## 2 Model and Method
The magnetization is observed from  
\[M=\langle S^{z}\rangle=\frac{1}{N}\sum_{ij}\Big{[}(S+0.5)\coth[(S+0.5)\beta E_ {ij})]-0.5\coth(0.5\beta E_{ij})\Big{]}. \tag{5}\]  
Here, \(S\) denotes the spin quantum number, \(\beta=1/k_{B}T\). \(E_{ij}\) represents the spin excitations obtained from the spin Green's function \(G_{ij}~{}=~{}\ll S_{i}^{+};S_{j}^{-}\gg\), calculated using the method of [PERSON] [25]. The following is a brief description. After a formal integration of the equation of motion for the retarding two-time GF  
\[G_{ij}(t)=\langle\langle S_{i}^{+}(t);S_{j}^{-}\rangle\rangle \tag{6}\]  
one obtains  
\[G_{ij}(t)=-i\theta(t)\langle[S_{i}^{+};S_{j}^{-}]\rangle\exp(-iE_{ij}(t)t), \tag{7}\]with \(\theta(x)=1\) for \(x>0\), \(\theta(x)=0\) for \(x<0\),  
\[E_{ij}(t)=E_{ij} - \frac{i}{t}\int_{0}^{t}dt^{\prime}t^{\prime}\Bigg{(}\frac{\langle[j_ {i}(t);j_{j}^{+}(t^{\prime})]\rangle}{\langle[S_{i}^{+}(t);S_{j}^{-}(t^{\prime} )]\rangle} \tag{8}\] \[- \frac{\langle[j_{i}(t);S_{j}^{-}(t^{\prime})]\rangle\langle[S_{i} ^{+}(t);j_{j}^{+}(t^{\prime})]\rangle}{\langle[S_{i}^{+}(t);S_{j}^{-}(t^{ \prime})]\rangle^{2}}\Bigg{)}\]  
and \(j_{i}(t)=\langle[S_{i}^{+}(t),H_{interaction}]\rangle\). The time-independent term  
\[H_{sp} = -\sum_{ij}(1-x)J_{1 ij}\mathbf{S}_{i}\cdot\mathbf{S}_{j}-\sum_{ij }xJ_{dij}\mathbf{S}_{i}\cdot\mathbf{S}_{j}-\sum_{il}J_{2 il}\mathbf{S}_{i}\cdot \mathbf{S}_{l} \tag{2}\]0  
is the excitation energy in the generalized Hartree-Fock approximation. The time-dependent term in Equation (8) includes damping effects. In CFO with easy-axis magnetic anisotropy, the observed ferroelectric coupling cannot be accounted for by the inverse Dzyaloshinskii-Moriya mechanism [26] or by the standard spin-current model [27; 28]. [PERSON] [29] proposed that in crystals exhibiting triclinic, monoclinic, or trigonal symmetry, ferroelectricity may arise from proper screw-type magnetic ordering via modulation of metal-ligand hybridization influenced by spin-orbit coupling. In the case of CFO, Fe\({}^{3+}\) ions possess three 3d electrons occupying a narrow, fully filled 3d t\({}_{2g}\) band in a high-spin configuration. According to [PERSON] [29], the d-p hybridization-altered by spin-orbit interactions in partially filled t\({}_{2g}\) orbitals can generate a microscopic electric polarization aligned along the axis of the magnetic helicoid. This form of local polarization can be expressed as follows:  
\[H_{sp} = -\sum_{ij}(1-x)J_{1 ij}\mathbf{S}_{i}\cdot\mathbf{S}_{j}-\sum_{ij }xJ_{dij}\mathbf{S}_{i}\cdot\mathbf{S}_{j}-\sum_{il}J_{2 il}\mathbf{S}_{i}\cdot \mathbf{S}_{l} \tag{2}\]1  
\(\mathbf{r}_{12}\) denotes the unit vector connecting adjacent ions \(i\) and \(j\). This framework effectively accounts for the emergence of ferroelectricity in systems such as CuFe\({}_{1-x}\)Al\({}_{x}\)O\({}_{2}\) (CFAO), which exhibits a proper screw-type magnetic order, and is likewise applicable to CFO. According to Arima's mechanism, a reversal in the helicity of the magnetic screw structure leads to a corresponding reversal in the electric polarization--a phenomenon that has been experimentally verified by CFAO [30]. Consequently, the ferroelectric behavior observed in the triangular-lattice antiferromagnet CFO holds promise for applications involving the detection and manipulation of spin chirality. Upon rotating the system about the \(c\)-axis by \(60^{0}\), such that the \(a\)-axis aligns with the [110] crystallographic direction, the unit vector \(\mathbf{e}_{ij}\) as well as the spin vectors \(\mathbf{S}_{i}\) and \(\mathbf{S}_{j}\), adopt the following coordinate representations:  
\[H_{sp} = -\sum_{ij}(1-x)J_{1 ij}\mathbf{S}_{i}\cdot\mathbf{S}_{j}-\sum_{ij }xJ_{dij}\mathbf{S}_{i}\cdot\mathbf{S}_{j}-\sum_{il}J_{2 il}\mathbf{S}_{i}\cdot \mathbf{S}_{l} \tag{2}\]2  
In the new coordinate system, the spins associated with the helical magnetic structure now reside within the \(xz\) plane. This configuration facilitates a more straightforward evaluation of the polarization using Equation (10).