \Autoref{eq:SAq0_QM_w}  gives the total quantum structure factor for the A-matrices expressed in \Autoref{eq:Aqw_qm}
%	\begin{eqnarray}
%	S^{\sf GS}_{\rm{A}}(\q=0, \omega)&=& \left(N-2 \sum_\k \left[\frac{A_{\k}}{\sqrt{A_{\k}^2-B_{\k}^2}} \right]\right)\delta(\omega) \nonumber\\
%	&&+\left.\frac{3}{2N}\left( \sum_\k \left[\frac{A_{\k}}{\sqrt{A_{\k}^2-B_{\k}^2}} \right]\right)^2\right.\delta(\omega) \; .\nonumber\\
%	\label{eq:SAq0_QM_w}
\section{System size dependence of the ordered moments }
\label{sec:Mermin_Wagner_Theorem}
We present in this Appendix, the details of the manufacturing of \Autoref{section:ordered.moment}. More precisely, we explain how we fitted the numerical data for the ordered moments and explain how we calculated the ordered moments from the analytical results.

In \Autoref{table:Tintervals}, we show the temperature intervals on which the  corresponding ordered parameters values are used for the fits of the slope $α(L)$ of the ordered parameters in  \Autoref{fig:ordered.moment}~\subref{fig:MW_theorem1}, for different system sizes.

We also present here how the ordered moments as expressed by \Autoref{eq:OM} and presented in \Autoref{fig:ordered.moment}~\subref{fig:MW_theorem2} are calculated . In order to compute \Autoref{eq:OM}, we need to perform a sum in k--space. We here also show that the sum scales logarithmically with the system size $L$ by explicitly calculating the coefficient $μ$ correspond to the logarithmic behavior [\Autoref{eq:fitL}]. To do this, we calculate the sum numerically for different system sizes $L$ and fit it according to \Autoref{eq:fitL} (as shown by the orange line in \Autoref{fig:ordered.moment}~\subref{fig:MW_theorem2}). Additionally, we also transform the sum into an integral and extract the logarithmic scaling behaviour.

\begin{center}
\begin{table}[t]
\begin{tabular}{ |c||c|c|  }
%\multicolumn{3}{|c|}{Temperature invervals} \\
\hline
System size L & $T_{min}$ & $T_{max}$\\
\hline
L=12 & 0.01 & 0.100177 \\
L=24 & 0.01 & 0.100177 \\
L=48 & 0.0252403 & 0.100177 \\
L=96 & 0.0343658 & 0.100177 \\
\hline
\end{tabular}
\caption{Temperature intervals used for fitting the parameter $\alpha(L)$ according to \protect\Autoref{eq:lowTfit2}.}\label{table:Tintervals}
\end{table}
\end{center}

The Brillouin zone is turned into a parallelogram of area $8π/√(3)$,
as it is spanned by the  reciprocal vectors $K_a$ and $K_b$ given in \Autoref{eq:def_ka_kb}. We then discretized it into $N=L^2$ tiles of dimension $δA$ given by
\be
\begin{matrix}
\delta \k_a=\frac{1}{L}\V{K}_a&\; , ~&\delta \k_b=\frac{1}{L}\V{K}_b\; ,
\end{matrix}
\ee
such that
\be
\delta A=\frac{8\pi}{\sqrt{3}L^2}\; .
\ee

In order to compare numerical with analytical results, we consider
\be
\frac{1}{N}\sum_{\k} I_{\k}\Rightarrow N \delta A \sum_{k_x,k_x}  I_{\k}=\int  I_{\k} \dd \k \; ,\label{eq:SumNum}
\ee
We can now sum over the k--space, numerically, or integrate, analytically.

In \Autoref{eq:SumNum}, we take as integrant the term expressed as a sum in the result obtained in  \Autoref{eq:OM}, as we wish to compute the temperature--dependent part of the ordered moment given in \Autoref{eq:OM}. Using \Autoref{eq:omega_vec_1}, we obtain
\be
I_{\k}= \frac{16 A_{\k}}{\epsilon_{\k,1}^2}\; ,
\ee
where $ϵ_,̨1$ is given in \Autoref{eq:omega_qm}.
We then compute the discrete sum numerically according to \Autoref{eq:SumNum} for the different system sizes, including the ones given in \Autoref{table:Tintervals}. When performing the sum,  we also avoid the origin $=̨(0,0)$, where $ϵ_,̨1$ vanishes, (indeed, $γ(0)=1$, and according to \Autoref{eq:DefAB},  $A_0=B_0$) and which is not included in the sum of \Autoref{eq:OM}.   For a specific system size, we then get a number as the results of the discrete sum obtained for that specific system size. These numbers are plotted as the red dots in  \Autoref{fig:ordered.moment}~\subref{fig:MW_theorem2}.

According to \Autoref{eq:lowTfit2}, we assume that the system size dependency should be of the form
\be
-\left.\frac{\dd S^{CL}_{\rm{Q}}(\q=\Gamma)}{\dd T}\right|_{T=0}=\frac{1}{N}\sum_{\k} I_{\k} \sim ~ C+\mu \log(L)+\frac{\nu}{L}+\frac{\xi}{L^2} \; .\label{eq:fitL}
\ee
We use \Autoref{eq:fitL} to fit the results obtained by computing the discrete sum in \Autoref{eq:SumNum}, i.e, the red dots in  \Autoref{fig:ordered.moment}~\subref{fig:MW_theorem2}. The fit is shown in in  \Autoref{fig:ordered.moment}~\subref{fig:MW_theorem2} by the orange line.

Additionally, we want to investigate how accurate the discrete sum is, compared to the integration, and how it depends on system size.
If we consider the integral version in 2--dimensions for polar coordinates, we can cut off to some small $k_s=4π/√(3)L$ in order to avoid the origin $=̨(0,0)$ as follows:
\be
\int I_{\k}\dd \k=2\pi\int_{0}^{\Lambda} I_{\k}k\dd k=2\pi\int_{0}^{k_s} I_{\k}k \dd k+2\pi\int_{k_s}^{\Lambda} I_{\k}k\dd k\; .
\ee
For the FQ state, where we chose, $J_1 = 0.0$ and $J_2 = -1.0$, the coefficients $A_$ and $B_$ [\Autoref{eq:DefAB}] and the dispersion relation $ϵ_,̨1$ [\Autoref{eq:omega_qm}] become
\begin{align*}
A_{\k}&=z\; ,\\
B_{\k}&=-z\gamma(\k\; ,)\\
\epsilon_{\k}^2&=z^2(1-\gamma(\k)^2)\numberthis\; .
\end{align*}
For the triangular lattice, the geometrical factor is given by \Autoref{eq:def_gamma_tri}, and for sufficiently small values of k, we can use the Taylor expansion on it. We obtain
\begin{subequations}
\begin{align}
\gamma(\k)&\simeq1-\frac{1}{4}(k_x^2+k_y^2)=1-\frac{1}{4}k^2\; ,\\
\epsilon_{\k}^2&\simeq z^2\frac{1}{2}k^2\; ,\\
I_{\k}=\frac{16 A_{\k}}{\epsilon_{\k,1}^2}&\simeq\frac{16 z}{z^2\frac{1}{2}k^2}=\frac{3	2}{zk^2}\; ,\\
2\pi\int_{k_s}^{k_f} I_{\k}k\dd k&\simeq 2\pi\int_{k_s}^{k_f} \frac{32}{zk^2}k\dd k \; .
\end{align}
\end{subequations}
Since $z=6$ for the triangular lattice, we have
\begin{align*}
&\frac{2\pi 16}{3}\int_{k_s}^{k_f} \frac{1}{k}\dd k= \frac{2\pi 16}{3}(\log(k_f)-\log(k_s)) \; ,\\
&= \frac{2\pi 16}{3}\log(L)+\frac{2\pi 16}{3}\log(k_f)-\frac{2\pi 16}{3}\log(\frac{4\pi}{\sqrt{3}})\; ,\numberthis\label{eq:OM_logL}
\end{align*}
where in the last line, we used the fact that we chose to cut off according to $k_s=4π/√(3)L$.
Before we fit the sum with the expression given by \Autoref{eq:fitL}, we need to account for correction coming from the tiling of the k-space as explained in \Autoref{eq:SumNum}. Therefore, we need to divide by
\be
\delta A*N=\frac{8 \pi^2}{\sqrt{3}}\; .
\ee
From \Autoref{eq:fitL}, we can obtain the value for the coefficient $μ$ for \Autoref{eq:OM_logL}, which we can compare with the fit from the values of the sum calculated numerically as shown in \Autoref{fig:ordered.moment}~\subref{fig:MW_theorem2}:
\be
\begin{matrix}
\mu_{ana}=\frac{\frac{2\pi 16}{3}}{\frac{8 \pi^2}{\sqrt{3}}}=\frac{4}{\sqrt{3}\pi}=0.735&\; , ~ & \mu_{num}=0.735\; .
\end{matrix}
\ee

\section{Useful Gaussian integrals}
\label{sec:table.of.integrals}

We present here useful Gaussian integrals that we used to calculate partition functions for the analytic derivations.
We namely used the following one--dimensional Gaussian integrals
\begin{subequations}
\begin{align}
\int  \exp{-\frac{1}{2}a x^{2}}\dd x&=\sqrt{\frac{2\pi}{a}}\label{eq:gauss_int_S1}\; ,\\
\int  \exp{-\frac{1}{2}a x^{2}}\exp{\pm bx}\dd x&=\sqrt{\frac{2\pi}{a}}\exp{\frac{b^2}{2 a}}\; ,\label{eq:gauss_int_S2}\\
\int  \exp{-\frac{1}{2}a x^{2}}\exp{\pm i b x}\dd x&=\sqrt{\frac{2\pi}{a}}\exp{-\frac{b^2}{2 a}}\; ,\label{eq:gauss_int_Q1}
\end{align}
which we can also generalize to a multi-dimensions integral with a source term
\be
\int \exp{-\frac{1}{2}x_iA_{ij}x_j+B_ix_i}\dd \vec{x}=\sqrt{\frac{(2\pi)^n}{\det A}}\exp{\frac{1}{2}\vec{B}^{T}A^{-1}\vec{B}}\; .\label{eq:gauss_int}
\ee
or more generally,
\be
\int \exp{-\frac{1}{2}x_i{A^{i}}_{j}x^j+B_{1,i}^{T}x^i+x_i B_{2}^{i}}\dd \vec{x}=\sqrt{\frac{(2\pi)^n}{\det A}}\exp{2\vec{B}_1^{T}A^{-1}\vec{B}_2}\; .\label{eq:gauss_int2}
\ee
\end{subequations}
where $n$ is the dimension of of the matrix $A$.
Below, we give the proof for \Autoref{eq:gauss_int2}.

