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+Linear spin-wave theory: dipole mode +\end_layout + +\begin_layout Author +Hao +\begin_inset space ~ +\end_inset + +Zhang, and Cristian +\begin_inset space ~ +\end_inset + +D. +\begin_inset space ~ +\end_inset + +Batista +\end_layout + +\begin_layout Section +Introduction +\end_layout + +\begin_layout Standard +In this note, we explain the implementation of the dipole mode of Sunny's + linear spin-wave theory (LSWT) module. + In particular, we will focus on the construction of the SWT Hamiltonian + in +\emph on +real +\emph default + space. + The remaining steps, Fourier transformation, SWT Hamiltonian's diagonalization + (Bogoliubov transformation), and the calculation of the dynamical spin + structure factor can be found in a separate note that explains the implementati +on of the SU( +\begin_inset Formula $N$ +\end_inset + +) mode. +\end_layout + +\begin_layout Standard +Like the well-known spin dynamics package +\family typewriter +spinW +\family default +, the starting point of Sunny's LSWT dipole mode is the Holstein-Primakoff + expansion, +\begin_inset Formula +\begin{align} +\tilde{S}_{i}^{+} & =\sqrt{2S-b_{i}^{\dagger}b_{i}^{\dagger}}b_{i}\approx\sqrt{2S}b_{i}\nonumber \\ +\tilde{S}_{i}^{-} & =b_{i}^{\dagger}\sqrt{2S-b_{i}^{\dagger}b_{i}^{\dagger}}\approx\sqrt{2S}b_{i}^{\dagger}\nonumber \\ +\tilde{S}_{i}^{z} & =S-b_{i}^{\dagger}b_{i},\label{eq:HP} +\end{align} + +\end_inset + +where +\begin_inset Formula $\tilde{\bm{S}}_{i}$ +\end_inset + + denotes the spin operators in the local reference frame. + Here the local reference frame is defined such that the spin dipole moment + is pointing along the local +\begin_inset Formula $\hat{z}$ +\end_inset + + direction. + As a result, the spin operator in the global (lab) frame +\begin_inset Formula $\bm{S}_{i}$ +\end_inset + + is related the that in the local frame through an SO(3) rotation: +\begin_inset Formula +\begin{equation} +{\bm{S}}_{i}=R_{i}\tilde{\bm{S}}_{i}\label{eq:local} +\end{equation} + +\end_inset + +To avoid any ambiguity, we write the SO(3) rotation in terms of Euler angles + +\begin_inset Formula +\begin{equation} +R=\begin{pmatrix}-\sin\phi & -\cos\phi\cos\theta & \cos\phi\sin\theta\\ +\cos\phi & -\sin\phi\cos\theta & \sin\phi\sin\theta\\ +0 & \sin\theta & \cos\theta +\end{pmatrix},\label{eq:rotation} +\end{equation} + +\end_inset + +where the two Euler angles can be obtained from the dipole moment +\begin_inset Formula $\langle\bm{S}\rangle=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ +\end_inset + +. +\end_layout + +\begin_layout Standard +The formulas for the bilinear interactions are exactly the same as those + that were implemented in +\family typewriter +spinW +\family default +. + However, as it will become clear in later sections of this note, the treatments + of the biquadratic interactions and the single-ion anisotropy are different + from the large- +\begin_inset Formula $S$ +\end_inset + + based +\family typewriter +spinW +\family default +. + As explained in +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset citation +LatexCommand cite +key "Hao21, Dahlbom22, Dahlbom22b" +literal "false" + +\end_inset + +, the better classical limit of a +\begin_inset Formula $N$ +\end_inset + +-level system ( +\begin_inset Formula $N=2S+1$ +\end_inset + +) should be the one based on the coherent states of SU( +\begin_inset Formula $N$ +\end_inset + +). + In particular, quantum corrections are organized in powers of +\begin_inset Formula $1/\lambda_{1}$ +\end_inset + + (the label of degenerate irreducible representations of SU( +\begin_inset Formula $N$ +\end_inset + +) +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset citation +LatexCommand cite +key "Hao21" +literal "false" + +\end_inset + +) instead of +\begin_inset Formula $1/S$ +\end_inset + +, and the classical limit is obtained by sending +\begin_inset Formula $\lambda_{1}$ +\end_inset + + to infinity instead of +\begin_inset Formula $S$ +\end_inset + +. + Our expansion is based on a renormalized classical spin Hamiltonian that + was introduced in +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset citation +LatexCommand cite +key "Dahlbom23" +literal "false" + +\end_inset + +. + It is identical to the traditional classical Hamiltonian except that nonlinear + terms have been renormalized by coefficients expressed in powers of +\begin_inset Formula $1/S$ +\end_inset + +. + These +\begin_inset Formula $1/S$ +\end_inset + + factors do not emerge from the introduction of higher-order quantum corrections +; instead, they are the consequence of group theoretical considerations + when