\textit{Proof:}
We assume $A$ to be a real symmetric $n×n$-matrix. This means that $A$  is orthogonally diagonalizable, i.e., it is similar to a diagonal matrix $D=(d^1_ 1, …, d^n_ n)$
\be
\V{A}=\V{S}\V{D}\V{S}^{-1}\label{eq:A_diag_D}\; ,
\ee
and the $n×n$ basis change matrix $S$ is orthogonal.
The  basis change matrix $S$ then satisfies
\be
\V{S}^T=\V{S}^{-1}\label{eq:S_ortho}\; ,
\ee
and the old coordinates are related to the new ones by
\begin{subequations}
\begin{align}
\vec{x}&=\V{S}\vec{y}\; ,\\
\vec{x}^{T}&=\vec{y}\V{S}^T\; ,\\
\dd \vec{x}&=\det\V{S}\dd \vec{y}=\dd \vec{y}\; ,
\end{align}
\end{subequations}
where in the last line we used the fact that the Jacobian matrix of the map $x⃗(y⃗)→Sy⃗$ is the matrix $S$ itself, and that its determinant is 1, since it is an orthogonal matrix.
The term in the exponential in \Autoref{eq:gauss_int2} then becomes
\begin{subequations}
\begin{align}
&=-\frac{1}{2}\vec{y}^{T}\V{D}\vec{y}+\vec{B}_1^T\V{S}\vec{y}+\vec{y}^{T}\V{S}^{-1}\vec{B}_{2}\; .
\end{align}
If we expand, we obtain:
\begin{align}
E&=\sum_{i}\left[-\frac{1}{2}y_{i} d_{~i}^{i}y^{i}+\sum_{\alpha}\left(B_{1,\alpha}^{T}S^{\alpha}_{~i} y^{i}+y_{i} (S^{-1})^{i}_{~\alpha}B_{2}^{\alpha}\right)\right]\; ,
\end{align}
\end{subequations}
where we used the fact that $D$ is a diagonal matrix.
For the $i^th$ term, we can complete the square as
\begin{align*}
&-\frac{1}{2}y_{i} d_{~i}^{i}y^{i}+\sum_{\alpha}\left(B_{1,\alpha}^{T}S^{\alpha}_{~i} y^{i}+y_{i} (S^{-1})^{i}_{~\alpha}B_{2}^{i}\right)\\
=&-\frac{1}{2} d_{i}^{i}(y_{i}-\frac{2}{d_{~i}^{i}}\sum_{\alpha}B_{1,\alpha}^{T}S^{\alpha}_{~i})(y^{i}-\frac{2}{d_{i}^{i}}\sum_{\beta}(S^{-1})^{i}_{~\beta}B_{2}^{\beta})\\
&+2\sum_{\alpha}\sum_{\beta}B_{1,\alpha}^{T}S^{\alpha}_{~i}\frac{1}{d_{i}^{i}}(S^{-1})^{i}_{~\beta}B_{2}^{\beta}\; .\numberthis\label{eq:square_completed}
\end{align*}
Using again the fact that $d_ i^i$ is the $i^th$ diagonal term of $D$, and inverting \Autoref{eq:A_diag_D}, we can rewrite  $1/d_i^i$ as
\be
\frac{1}{d_{i}^{i}}=(D^{-1})^{i}_{~i}=\sum_{\mu,\nu}(S^{-1})^{i}_{~\mu}(A^{-1})^{\mu}_{~\nu}(S)^{\nu}_{~i}\; .\label{eq:1_over_d}
\ee
Performing the variable change as
\begin{subequations}
\begin{align}
z_i&=y_{i}-\frac{2}{d_{~i}^{i}}\sum_{\alpha}B_{1,\alpha}^{T}S^{\alpha}_{~i}\; ,\\
z^i&=y^{i}-\frac{2}{d_{i}^{i}}\sum_{\beta}(S^{-1})^{i}_{~\beta}B_{2}^{\beta}\; ,\\
\dd \vec{z}&=\dd \vec{y}\; ,
\end{align}
\end{subequations}
and inserting \Autoref{eq:1_over_d} into the last term of \Autoref{eq:square_completed}, and summing over all the components, we obtain
\begin{subequations}
\begin{align}
E=&-\frac{1}{2}\vec{z}^{T}\V{D}\vec{z}+2\vec{B}_1^T(\V{A}^{-1})\vec{B}_{2}\; .
\end{align}
\end{subequations}
We now have the product of n Gaussian integrals of  the form of \Autoref{eq:gauss_int_S1}
\begin{align}
\int& \exp{-\frac{1}{2}x_i{A^{i}}_{j}x^j+B_{1,i}^{T}x^i+x_i B_{2}^{i}}\dd \vec{x}=\nonumber\\
&=\left[\prod_i\int\exp{-\frac{1}{2}z_i{d^{i}}_{i}z^i}\dd z_i \right]\exp{2\vec{B}_1^T(\V{A}^{-1})\vec{B}_{2}}\nonumber\\
&=\sqrt{\frac{(2\pi)^n}{\prod_i{d^{i}}_{~i}}}\exp{2\vec{B}_1^T(\V{A}^{-1})\vec{B}_{2}}\; .
\end{align}
We then use the fact that
\be
\det\V{A}=\det\V{S}\det\V{D}\det\V{S}^{-1}=\det\V{D}=\prod_i{d^{i}}_{~i}\; ,
\ee
to obtain \Autoref{eq:gauss_int2}
\section{Application to an easy--plane ferromagnet}
\label{section:anisotropy_FM}

%\subsection{Easy-Plane  and  Single--Ion Anisotropy in the Spin-1 FM Heisenberg Hamiltonian }
We here wish to apply the $u(3)$ formalism and its representation in terms of the A-matrices to the Heisenberg ferromagnetic easy-plane anisotropic model.
The A-matrices are especially useful to work with on the TR-invariant basis, and relatively easy to use when the ground state is quadrupolar. However, some attention is demanded when working with systems where dipoles rather than quadrupoles order. We demonstrate here how one can carefully apply our method for dipolar ordering. Additionally, as explained in \Autoref{section:anisotropy}, we make the interactions anisotropic.
We show results for the zero--temperature quantum structure factors for dipole, quadrupole and A-matrix moments applied to the ferromagnetic (FM) state on the triangular lattice for the anisotropic Heisenberg Hamiltonian (BBQ Hamiltonian [\Autoref{eq:BBQ.model}] with anisotropic $J_1$ and $J_2=0$), with single--ion anisotropy.

% Fig. 22 - magnetic basis a la Kim

\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{basis_x_2.png}
\caption{
Eigenstates of $\ops{i}{x}$ and basis states of $\mathcal{B}^x$
%\rewrite{\bf [Include revised figure from Thesis].}
\label{fig:basis_x}
\end{figure}
such that
\begin{subequations}
\begin{align}
\bra{1^x} \ops{i}{x}\ket{1^x}&=1\; ,\\
\bra{0^x} \ops{i}{x}\ket{0^x}&=0\; ,\\
\bra{\overline{1^x}} \ops{i}{x}\ket{\overline{1^x}}&=-1\; ,\\
\bra{\alpha} \ops{i}{\mu}\ket{\alpha}&=0 ~~~~ \textrm{for } \ket{\alpha} \in 	\mathcal{B}^x \textrm{ and }  \mu=y,z\; .
\end{align}
\end{subequations}

We consider the following
\be
\Ham=\Ham^{\rm{EP}}+\Ham^{\rm{SI}}\; . \label{eq:Ham_EP_SI_FM}
\ee
$^EP$ represents the Heisenberg Hamiltonian for spin-1 with easy--plane Heisenberg anisotropic exchange couplings $J$
\be
\Ham^{\rm{EP}}=\sum_{\nn{i,j}}\left[\opsb{i}\cdot\textbf{J}\cdot\opsb{j}\right]\; , \label{eq:Ham_EP_FM}
\ee
where the spin dipole operator $i$ is defined in \Autoref{eq:Sdipole}, and where $J$ corresponds to the usual nearest neighbor spin-spin coupling tensor.
$^SI$ accounts for single--ion anisotropy and is given by
\be
\Ham^{\rm{SI}}=\sum_{i}\opsb{i}\textbf{D}\opsb{i}\label{eq:Ham_SI_FM}\; ,
\ee
where $D$ corresponds to the usual single site spin-spin coupling tensor.
We assume  the spin-spin coupling tensors $J$ and $D$ to only have diagonal components:
\be
\Ham^{\rm{EP}}=\sum_{\nn{i,j}}\left[J^{xx}\ops{i}{x}\ops{j}{x}+J^{yy}\ops{i}{y}\ops{j}{y}+J^{zz}\ops{i}{z}\ops{j}{z}\right]\; , \label{eq:Ham_EP_FM_2}
\ee
\be
\Ham^{\rm{SI}}=\sum_{i}\left[D^{xx}\ops{i}{x}\ops{i}{x}+D^{yy}\ops{i}{y}\ops{i}{y}+D^{zz}\ops{i}{z}\ops{i}{z}\right]\; . \label{eq:Ham_SI_FM_2}
\ee
We also assume the coupling constants to be negative and the order to be ferromagnetic:
\be
J^{\alpha\alpha}<0\; ,\label{eq:J_FM}
\ee
\be
D^{\alpha\alpha}<0\; .\label{eq:D_FM}
\ee
We can assume the ground state to be a state with the spin pointing somewhere in the xy-plane, and we can choose it to be pointing along the x-axis:
\be
\ket{GS}=\ket{1^x}\; .\label{eq:gs_x}
\ee
As a basis, we choose the eigenstates of $ix$:
\begin{align}
\mathcal{B}^x&=\left\lbrace\ket{1^x},\ket{0^x},\ket{\overline{1^x}}\right\rbrace\; ,\label{eq:Bx}
\end{align}
as represented in \Autoref{fig:basis_x},

Even though the  A-matrices are deeply linked to the time-reversal (TR) invariant basis, we will here mostly focus on the basis $ℬ^x$ [\Autoref{eq:Bx}] and then transform the required quantities accordingly.

% Fig. 23 - rotations starting from dipole state

\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{FM_anisotropy.png}
\caption{
Fluctuations created by the generators $\opa{}{x}{y}$, $\opa{}{x}{z}$, $\opa{}{y}{x}$, and $\opa{}{z}{x}$ , according to \protect\Autoref{eq:defR}, for an angle $\phi_{\alpha\beta}=\frac{\pi}{8}$.
% \rewrite{\bf [Include revised figure from Thesis].}
\label{fig:fluct}
\end{figure}

To do that, we remember that the spin dipole moments can be rewritten in terms of the A matrices [\Autoref{eq:dipole.in.terms.of.A}], expressed in the time-reversal (TR) invariant basis [\Autoref{eq:TRB}] as shown in \Autoref{fig:TR.basis}.
Using \Autoref{eq:dipole.in.terms.of.A}, the terms of the easy--plane anisotropic Hamiltonian [\Autoref{eq:Ham_EP_FM_2}], in terms of the A-matrices, becomes
\begin{subequations}
\begin{align}
\ops{i}{x}\ops{j}{x}&=-(\opa{i}{y}{z}-\opa{i}{z}{y})(\opa{j}{y}{z}-\opa{j}{z}{y})\; ,\\
\ops{i}{y}\ops{j}{y}&=-(\opa{i}{z}{x}-\opa{i}{x}{z})(\opa{j}{z}{x}-\opa{j}{x}{z})\; ,\\
\ops{i}{z}\ops{j}{z}&=(\opa{i}{x}{y}-\opa{i}{y}{x})(\opa{j}{x}{y}-\opa{j}{y}{x})\; .
\end{align} \label{eq:SS_EP}
\end{subequations}
For the single ion terms, we use \Autoref{eq:DefQ} and \Autoref{eq:quadrupole.in.terms.of.A} to rewrite the terms $iαiα$ of the single--ion anisotropic Hamiltonian [\Autoref{eq:Ham_SI_FM_2}] in the function of the A-matrices, as
\begin{subequations}
\begin{align}
\ops{i}{x}\ops{i}{x}&=-\frac{2}{3}\opa{i}{x}{x}+\frac{1}{3}\opa{i}{y}{y}+\frac{1}{3}\opa{i}{z}{z}+\frac{2}{3}\; ,\\
\ops{i}{y}\ops{i}{y}&=-\frac{2}{3}\opa{i}{y}{y}+\frac{1}{3}\opa{i}{x}{x}+\frac{1}{3}\opa{i}{z}{z}+\frac{2}{3}\; ,\\
\ops{i}{z}\ops{i}{z}&=-\frac{2}{3}\opa{i}{z}{z}+\frac{1}{3}\opa{i}{y}{y}+\frac{1}{3}\opa{i}{x}{x}+\frac{2}{3}\; .
\end{align} \label{eq:SS_SI}
\end{subequations}
Using \Autoref{eq:SS_EP} and \Autoref{eq:SS_SI}, the total Hamiltonian [\Autoref{eq:Ham_EP_SI_FM}] in terms of the A-matrices then becomes
\begin{align*}
\Ham=\sum_{\nn{i,j}}&\left[-J^{xx}(\opa{i}{y}{z}-\opa{i}{z}{y})(\opa{j}{y}{z}-\opa{j}{z}{y}))\right.\\
&\left.+D^{zz}(-\frac{2}{3}\opa{i}{z}{z}+\frac{1}{3}\opa{i}{y}{y}+\frac{1}{3}\opa{i}{x}{x}+\frac{2}{3})\right]\; . \label{eq:HamU3_EP_SI}\numberthis
\end{align*}

If we define the basis change $Λ_3$ to be the basis change matrix between $ℬ_2$ and $ℬ^x$, such that  if a state $|ϕ⟩_ℬ_2$ is given in the TR invariant basis $ℬ_2$, in the basis $ℬ^x$ , its components are given by
\be
\ket{\phi}_{\mathcal{B}^x}= \Lambda_3\ket{\phi}_{\mathcal{B}_{2}}\; . \label{eq:Def_lambda3}
\ee
We found that the basis change matrix  $Λ_3$ yields
\be
\Lambda_3=\begin{pmatrix}
0 &  \frac{1}{\sqrt{2}} &  \frac{i}{\sqrt{2}}\\
-i & 0 & 0\\
0 & \frac{1}{\sqrt{2}} &- \frac{i}{\sqrt{2}}
\end{pmatrix}\; .\label{eq:lambda2}
\ee
An operator $O_ℬ_2$ given in the TR invariant basis $ℬ_2$
is expressed as
\be
\oph{O}_{\mathcal{B}^x}= \Lambda_3\oph{O}_{\mathcal{B}_{2}} \Lambda_3^{\dagger} \; , \label{eq:O_x_TR}
\ee
in the basis $ℬ^x$.

We will start working the basis $ℬ^x$, where everything is simple, since the ground state is one of the basis states and the orthogonal fluctuations can be expressed in terms of the other orthogonal basis states.
Indeed, the ground state matrix takes the simple form
\be
\begin{pmatrix}
1& 0 & 0 \\
0 & 0 & 0 \\
0& 0 & 0 \\
\end{pmatrix}\; ,\label{eq:A0}
\ee
since the ground state is simply the state  $|1^x⟩$ [\Autoref{eq:gs_x}]
or expressed in terms of director components
\be
{\textbf{d}^{\dagger}_0}_{\mathcal{B}^x}=\begin{pmatrix}1\\0\\0\end{pmatrix}\; .
\ee
We can generate orthogonal fluctuations by application of the exponential map given in \Autoref{eq:defR}.
The new state describing the fluctuations around the ground state is given by
\be
{\textbf{d}^{\dagger}}(\phi)=\hat{R}(\phi){\textbf{d}^{\dagger}_0}\; .
\ee
The A matrix  transforms according to \Autoref{eq:NewA}.
Only the generators $xx$, $xy$, $xz$, $yx$, and $zx$ will have non zero contribution when applied to the ground state matrix [\Autoref{eq:A0}]. \Autoref{fig:fluct} represents the action of the generators on the ground state. We can see, for example, that the generator $xy$, will create a fluctuation along $|0^x⟩$, i.e. an $$ boson, and will induce the new state to exhibit some quadrupolar features.

Using the constraint on the trace of A-matrices [\Autoref{eq:traceA.on.A.matrix}], we express the contribution from $xx$ in terms of the others components in order to ensure the length of the spin to be $S=1$ (which is equivalent to constraining the trace of A to be equal to 1), so that we properly restrict to $su(3)$ and make sure that we are correctly representing a spin-1. We obtain
\begin{align}
\textbf{A}(\phi)_{\mathcal{B}^x}&=\begin{pmatrix}
1 -\phi_{xy} \phi_{yx}-\phi_{xz} \phi_{zx}& i \phi_{xy} & i \phi_{xz} \\
-i \phi_{yx} & \phi_{xy} \phi_{yx} & \phi_{xz} \phi_{yx} \\
-i \phi_{zx} & \phi_{xy} \phi_{zx} & \phi_{xz} \phi_{zx} \\
\end{pmatrix}\; .
\end{align}
We can then easily introduce bosonic fluctuations  by
\begin{subequations}
\begin{align}
i \phi_{xy}&=\ba{}\; ,\\
-i \phi_{yx}&=\bad{}\; ,\\
i \phi_{xz} &= \bb{}\; ,\\
-i \phi_{zx} &= \bbd{}\; ,
\end{align}
\end{subequations}
such that we get
\be
\ophb{A}_{\mathcal{B}^x}=
\begin{pmatrix}
1-\bad{i}\ba{i}- \bbd{i}\bb{i}&\ba{i} & \bb{i}\\
\bad{i} &\bad{i}\ba{i} &\bad{i} \bb{i}\\
\bbd{i} & \ba{i}\bbd{i} & \bbd{i}\bb{i}
\end{pmatrix}\; .\label{eq:A_bos_tilda}
\ee
According to \Autoref{eq:O_x_TR}, the A matrices expressed in the TR invariant basis $ℬ_2$ are given by
\begin{widetext}
\begin{align*}
\ophb{A}_{B_{2}}&=\Lambda_3^{\dagger}\ophb{A}_{\mathcal{B}^x}\Lambda_3\\
\bad{i}\ba{i} &\frac{i}{\sqrt{2}} \bad{i} +\frac{i}{\sqrt{2}}\bad{i}\bb{i} &-\frac{1}{\sqrt{2}} \bad{i} +\frac{1}{\sqrt{2}}\bad{i}\bb{i}\\
-\frac{i}{\sqrt{2}} \ba{i} -\frac{i}{\sqrt{2}}\ba{i}\bbd{i} &  \frac{1}{2} + \frac{1}{2}\bbd{i}+ \frac{1}{2}\bb{i}- \frac{1}{2}\bad{i}\ba{i}  & \frac{i}{2}+ \frac{i}{2}\bbd{i}- \frac{i}{2}\bb{i}-\frac{i}{2}\bad{i}\ba{i}-i \bbd{i}\bb{i}\\
-\frac{1}{\sqrt{2}} \ba{i} +\frac{1}{\sqrt{2}}\ba{i}\bbd{i} &-\frac{i}{2}- \frac{i}{2}\bb{i}+ \frac{i}{2}\bbd{i}+ \frac{i}{2}\bad{i}\ba{i}+i \bbd{i}\bb{i} &  \frac{1}{2} - \frac{1}{2}\bbd{i}- \frac{1}{2}\bb{i}- \frac{1}{2}\bad{i}\ba{i}
\end{pmatrix}\; .\numberthis\label{eq:A_TR}
\end{align*}
\end{widetext}
Inserting \Autoref{eq:A_TR} into \Autoref{eq:SS_SI}, we get the single--ion terms in the function of the bosons
\begin{align*}
\ops{i}{x}\ops{i}{x}&=1-\bad{i}\ba{i}\; ,\\
\ops{i}{y}\ops{i}{y}&=\frac{1}{2}(1+\bad{i}\ba{i}-\bbd{i}-\bb{i})\; ,\numberthis \label{eq:SS_SI_boson}\\
\ops{i}{z}\ops{i}{z}&=\frac{1}{2}(1+\bad{i}\ba{i}+\bbd{i}+\bb{i})\; .
\end{align*}
We notice that if $D^yy$ is not equal to $D^zz$, then the Hamiltonian [\Autoref{eq:HamU3_EP_SI}] has single bosons terms, meaning that the state about which we expanded the fluctuations is not the ground state any more.
Therefore, to be consistent with the easy-plane FM order and the ground state [\Autoref{eq:gs_x}],  we choose
\be
D^{\perp}=D^{yy}=D^{zz}  \textrm{~with~} \norm{D^{\perp}}<\norm{D^{xx}}\; .
\ee