comparing two different classical limits. + The details of the group theoretical considerations can be found in +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset citation +LatexCommand cite +key "Dahlbom23" +literal "false" + +\end_inset + +. +\end_layout + +\begin_layout Standard +The structure of this note is organized as follows. + We first review the construction of the LSWT Hamiltonian for the bilinear + interactions (the same as in +\family typewriter +spinW +\family default +). + Then we consider the biquadratic interaction that includes the proper renormali +zations (note that +\family typewriter +spinW +\family default + does not take into account the quantum effects captured by our approach, + which would lead to wrong answers). + Finally, we apply the group theoretical treatment to write down the LSWT + Hamiltonian for +\emph on +arbitary +\emph default + single-ion anisotropy (note that +\family typewriter +spinW +\family default + can only handle quadratic single-ion anisotropy plus it does not include + the proper renormalization. +\end_layout + +\begin_layout Section +Bilinear interaction +\end_layout + +\begin_layout Standard +\begin_inset CommandInset label +LatexCommand label +name "sec:blinear" + +\end_inset + + Let us first consider the case of a general bilinear interaction: +\begin_inset Formula +\begin{equation} +\mathcal{H}_{{\rm Bil}}=J_{ab}S_{1}^{a}S_{2}^{b},\label{eq:bil} +\end{equation} + +\end_inset + +where we are using the convention of summation over repeated indices +\begin_inset Formula $a,b=x,y,z$ +\end_inset + +. + After performing the rotation +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:local" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + to the local reference frame, we obtain: +\begin_inset Formula +\begin{equation} +\mathcal{H}_{{\rm Bil}}=J_{ab}R_{1}^{ca}{\tilde{S}}_{1}^{c}R_{2}^{db}\tilde{S}_{2}^{d}.\label{eq:billoc} +\end{equation} + +\end_inset + +The next step is to express the spin operators in terms of the HP bosons + by using Eq. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:HP" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, to obtain the result: +\begin_inset Formula +\begin{equation} +\mathcal{H}_{{\rm Bil}}^{{\rm SW}}=Pb_{1}b_{2}+P^{*}b_{1}^{\dagger}b_{2}^{\dagger}+Qb_{1}^{\dagger}b_{2}+Q^{*}b_{2}^{\dagger}b_{1}-R_{ij}^{33}(b_{1}^{\dagger}b_{1}+b_{2}^{\dagger}b_{2}), +\end{equation} + +\end_inset + +where +\begin_inset Formula $R_{ij}=SR_{i}^{T}J_{ab}R_{j}$ +\end_inset + + ( +\begin_inset Formula $R_{i},R_{j}\in\mathrm{SO}(3)$ +\end_inset + + are the SO(3) rotations for site +\begin_inset Formula $i$ +\end_inset + +, +\begin_inset Formula $j$ +\end_inset + + defined in Eq. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:rotation" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +) with +\begin_inset Formula +\begin{align} +P & =\frac{1}{2}\left(R_{ij}^{11}-R_{ij}^{22}+i(-R_{ij}^{12}-R_{ij}^{21})\right)\nonumber \\ +Q & =\frac{1}{2}\left(R_{ij}^{11}+R_{ij}^{22}+i(-R_{ij}^{12}+R_{ij}^{21})\right) +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +% +\backslash +begin{equation} +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +% +\backslash +mathcal{H}^{ +\backslash +rm SW}_{ +\backslash +rm Bil} = [K_0 + K_{ +\backslash +mu} b^{ +\backslash +dagger}_{ +\backslash +mu} + K^*_{ +\backslash +mu} b^{ +\backslash +;}_{ +\backslash +mu} + F_{ +\backslash +mu +\backslash +nu} b^{ +\backslash +dagger}_{ +\backslash +mu} b^{ +\backslash +;}_{ +\backslash +nu} + G_{ +\backslash +mu +\backslash +nu} b^{ +\backslash +dagger}_{ +\backslash +mu} b^{ +\backslash +dagger}_{ +\backslash +nu} + G^*_{ +\backslash +mu +\backslash +nu} +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +% b^{ +\backslash +;}_{ +\backslash +nu} b^{ +\backslash +;}_{ +\backslash +mu} ] +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +% +\backslash +end{equation} +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +%with +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +% +\backslash +begin{eqnarray} +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +%K_0 &=& S^2 J_{ab} R_{1}^{z a} R_{2}^{z b}, +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +% +\backslash +nonumber +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +%K_1 &=& S_2 +\backslash +sqrt{ +\backslash +frac{ S_1}{2}} J_{ab} (R_{1}^{x a} + i R_{1}^{y a}) R_{2}^{z b} +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +% +\backslash +nonumber +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +%K_2 &=& S_1 +\backslash +sqrt{ +\backslash +frac{ S_2}{2}} J_{ab} R_{2}^{z a} (R_{2}^{x b} + i R_{2}^{y