After inserting \Autoref{eq:A_TR} into the total Hamiltonian [\Autoref{eq:HamU3_EP_SI}], only keeping fluctuations
up to 2nd order, and performing a Fourier transform, the Hamiltonian [ \Autoref{eq:HamU3_EP_SI}] becomes
\begin{align*}
\Ham=&\frac{1}{2}\sum_{{\k}}\left[\begin{pmatrix}
\bad{{\k}}, &\ba{-{\k}}
\end{pmatrix}
\begin{pmatrix}
A_{\k} & B_{\k} \\
B_{\k}  & A_{\k} \\
\end{pmatrix}
\begin{pmatrix}
\ba{{\k}}\\
\bad{-{\k}}\\
\end{pmatrix}\right.\\
\bbd{{\k}}, &\bb{-{\k}}
\end{pmatrix}
\begin{pmatrix}
C_{\k} & 0 \\
0  & C_{\k} \\
\end{pmatrix}
\begin{pmatrix}
\bb{{\k}}\\
\bbd{-{\k}}\\
\end{pmatrix}\right]\\
&+\frac{1}{2}NzJ^{xx}+N(D^{xx}+D^{\perp})\; ,\numberthis
\label{eq:HamU3FT_SI}
\end{align*}
\be\begin{matrix}
A_{\k}=-J^{xx}z+\frac{1}{2}z(J^{yy}+J^{zz})\gamma(\k)+(D^{\perp}-D^{xx})\; ,\\
B_{\k}=\frac{1}{2}z(J^{zz}-J^{yy})\gamma(\k)\; ,\\
C_{\k}=-2zJ^{xx}\; .
\end{matrix}\label{eq:Def_A_B_C_FM}
\ee
Similarly to the FQ case, we need to solve an eigensystem  analogous to \Autoref{eq:quantum.eigensystem}. The dispersion relations for $$ and $$ can be found by imposing them to have bosonic commutation relations [\Autoref{eq:Uinv}], and diagonalizing
\be
\sigma_z 	\begin{pmatrix}
A_{\k} & B_{\k} \\
B_{\k}  & A_{\k} \\
\end{pmatrix}=
\begin{pmatrix}
1 & 0 \\
0  & -1 \\
\end{pmatrix}
\begin{pmatrix}
A_{\k} & B_{\k} \\
B_{\k}  & A_{\k} \\
\end{pmatrix}=
\begin{pmatrix}
A_{\k} & B_{\k} \\
-B_{\k}  & -A_{\k} \\
\end{pmatrix}\; ,
\ee
where the multiplication by $ σ_z$ imposes the bosonic commutation relations.
The eigenvalues $ϵ_$ are given by
\be
\begin{matrix}
\epsilon_{\k,1}=+\sqrt{A_{\k}^2-B_{\k}^2} &,~&\epsilon_{\k,2}=-\sqrt{A_{\k}^2-B_{\k}^2}
\end{matrix}\label{eq:omega_a_U3_SI}\; .
\ee
The dispersion relations for the $$ and $$ are obtained by diagonalizing
\be
\sigma_z 		\begin{pmatrix}
C_{\k} & 0 \\
0  & C_{\k} \\
\end{pmatrix}=
\begin{pmatrix}
C_{\k} & 0 \\
0  & -C_{\k} \\
\end{pmatrix}\; .
\ee
The eigenvalues $ϵ_$ are given by
\be
\begin{matrix}
\epsilon_{\k,3}=-C_{\k} &,~&\epsilon_{\k,4}=+C_{\k}
\end{matrix}\label{eq:omega_b_U3_SI}\; .
\ee
Because the coupling constants are negative, the physical results are
\be
\begin{matrix}
\epsilon_{\k,1}=+\sqrt{A_{\k}^2-B_{\k}^2} &,~&\epsilon_{\k,3}=-C_{\k}=2z\norm{J^{xx}}
\end{matrix}\label{eq:omega_2_U3_SI}\; ,
\ee
where $A_$, $ B_$, and  $C_$ are given in \Autoref{eq:Def_A_B_C_FM}.

Following the same procedure as for the FQ state in \Autoref{section:quantum.theory}, we calculate dynamical structure factors for the anisotropic FM case. We start by finding the Bogoliubov transformation that diagonalizes \Autoref{eq:HamU3FT_SI}. Following the steps given in \hyperref[sec:bogolioubov_transfomation]{Appendix~}\ref{sec:bogolioubov_transfomation}, we get
\begin{subequations}
\begin{align}
\ba{\k}{}=&\frac{1}{\sqrt{\Delta_{\k}^2- B_{\k}^2}}(\Delta_{\k}\balpha{\k}-B_{\k}\balphad{-\k})\; ,\label{eq:Bogo_FM_a_1}\\
\bad{-\k}{}=&\frac{1}{\sqrt{\Delta_{\k}^2- B_{\k}^2}}(-B_{\k}\balpha{\k}+\Delta_{\k}\balphad{-\k})\; ,\\
\bad{\k}{}=&\frac{1}{\sqrt{\Delta_{\k}^2- B_{\k}^2}}(\Delta_{\k}\balphad{\k}-B_{\k}\balpha{-\k})\; ,\\
\ba{-\k}{}=&\frac{1}{\sqrt{\Delta_{\k}^2- B_{\k}^2}}(-B_{\k}\balphad{\k}+\Delta_{\k}\balpha{-\k})\; , \label{eq:Bogo_FM_a_4}
\end{align}\label{eq:Bogo_FM_a}
\end{subequations}
\begin{subequations}
\begin{align}
\bb{\k}{}&=\bbeta{\k}\; ,\\
\bbd{-\k}{}&=\bbetad{-\k}\; ,\\
\bbd{\k}{}&=\bbetad{\k}\; ,\\
\bb{-\k}{}&=\bbeta{-\k}\; ,
\end{align}\label{eq:Bogo_FM_b}
\end{subequations}
where $Δ_$ is given in \Autoref{eq:Delta}, and where $A_$ and $ B_$ are given in \Autoref{eq:Def_A_B_C_FM}.

We follow now the calculations outlined in \Autoref{sec:structure_factors_quantum_qn0} in order to calculate the quantum structure factors.
Since we are working in the Bogoliubov representation, the ground state $|GS⟩$ is the vacuum state $|vac⟩$ for the Bogoliubov bosons. The structure factors are given by \Autoref{eq:SqOF}. We calculate $⟨vac|Ô_^α|μ⟩$ with \mbox{$\ket{\mu}= \balphad{\k}\ket{\rm{vac}}\oplus\bbetad{\k}\ket{\rm{vac}}$} and
$Ô_^α=α$ with $α=x,y,z$, for the dipole structure factor for instance.
Using \Autoref{eq:dipole.in.terms.of.A}, \Autoref{eq:quadrupole.in.terms.of.A} and \Autoref{eq:A_TR}, we can rewrite the spin dipole, the spin quadropole, and the A-matrix operators in terms of the bosons up to linear order, and after performing a Fourier transform, we can rewrite them in terms of the Bogoliubov bosons using \Autoref{eq:Bogo_FM_a} and \Autoref{eq:Bogo_FM_b}. This allows to easily calculate the structure factors [\Autoref{eq:SqOF}].

%    Fig. 24 - Easy Plane FM

\begin{figure*}[h]
\begin{center}
\begin{minipage}[b]{\textwidth}
\subfloat[$~~~~~~~~~~~S^{\sf QM}_{\rm{A}}(\q, \omega)~~~~~~~~~~~$ $\left(\right. J^{xx}=J^{yy}=J^{zz}\left.\right)$	 	\label{fig:Aqw_FM_iso}]{
\includegraphics[width=0.3\linewidth]{Aqw_FM_Iso.jpeg}
\subfloat[$~~~~~~~~~~~S^{\sf QM}_{\rm{Q}}(\q, \omega)~~~~~~~~~~~$ $\left(\right. J^{xx}=J^{yy}=J^{zz} \left.\right)$			\label{fig:Qqw_FM_iso}]{
\includegraphics[width=0.3\linewidth]{Qqw_FM_Iso.jpeg}
\subfloat[$~~~~~~~~~~~S^{\sf QM}_{\rm{S}}(\q, \omega)~~~~~~~~~~~$ $\left(\right. J^{xx}=J^{yy}=J^{zz} \left.\right)$			\label{fig:Sqw_FM_iso}]{
\includegraphics[width=0.3\linewidth]{Sqw_FM_Iso.jpeg}
\subfloat[$~~~~~~~~~~~S^{\sf QM}_{\rm{A}}(\q, \omega)~~~~~~~~~~~$ $\left(\right. J^{yy}=J^{zz}=0.8J^{xx} \left.\right)$			\label{fig:Sqw_FM_A_EP}]{
\includegraphics[width=0.3\linewidth]{Aqw_FM_EP_JyJz08.jpeg}
\subfloat[$~~~~~~~~~~~S^{\sf QM}_{\rm{Q}}(\q, \omega)~~~~~~~~~~~$ $\left(\right. J^{yy}=J^{zz}=0.8J^{xx} \left.\right)$			\label{fig:Sqw_FM_Q_EP}]{
\includegraphics[width=0.3\linewidth]{Qqw_FM_EP_JyJz08.jpeg}
\subfloat[$~~~~~~~~~~~~S^{\sf QM}_{\rm{S}}(\q, \omega)~~~~~~~~~~~~$ $ \left(\right.J^{yy}=J^{zz}=0.8J^{xx} \left.\right)$  		\label{fig:Sqw_FM_S_EP}]{
\includegraphics[width=0.3\linewidth]{Sqw_FM_EP_JyJz08.jpeg}
\subfloat[$~~~~~~~~~~~~~S^{\sf QM}_{\rm{A}}(\q, \omega)~~~~~~~~~~~~~$ $ \left(\right. D^{\perp}=0.5D^{x}\left.\right)$			\label{fig:Sqw_FM_A_SI}]{
\includegraphics[width=0.3\linewidth]{Aqw_FM_SI_Dx1_Dyz05.jpeg}
\subfloat[$~~~~~~~~~~~~~S^{\sf QM}_{\rm{Q}}(\q, \omega)~~~~~~~~~~~~~$ $ \left(\right. D^{\perp}=0.5D^{x}\left.\right)$			\label{fig:Sqw_FM_Q_SI}]{
\includegraphics[width=0.3\linewidth]{Qqw_FM_SI_Dx1_Dyz05.jpeg}
\subfloat[$~~~~~~~~~~~~~~S^{\sf QM}_{\rm{S}}(\q, \omega)~~~~~~~~~~~~~~$ $  \left(\right. D^{\perp}=0.5D^{x}\left.\right)$  		\label{fig:Sqw_FM_S_SI}]{
\includegraphics[width=0.3\linewidth]{Sqw_FM_SI_Dx1_Dyz05.jpeg}
\subfloat[$~~~~~~~~~~~~~S^{\sf QM}_{\rm{A}}(\q, \omega)~~~~~~~~~~~~~$ $  \left(\right.J^{yy}=D^{x}=J^{xx} \left.\right. ,$ $\left.\right.J^{zz}=D^{\perp}=0.8J^{xx}\left.\right)$
\label{fig:Sqw_FM_A_EP_SI}]{
\includegraphics[width=0.3\linewidth]{Aqw_FM_EP_SI_Jz08_Dyz08.jpeg}
\subfloat[$~~~~~~~~~~~~~S^{\sf QM}_{\rm{Q}}(\q, \omega)~~~~~~~~~~~~~$ $  \left(\right.J^{yy}=D^{x}=J^{xx}\left.\right. ,$ $\left.\right.J^{zz}=D^{\perp}=0.8J^{xx}\left.\right)$			\label{fig:Sqw_FM_Q_EP_SI}]{
\includegraphics[width=0.3\linewidth]{Qqw_FM_EP_SI_Jz08_Dyz08.jpeg}
\subfloat[$~~~~~~~~~~~~~S^{\sf QM}_{\rm{S}}(\q, \omega)~~~~~~~~~~~~~$ $ \left(\right.J^{yy}=D^{x}=J^{xx}\left.\right. ,$ $\left.\right.J^{zz}=D^{\perp}=0.8J^{xx}\left.\right)$
\label{fig:Sqw_FM_S_EP_SI}]{
\includegraphics[width=0.3\linewidth]{Sqw_FM_EP_SI_Jz08_Dyz08.jpeg}
\end{minipage}
\end{center}
\caption{
Dynamical structure factors obtained from zero-T quantum calculations
for $S_{\rm{A}}(\q, \omega)$ (A-matrices consisting of a mixture of dipolar and
quadrupolar moments), $S_{\rm{Q}}(\q, \omega)$  (quadrupolar moments) and
$S_{\rm{S}}(\q, \omega)$  (dipolar moments) for the ferromagnetic (FM) phase
of the BBQ model on the triangular lattice (\protect\Autoref{eq:HamA1}) with
$J_1$ being considered as Heisenberg anisotropic exchange interactions
[\protect\Autoref{eq:Ham_EP_FM_2}] and $J_2=0$, and with an additional single
ion anisotropic exchange Hamiltonian [\protect\Autoref{eq:Ham_SI_FM_2}].
\protect\subref{fig:Aqw_FM_iso}--\protect\subref{fig:Sqw_FM_iso}
Dynamical structure factors obtained for the isotropic FM state of the Heisenberg
Hamiltonian [\protect\Autoref{eq:Ham_EP_FM_2}] where $J^{xx}=J^{yy}=J^{zz}=-1$
without any single--ion anisotropy [\protect\Autoref{eq:Ham_SI_FM_2}], $D^{\perp}=D^{x}=0$.
\protect\subref{fig:Sqw_FM_A_EP}--\protect\subref{fig:Sqw_FM_S_EP}
Dynamical structure factors obtained for the easy-plane anisotropic FM state of the
Heisenberg Hamiltonian [\protect\Autoref{eq:Ham_EP_FM_2}] where
$J^{yy}=J^{zz}=0.8J^{xx}$ and $J^{xx}=-1$ without any single--ion anisotropy
[\protect\Autoref{eq:Ham_SI_FM_2}], $D^{\perp}=D^{x}=0$.
\protect\subref{fig:Sqw_FM_A_SI}--\protect\subref{fig:Sqw_FM_S_SI}
Dynamical structure factors obtained for the isotropic FM state of the
Heisenberg Hamiltonian [\protect\Autoref{eq:Ham_EP_FM_2}] where
$J^{yy}=J^{zz}=J^{xx}=-1$ with single--ion anisotropy [\protect\Autoref{eq:Ham_SI_FM_2}],
$D^{\perp}=0.5D^{x}$ and $D^{x}=J^{xx}$.
\protect\subref{fig:Sqw_FM_A_EP_SI}--\protect\subref{fig:Sqw_FM_S_EP_SI}
Dynamical structure factors obtained for the easy-plane anisotropic FM state
of the Heisenberg Hamiltonian [\protect\Autoref{eq:Ham_EP_FM_2}]
where $J^{zz}=0.8J^{xx}$ and $J^{yy}=J^{xx}=-1$  with single--ion anisotropy
[\protect\Autoref{eq:Ham_SI_FM_2}], $D^{\perp}=0.8D^{x}$ and $D^{x}=J^{xx}$.
\label{fig:comparison_dispersion_intensities_FM_ani}
\end{figure*}

Using \Autoref{eq:SO_qn0}, the dynamical spin dipole structure factor, defined by \Autoref{eq:SqSF_app}, is given by
\begin{align}
S^{\sf FM}_{\rm{S}}(\q, \omega)&=\frac{A_{\q}}{\sqrt{A_{\q}^2-B_{\q}^2}}\delta(\omega-\epsilon_{\q,1})+S^{\sf GS_{\sf{FM}}}_{\rm{S}}(\q=0, \omega)\; .\label{eq:Sqw_qm_FM}
\end{align}
%where we used \Autoref{eq:Delta}, and where $\omega_{\q,1}$ is given in \Autoref{eq:omega_2_U3_SI}
The dynamical spin quadrupole structure factor, as given by \Autoref{eq:dynamical.quadrupole.structure.factor_app}, yields
\begin{align*}
S^{\sf FM}_{\rm{Q}}(\q, \omega)&=2\frac{A_{\q}}{\sqrt{A_{\q}^2-B_{\q}^2}}\delta(\omega-\epsilon_{\q,1})+4\delta(\omega-\epsilon_{\q,3})\\
&+S^{\sf GS_{\sf{FM}}}_{\rm{Q}}(\q=0, \omega)\; .\numberthis\label{eq:Qqw_qm_FM}
\end{align*}

The total dynamical factor for the $$ operators defined in \Autoref{eq:dynamical.A.matrix.structure.factor_app} becomes
\begin{align*}
S_{\rm{A}}^{\sf FM}(\q,\omega)&=\frac{A_{\q}}{\sqrt{A_{\q}^2-B_{\q}^2}}\delta(\omega-\epsilon_{\q,1})+\delta(\omega-\epsilon_{\q,3})\\
&+S^{\sf GS_{\sf{FM}}}_{\rm{A}}(\q=0, \omega)\; ,\numberthis\label{eq:Aqw_qm_FM}
\end{align*}
where we explicitly summed over the indexes $α$ and $β$ and where the terms of the form $S^GS_FM_O(=0, ω)$ represent the ground state  and zero--point energy contribution to the structure factors at $=0$, but are not calculated here, for simplicity reasons.
For these 3 results, \Autoref{eq:Sqw_qm_FM}, \Autoref{eq:Qqw_qm_FM}, and \Autoref{eq:Aqw_qm_FM}, we used \Autoref{eq:Delta}, and $ϵ_,1$, and $ϵ_,3$ are given in \Autoref{eq:omega_2_U3_SI}.

We also check that the sum rule \Autoref{eq:sum.rule.AQS} is indeed satisfied after noticing that the constant terms in \Autoref{eq:sum.rule.AQS} would only contribute for $=0$ and at equal time, and can therefore be neglected.
These results are identical to results that one can obtain by performing a conventional multi-bosons expansion.

In \Autoref{fig:comparison_dispersion_intensities_FM_ani}, we show results for the dynamical structure factors [\Autoref{eq:Sqw_qm_FM}, \Autoref{eq:Qqw_qm_FM}, and \Autoref{eq:Aqw_qm_FM}] for the ferromagnetic state for the anisotropic Heisenberg Hamiltonian with single--ion anisotropy [\Autoref{eq:Ham_EP_SI_FM}] on the triangular lattice.
We first notice that the quadrupolar band $ϵ_,3$, which corresponds to the $ΔS=2$ excitation band associated with the $$ boson,  is gapped and non--dispersive. Because it essentially corresponds to the excitation obtained by applying the lowering operator $S^+$ twice, it is quadrupolar in nature and will only contribute to the quadrupolar structure factor channel. Moreover, such a quadrupolar excitation from a FM ground state has a finite energy cost, and it also doesn't have any neighboring quadrupoles to interact with, so it is therefore localized.
The isotropic FM Heisenberg case without single--ion anisotropy is presented in \Autoref{fig:comparison_dispersion_intensities_FM_ani}~\subref{fig:Aqw_FM_iso}--\subref{fig:Sqw_FM_iso}.
As shown in \Autoref{fig:comparison_dispersion_intensities_FM_ani}~\subref{fig:Sqw_FM_A_EP}--\subref{fig:Sqw_FM_S_EP}, we note that the introduction of easy--plane anisotropy with $J^yy=J^zz≠J^xx$ creates a gap and lifts the dispersion relation according to \Autoref{eq:omega_2_U3_SI} and \Autoref{eq:Def_A_B_C_FM}.
In \Autoref{fig:comparison_dispersion_intensities_FM_ani}~\subref{fig:Sqw_FM_A_SI}--\subref{fig:Sqw_FM_S_SI}, we see that introducing single--ion anisotropy with $D^⊥≠D^xx$ also creates a gap and lifts the dispersion relation again according to \Autoref{eq:omega_2_U3_SI} and \Autoref{eq:Def_A_B_C_FM}.
In \Autoref{fig:comparison_dispersion_intensities_FM_ani}~\subref{fig:Sqw_FM_A_EP_SI}--\subref{fig:Sqw_FM_S_EP_SI}, we display the interplay of easy-plane and single--ion anisotropy.

\newpage

\end{appendices}
\newpage
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\BibitemOpen
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10.1103/PhysRev.86.694} {\bibfield  {journal} {\bibinfo  {journal} {Phys.
Rev.}\ }\textbf {\bibinfo {volume} {86}},\ \bibinfo {pages} {694--701}
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\BibitemOpen
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exchange}},}\ }\href@noop {} {\bibfield  {journal} {\bibinfo  {journal}
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\BibitemOpen
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\BibitemOpen
\bibfield  {author} {\bibinfo {author} {\bibfnamefont {Mustansir}\
\bibnamefont {Barma}},\ }\bibfield  {title} {\enquote {\bibinfo {title}
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{\doibase 10.1103/PhysRevB.12.2710} {\bibfield  {journal} {\bibinfo
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Nematics}},}\ }\href
{http://www.jetp.ac.ru/cgi-bin/e/index/e/60/2/p267?a=list} {\bibfield
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\bibinfo {pages} {467--475} (\bibinfo {year} {1984})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Harada}\ and\ \citenamefont
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order in isotropic Heisenberg models with biquadratic interaction}},}\ }\href
{\doibase 10.1103/PhysRevB.65.052403} {\bibfield  {journal} {\bibinfo
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\bibitem [{\citenamefont {Tsunetsugu}\ and\ \citenamefont
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{\bibinfo {title} {{Spin Nematic Phase in S=1 Triangular
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\bibitem [{\citenamefont {L\"auchli}\ \emph {et~al.}(2006)\citenamefont
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}\bibfield  {title} {\enquote {\bibinfo {title} {{Exotic $S=1$ spin-liquid
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}\bibfield  {title} {\enquote {\bibinfo {title} {{Paired chiral spin liquid
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{\doibase 10.1103/PhysRevB.86.224409} {\bibfield  {journal} {\bibinfo
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\bibitem [{\citenamefont {Xu}\ \emph {et~al.}(2012)\citenamefont {Xu},
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\BibitemOpen
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{\bibfnamefont {Matthew P.~A.}\ \bibnamefont {Fisher}},\ }\bibfield  {title}
{\enquote {\bibinfo {title} {Spin liquid phases for spin-1 systems on the
triangular lattice},}\ }\href {\doibase 10.1103/PhysRevLett.108.087204}
{\bibfield  {journal} {\bibinfo  {journal} {Phys. Rev. Lett.}\ }\textbf
{\bibinfo {volume} {108}},\ \bibinfo {pages} {087204} (\bibinfo {year}
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}\bibfield  {title} {\enquote {\bibinfo {title} {{Frustrated Quantum Critical
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}\href {\doibase 10.1103/PhysRevLett.109.016402} {\bibfield  {journal}
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\bibinfo {pages} {016402} (\bibinfo {year} {2012})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Hwang}\ \emph {et~al.}(2013)\citenamefont {Hwang},
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and\ \bibinfo {author} {\bibfnamefont {Yong~Baek}\ \bibnamefont {Kim}},\
}\bibfield  {title} {\enquote {\bibinfo {title} {Three-dimensional nematic
spin liquid in a stacked triangular lattice 6$h$-b structure},}\ }\href
{\doibase 10.1103/PhysRevB.87.235103} {\bibfield  {journal} {\bibinfo
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\bibitem [{\citenamefont {Buessen}\ \emph {et~al.}(2018)\citenamefont
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{Trebst}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Quantum spin
liquids in frustrated spin-1 diamond antiferromagnets},}\ }\href {\doibase
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{057201} (\bibinfo {year} {2018})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Nakatsuji}\ \emph {et~al.}(2005)\citenamefont
{Nakatsuji}, \citenamefont {Nambu}, \citenamefont {Tonomura}, \citenamefont
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{Tsunetsugu}, \citenamefont {Qiu},\ and\ \citenamefont
\BibitemOpen
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\bibinfo {author} {\bibfnamefont {Seth}\ \bibnamefont {Jonas}}, \bibinfo
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{\bibfnamefont {Hirokazu}\ \bibnamefont {Tsunetsugu}}, \bibinfo {author}
{\bibfnamefont {Yiming}\ \bibnamefont {Qiu}}, \ and\ \bibinfo {author}
{\bibfnamefont {Yoshiteru}\ \bibnamefont {Maeno}},\ }\bibfield  {title}
{\enquote {\bibinfo {title} {Spin disorder on a triangular lattice},}\ }\href
{\doibase 10.1126/science.1114727} {\bibfield  {journal} {\bibinfo  {journal}
{Science}\ }\textbf {\bibinfo {volume} {309}},\ \bibinfo {pages} {1697--1700}
(\bibinfo {year} {2005})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Nambu}\ \emph {et~al.}(2006)\citenamefont {Nambu},
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\bibnamefont {Maeno}},\ }\bibfield  {title} {\enquote {\bibinfo {title}
{{Coherent Behavior and Nonmagnetic Impurity Effects of Spin Disordered State
in ${\mathrm{NiGa}}_{2}{\mathrm{S}}_{4}$}},}\ }\href
{https://doi.org/10.1143/JPSJ.75.043711} {\bibfield  {journal} {\bibinfo
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{volume} {75}},\ \bibinfo {pages} {043711} (\bibinfo {year}
{2006})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Bhattacharjee}\ \emph {et~al.}(2006)\citenamefont
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\BibitemOpen
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{Shenoy}}, \ and\ \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
{Senthil}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {{Possible
ferro-spin nematic order in
$\mathrm{Ni}{\mathrm{Ga}}_{2}{\mathrm{S}}_{4}$}},}\ }\href {\doibase
10.1103/PhysRevB.74.092406} {\bibfield  {journal} {\bibinfo  {journal} {Phys.
Rev. B}\ }\textbf {\bibinfo {volume} {74}},\ \bibinfo {pages} {092406}
(\bibinfo {year} {2006})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Valentine}\ \emph {et~al.}(2020)\citenamefont
{Valentine}, \citenamefont {Higo}, \citenamefont {Nambu}, \citenamefont
{Chaudhuri}, \citenamefont {Wen}, \citenamefont {Broholm}, \citenamefont
{Nakatsuji},\ and\ \citenamefont {Drichko}}]{Valentine2020}%
\BibitemOpen
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\bibnamefont {Valentine}}, \bibinfo {author} {\bibfnamefont {Tomoya}\
\bibnamefont {Higo}}, \bibinfo {author} {\bibfnamefont {Yusuke}\ \bibnamefont
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{Chaudhuri}}, \bibinfo {author} {\bibfnamefont {Jiajia}\ \bibnamefont {Wen}},
\bibinfo {author} {\bibfnamefont {Collin}\ \bibnamefont {Broholm}}, \bibinfo
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{title} {\enquote {\bibinfo {title} {{Impact of the Lattice on Magnetic
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Antiferromagnet ${\mathrm{NiGa}}_{2}{\mathrm{S}}_{4}$}},}\ }\href {\doibase
10.1103/PhysRevLett.125.197201} {\bibfield  {journal} {\bibinfo  {journal}
{Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {125}},\ \bibinfo {pages}
{197201} (\bibinfo {year} {2020})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Gao}\ \emph {et~al.}(2020)\citenamefont {Gao},
\citenamefont {Yao}, \citenamefont {Li},\ and\ \citenamefont
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\bibfield  {author} {\bibinfo {author} {\bibfnamefont {Yong-Hao}\
\bibnamefont {Gao}}, \bibinfo {author} {\bibfnamefont {Xu-Ping}\ \bibnamefont
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\bibinfo {author} {\bibfnamefont {Gang}\ \bibnamefont {Chen}},\ }\bibfield
{title} {\enquote {\bibinfo {title} {Spin-1 pyrochlore antiferromagnets:
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10.1007/s11467-020-0974-4} {\bibfield  {journal} {\bibinfo  {journal}
{Frontiers of Physics}\ }\textbf {\bibinfo {volume} {15}},\ \bibinfo {pages}
{63201} (\bibinfo {year} {2020})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Plumb}\ \emph {et~al.}(2019)\citenamefont {Plumb},
\citenamefont {Changlani}, \citenamefont {Scheie}, \citenamefont {Zhang},
\citenamefont {Krizan}, \citenamefont {Rodriguez-Rivera}, \citenamefont
{Qiu}, \citenamefont {Winn}, \citenamefont {Cava},\ and\ \citenamefont
\BibitemOpen
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\bibinfo {author} {\bibfnamefont {Shu}\ \bibnamefont {Zhang}}, \bibinfo
{author} {\bibfnamefont {J.~W.}\ \bibnamefont {Krizan}}, \bibinfo {author}
{\bibfnamefont {J.~A.}\ \bibnamefont {Rodriguez-Rivera}}, \bibinfo {author}
{\bibfnamefont {Yiming}\ \bibnamefont {Qiu}}, \bibinfo {author}
{\bibfnamefont {B.}~\bibnamefont {Winn}}, \bibinfo {author} {\bibfnamefont
{R.~J.}\ \bibnamefont {Cava}}, \ and\ \bibinfo {author} {\bibfnamefont
{C.~L.}\ \bibnamefont {Broholm}},\ }\bibfield  {title} {\enquote {\bibinfo
{title} {{Continuum of quantum fluctuations in a three-dimensional S = 1
Heisenberg magnet}},}\ }\href {\doibase 10.1038/s41567-018-0317-3} {\bibfield
{journal} {\bibinfo  {journal} {Nature Physics}\ }\textbf {\bibinfo {volume}
{15}},\ \bibinfo {pages} {54--59} (\bibinfo {year} {2019})}\BibitemShut
\bibitem [{\citenamefont {Zhang}\ \emph {et~al.}(2019)\citenamefont {Zhang},
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{Tchernyshyov},\ and\ \citenamefont {Moessner}}]{Zhang2019}%
\BibitemOpen
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{Plumb}}, \bibinfo {author} {\bibfnamefont {Oleg}\ \bibnamefont
{Tchernyshyov}}, \ and\ \bibinfo {author} {\bibfnamefont {Roderich}\
\bibnamefont {Moessner}},\ }\bibfield  {title} {\enquote {\bibinfo {title}
{{Dynamical Structure Factor of the Three-Dimensional Quantum Spin Liquid
Candidate ${\mathrm{NaCaNi}}_{2}{\mathrm{F}}_{7}$}},}\ }\href {\doibase
10.1103/PhysRevLett.122.167203} {\bibfield  {journal} {\bibinfo  {journal}
{Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {122}},\ \bibinfo {pages}
{167203} (\bibinfo {year} {2019})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Chamorro}\ \emph {et~al.}(2018)\citenamefont
{Chamorro}, \citenamefont {Ge}, \citenamefont {Flynn}, \citenamefont
{Subramanian}, \citenamefont {Mourigal},\ and\ \citenamefont
\BibitemOpen
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\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Flynn}}, \bibinfo
{author} {\bibfnamefont {M.~A.}\ \bibnamefont {Subramanian}}, \bibinfo
{author} {\bibfnamefont {M.}~\bibnamefont {Mourigal}}, \ and\ \bibinfo
{author} {\bibfnamefont {T.~M.}\ \bibnamefont {McQueen}},\ }\bibfield
{title} {\enquote {\bibinfo {title} {{Frustrated spin one on a diamond
lattice in ${\mathrm{NiRh}}_{2}{\mathrm{O}}_{4}$}},}\ }\href {\doibase
10.1103/PhysRevMaterials.2.034404} {\bibfield  {journal} {\bibinfo  {journal}
{Phys. Rev. Materials}\ }\textbf {\bibinfo {volume} {2}},\ \bibinfo {pages}
{034404} (\bibinfo {year} {2018})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Miiller}\ \emph {et~al.}(2011)\citenamefont
{Miiller}, \citenamefont {Christensen}, \citenamefont {Khan}, \citenamefont
{Sharma}, \citenamefont {Macquart}, \citenamefont {Avdeev}, \citenamefont
{McIntyre}, \citenamefont {Piltz},\ and\ \citenamefont {Ling}}]{Miiller2011}%
\BibitemOpen
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\bibnamefont {Miiller}}, \bibinfo {author} {\bibfnamefont {Mogens}\
\bibnamefont {Christensen}}, \bibinfo {author} {\bibfnamefont {Arfhan}\
\bibnamefont {Khan}}, \bibinfo {author} {\bibfnamefont {Neeraj}\ \bibnamefont
{Sharma}}, \bibinfo {author} {\bibfnamefont {Ren{\'e}~B.}\ \bibnamefont
{Macquart}}, \bibinfo {author} {\bibfnamefont {Maxim}\ \bibnamefont
{Avdeev}}, \bibinfo {author} {\bibfnamefont {Garry~J.}\ \bibnamefont
{McIntyre}}, \bibinfo {author} {\bibfnamefont {Ross~O.}\ \bibnamefont
{Piltz}}, \ and\ \bibinfo {author} {\bibfnamefont {Chris~D.}\ \bibnamefont
{Ling}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {{YCa3(VO)3(BO3)4:
A Kagom{\'e} Compound Based on Vanadium(III) with a Highly Frustrated Ground
State}},}\ }\bibfield  {booktitle} {\emph {\bibinfo {booktitle} {Chemistry of
Materials}},\ }\href {\doibase 10.1021/cm1034003} {\bibfield  {journal}
{\bibinfo  {journal} {Chemistry of Materials}\ }\textbf {\bibinfo {volume}
{23}},\ \bibinfo {pages} {1315--1322} (\bibinfo {year} {2011})}\BibitemShut
\bibitem [{\citenamefont {Cheng}\ \emph {et~al.}(2011)\citenamefont {Cheng},
\citenamefont {Li}, \citenamefont {Balicas}, \citenamefont {Zhou},
\citenamefont {Goodenough}, \citenamefont {Xu},\ and\ \citenamefont
\BibitemOpen
\bibfield  {author} {\bibinfo {author} {\bibfnamefont {J.~G.}\ \bibnamefont
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{author} {\bibfnamefont {L.}~\bibnamefont {Balicas}}, \bibinfo {author}
{\bibfnamefont {J.~S.}\ \bibnamefont {Zhou}}, \bibinfo {author}
{\bibfnamefont {J.~B.}\ \bibnamefont {Goodenough}}, \bibinfo {author}
{\bibfnamefont {Cenke}\ \bibnamefont {Xu}}, \ and\ \bibinfo {author}
{\bibfnamefont {H.~D.}\ \bibnamefont {Zhou}},\ }\bibfield  {title} {\enquote
{\bibinfo {title} {{High-Pressure Sequence of
${\mathrm{Ba}}_{3}{\mathrm{NiSb}}_{2}{\mathrm{O}}_{9}$ Structural Phases: New
$S=1$ Quantum Spin Liquids Based on ${\mathrm{Ni}}^{2+}$}},}\ }\href
{\doibase 10.1103/PhysRevLett.107.197204} {\bibfield  {journal} {\bibinfo
{journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {107}},\ \bibinfo
{pages} {197204} (\bibinfo {year} {2011})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Quilliam}\ \emph {et~al.}(2016)\citenamefont
{Quilliam}, \citenamefont {Bert}, \citenamefont {Manseau}, \citenamefont
{Darie}, \citenamefont {Guillot-Deudon}, \citenamefont {Payen}, \citenamefont
{Baines}, \citenamefont {Amato},\ and\ \citenamefont
\BibitemOpen
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{author} {\bibfnamefont {C.}~\bibnamefont {Darie}}, \bibinfo {author}
{\bibfnamefont {C.}~\bibnamefont {Guillot-Deudon}}, \bibinfo {author}
{\bibfnamefont {C.}~\bibnamefont {Payen}}, \bibinfo {author} {\bibfnamefont
{C.}~\bibnamefont {Baines}}, \bibinfo {author} {\bibfnamefont
{A.}~\bibnamefont {Amato}}, \ and\ \bibinfo {author} {\bibfnamefont
{P.}~\bibnamefont {Mendels}},\ }\bibfield  {title} {\enquote {\bibinfo
{title} {{Gapless quantum spin liquid ground state in the spin-1
6HB-${\mathrm{Ba}}_{3}{\mathrm{NiSb}}_{2}{\mathrm{O}}_{9}$}},}\ }\href
{\doibase 10.1103/PhysRevB.93.214432} {\bibfield  {journal} {\bibinfo
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\bibitem [{\citenamefont {Fak}\ \emph {et~al.}(2017)\citenamefont {Fak},
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}\bibfield  {title} {\enquote {\bibinfo {title} {Spin and quadrupolar orders
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}\href {\doibase 10.1103/PhysRevB.93.235148} {\bibfield  {journal} {\bibinfo
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}\bibfield  {title} {\enquote {\bibinfo {title} {{Possible nematic spin
liquid in spin-1 antiferromagnetic system on the square lattice: Implications
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atoms in optical lattices: Symmetry breaking and fractionalization},}\ }\href
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{Demler}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Spin-exchange
interactions of spin-one bosons in optical lattices: Singlet, nematic, and
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\bibitem [{\citenamefont {Stamper-Kurn}\ and\ \citenamefont
\BibitemOpen
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\bibnamefont {Ueda}},\ }\bibfield  {title} {\enquote {\bibinfo {title}
{Spinor bose gases: Symmetries, magnetism, and quantum dynamics},}\ }\href
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\bibitem [{\citenamefont {de~Forges~de Parny}\ \emph
{et~al.}(2014)\citenamefont {de~Forges~de Parny}, \citenamefont {Yang},\ and\
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\bibnamefont {Mila}},\ }\bibfield  {title} {\enquote {\bibinfo {title}
{{Anderson Tower of States and Nematic Order of Spin-1 Bosonic Atoms on a 2D
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{2014})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Zibold}\ \emph {et~al.}(2016)\citenamefont {Zibold},
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{\bibfnamefont {F.}~\bibnamefont {Gerbier}},\ }\bibfield  {title} {\enquote
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condensates},}\ }\href {\doibase 10.1103/PhysRevA.93.023614} {\bibfield
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\bibitem [{\citenamefont {White}\ and\ \citenamefont
\BibitemOpen
\bibfield  {author} {\bibinfo {author} {\bibfnamefont {Steven~R.}\
\bibnamefont {White}}\ and\ \bibinfo {author} {\bibfnamefont {Ian}\
\bibnamefont {Affleck}},\ }\bibfield  {title} {\enquote {\bibinfo {title}
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}\href {\doibase 10.1103/PhysRevB.77.134437} {\bibfield  {journal} {\bibinfo
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{pages} {134437} (\bibinfo {year} {2008})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Zhao}\ \emph {et~al.}(2012)\citenamefont {Zhao},
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{T.}~\bibnamefont {Xiang}},\ }\bibfield  {title} {\enquote {\bibinfo {title}
{{Plaquette order and deconfined quantum critical point in the spin-1
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{\doibase 10.1103/PhysRevB.85.134416} {\bibfield  {journal} {\bibinfo
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\bibfield  {author} {\bibinfo {author} {\bibfnamefont {Ido}\ \bibnamefont
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{Corboz}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {{A tensor
network study of the complete ground state phase diagram of the spin-1
bilinear-biquadratic Heisenberg model on the square lattice}},}\ }\href
{\doibase 10.21468/SciPostPhys.3.4.030} {\bibfield  {journal} {\bibinfo
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\BibitemOpen
\bibfield  {author} {\bibinfo {author} {\bibfnamefont {Ribhu~K.}\
\bibnamefont {Kaul}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {{Spin
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}\href {\doibase 10.1103/PhysRevB.86.104411} {\bibfield  {journal} {\bibinfo
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\bibitem [{\citenamefont {V\"oll}\ and\ \citenamefont
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\bibfield  {author} {\bibinfo {author} {\bibfnamefont {Annika}\ \bibnamefont
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{Wessel}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {{Spin dynamics
of the bilinear-biquadratic $S=1$ Heisenberg model on the triangular lattice:
A quantum Monte Carlo study}},}\ }\href {\doibase 10.1103/PhysRevB.91.165128}
{\bibfield  {journal} {\bibinfo  {journal} {Phys. Rev. B}\ }\textbf {\bibinfo
{volume} {91}},\ \bibinfo {pages} {165128} (\bibinfo {year}
{2015})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Stoudenmire}\ \emph {et~al.}(2009)\citenamefont
{Stoudenmire}, \citenamefont {Trebst},\ and\ \citenamefont
\BibitemOpen
\bibfield  {author} {\bibinfo {author} {\bibfnamefont {E.~M.}\ \bibnamefont
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{Balents}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {{Quadrupolar
correlations and spin freezing in $S=1$ triangular lattice
antiferromagnets}},}\ }\href {\doibase 10.1103/PhysRevB.79.214436} {\bibfield
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\bibitem [{\citenamefont {Bloch}(1946)}]{Bloch1946}%
\BibitemOpen
\bibfield  {author} {\bibinfo {author} {\bibfnamefont {F.}~\bibnamefont
{Bloch}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Nuclear
induction},}\ }\href {\doibase 10.1103/PhysRev.70.460} {\bibfield  {journal}
{\bibinfo  {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {70}},\
\bibinfo {pages} {460--474} (\bibinfo {year} {1946})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Penc}\ and\ \citenamefont
\BibitemOpen
\bibfield  {author} {\bibinfo {author} {\bibfnamefont {Karlo}\ \bibnamefont
{Penc}}\ and\ \bibinfo {author} {\bibfnamefont {Andreas~M.}\ \bibnamefont
{L{\"a}uchli}},\ }\enquote {\bibinfo {title} {Spin nematic phases in quantum
spin systems},}\ in\ \href {\doibase 10.1007/978-3-642-10589-0_13} {\emph
{\bibinfo {booktitle} {Introduction to Frustrated Magnetism: Materials,
Experiments, Theory}}},\ \bibinfo {editor} {edited by\ \bibinfo {editor}
{\bibfnamefont {Claudine}\ \bibnamefont {Lacroix}}, \bibinfo {editor}
{\bibfnamefont {Philippe}\ \bibnamefont {Mendels}}, \ and\ \bibinfo {editor}
{\bibfnamefont {Fr{\'e}d{\'e}ric}\ \bibnamefont {Mila}}}\ (\bibinfo
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\bibfield  {author} {\bibinfo {author} {\bibfnamefont {Claude}\ \bibnamefont
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{Zuber}},\ }\href@noop {} {\emph {\bibinfo {title} {Quantum field theory /
Claude Itzykson and Jean-Bernard Zuber}}}\ (\bibinfo  {publisher}
{McGraw-Hill International Book Co New York},\ \bibinfo {year} {1980})\ pp.\
\bibinfo {pages} {xi, 515 p. :}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Balla}(2014)}]{Balla2014-Thesis}%
\BibitemOpen
\bibfield  {author} {\bibinfo {author} {\bibfnamefont {Peter}\ \bibnamefont
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(\bibinfo {year} {2014})\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Remund}(2015)}]{Remund2015-Thesis}%
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\bibfield  {author} {\bibinfo {author} {\bibfnamefont {Kimberly}\
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\bibinfo  {school} {Ecole Polytechnique Federal Lausanne} (\bibinfo {year}
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\bibitem [{\citenamefont {Zhang}\ and\ \citenamefont
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{Zhang}}\ and\ \bibinfo {author} {\bibfnamefont {Cristian~D.}\ \bibnamefont
{Batista}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Classical spin
dynamics based on $\mathrm{SU}(n)$ coherent states},}\ }\href {\doibase
10.1103/PhysRevB.104.104409} {\bibfield  {journal} {\bibinfo  {journal}
{Phys. Rev. B}\ }\textbf {\bibinfo {volume} {104}},\ \bibinfo {pages}
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\bibitem [{\citenamefont {Ueda}\ \emph {et~al.}(2016)\citenamefont {Ueda},
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{Akagi}}, \ and\ \bibinfo {author} {\bibfnamefont {Nic}\ \bibnamefont
{Shannon}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Quantum
solitons with emergent interactions in a model of cold atoms on the
triangular lattice},}\ }\href {\doibase 10.1103/PhysRevA.93.021606}
{\bibfield  {journal} {\bibinfo  {journal} {Phys. Rev. A}\ }\textbf {\bibinfo
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\bibitem [{\citenamefont {Landau}\ and\ \citenamefont
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\bibfield  {author} {\bibinfo {author} {\bibfnamefont {L.~D.}\ \bibnamefont
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{Lifshitz}},\ }\href@noop {} {\emph {\bibinfo {title} {Quantum Mechanics
(Non-relativistic Theory}}},\ \bibinfo {edition} {3rd}\ ed.\ (\bibinfo
{publisher} {Butterworth--Heineman, Oxford},\ \bibinfo {year} {1977})\ p.\
\bibinfo {pages} {p199}\BibitemShut {NoStop}%
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\BibitemOpen
\bibfield  {author} {\bibinfo {author} {\bibfnamefont {A}~\bibnamefont
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\bibitem [{\citenamefont {Landau}\ and\ \citenamefont
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\bibfield  {author} {\bibinfo {author} {\bibfnamefont {David~P.}\
\bibnamefont {Landau}}\ and\ \bibinfo {author} {\bibfnamefont {Kurt}\
\bibnamefont {Binder}},\ }\href {\doibase 10.1017/CBO9781139696463} {\emph
{\bibinfo {title} {A Guide to Monte Carlo Simulations in Statistical
Physics}}},\ \bibinfo {edition} {4th}\ ed.\ (\bibinfo  {publisher} {Cambridge
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\bibitem [{\citenamefont {Moessner}\ and\ \citenamefont
\BibitemOpen
\bibfield  {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
{Moessner}}\ and\ \bibinfo {author} {\bibfnamefont {J.~T.}\ \bibnamefont
{Chalker}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {{Properties of
a Classical Spin Liquid: The Heisenberg Pyrochlore Antiferromagnet}},}\
}\href {http://link.aps.org/doi/10.1103/PhysRevLett.80.2929} {\bibfield
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\bibitem [{\citenamefont {Pohle}\ \emph {et~al.}(2021)\citenamefont {Pohle},
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{title} {\enquote {\bibinfo {title} {{Theory of
${\mathrm{Ca}}_{10}{\mathrm{Cr}}_{7}{\mathrm{O}}_{28}$ as a bilayer
breathing-kagome magnet: Classical thermodynamics and semiclassical
dynamics}},}\ }\href {\doibase 10.1103/PhysRevB.104.024426} {\bibfield
{journal} {\bibinfo  {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume}
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\bibitem [{\citenamefont {Gilbert}(2004)}]{Gilbert2004}%
\BibitemOpen
\bibfield  {author} {\bibinfo {author} {\bibfnamefont {T.L.}\ \bibnamefont
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{\doibase 10.1109/TMAG.2004.836740} {\bibfield  {journal} {\bibinfo
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\bibitem [{\citenamefont {Nelson}(1967)}]{Nelson1967}%
\BibitemOpen
\bibfield  {author} {\bibinfo {author} {\bibfnamefont {T.~J.}\ \bibnamefont
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harmonic functions for the group su(3) as specialized matrix elements of a
general finite transformation},}\ }\href {\doibase 10.1063/1.1705289}
{\bibfield  {journal} {\bibinfo  {journal} {Journal of Mathematical Physics}\
}\textbf {\bibinfo {volume} {8}},\ \bibinfo {pages} {857--863} (\bibinfo
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{Wang}},\ }\enquote {\bibinfo {title} {Lie groups},}\ in\ \href {\doibase
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