b}) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +% +\backslash +end{eqnarray} +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +% +\backslash +begin{eqnarray} +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +%F_{11} &=& +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +% +\backslash +nonumber +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +%F_{22} &=& +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +% +\backslash +nonumber +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +%F_{21} &=& +\backslash +frac{ +\backslash +sqrt{S_1 S_2}}{2} +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +% +\backslash +nonumber +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +%F_{12} &=& +\backslash +frac{ +\backslash +sqrt{S_1 S_2}}{2} +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +% +\backslash +end{eqnarray} +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +%and +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +% +\backslash +begin{eqnarray} +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +%G_{11} &=& 0 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +% +\backslash +nonumber +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +%G_{22} &=& 0 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +% +\backslash +nonumber +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +%G_{21} &=& +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +% +\backslash +nonumber +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +%G_{12} &=& +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +% +\backslash +end{eqnarray} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Section +Biquadratic interactions +\end_layout + +\begin_layout Standard +The goal of this short section is to derive the general expression for the + contribution of a scalar biquadratic interaction to the spin wave Hamiltonian. + We will then assume that the spin Hamiltonian includes a term of the form: + +\begin_inset Formula +\begin{equation} +\mathcal{H}_{{\rm B}}=K_{12}(\bm{S}_{1}\cdot\bm{S}_{2})^{2}.\label{eq:biq} +\end{equation} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +%where $ +\backslash +bm{S}_1 +\backslash +equiv (S^x_1, S^y_1, S^z_1)$ represents a spin $S_1$ at site 1 and $ +\backslash +bm{S}_2 +\backslash +equiv (S^x_2, S^y_2, S^z_2)$ represents a spin $S_2$ at site 2. + We will assume that the ordered moments point along the directions $ +\backslash +tilde{ +\backslash +bm z}_1$ and $ +\backslash +tilde{ +\backslash +bm z}_2$ directions and that $R_1$ and $R_2$ are the local rotation operators + that rotate the global quantization ${ +\backslash +bm z}$-axis into $ +\backslash +tilde{ +\backslash +bm z}_1$ and $ +\backslash +tilde{ +\backslash +bm z}_2$, respectively: +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +% +\backslash +begin{equation} +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +%R_1 +\backslash +tilde{ +\backslash +bm z}_1 = { +\backslash +bm z}, +\backslash +quad R_2 +\backslash +tilde{ +\backslash +bm z}_2 = { +\backslash +bm z}. +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +% +\backslash +label{eq:quant-axis} +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +% +\backslash +end{equation} +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + +As before, the first step of a spin wave calculation is to rotate the local + reference frames into new frames where +\begin_inset Formula $\tilde{\bm{z}}_{1}$ +\end_inset + + and +\begin_inset Formula $\tilde{\bm{z}}_{2}$ +\end_inset + + are the local quantization axes: and rewrite +\begin_inset Formula $\mathcal{H}_{{\rm B}}$ +\end_inset + + in terms of +\begin_inset Formula $\tilde{\bm{S}}_{1}$ +\end_inset + + and +\begin_inset Formula $\tilde{\bm{S}}_{2}$ +\end_inset + +: +\begin_inset Formula +\begin{equation} +\mathcal{H}_{{\rm B}}=K_{12}(R_{1}\tilde{\bm{S}}_{1}\cdot R_{2}\tilde{\bm{S}}_{2})^{2}=K_{12}(\tilde{\bm{S}}_{1}\cdot R_{1}^{T}R_{2}\tilde{\bm{S}}_{2})^{2}, +\end{equation} + +\end_inset + +where we have used the invariance of the scalar product under global rotations: + +\begin_inset Formula $R\tilde{\bm{S}}_{1}\cdot R\tilde{\bm{S}}_{2}=\tilde{\bm{S}}_{1}\cdot\tilde{\bm{S}}_{2}$ +\end_inset + + +\end_layout + +\begin_layout Standard +The next step is to implement the HP transformation and keep terms up to + quadratic order in the HP bosons +\begin_inset Formula $b_{1}^{(\dagger)}$ +\end_inset + + and +\begin_inset Formula $b_{2}^{(\dagger)}$ +\end_inset + +. + For that purpose, it is enough the expand the spin operators +\begin_inset Formula $\tilde{\bm{S}}_{1}$ +\end_inset + + and +\begin_inset Formula $\tilde{\bm{S}}_{2}$ +\end_inset + + up to quadratic order in the HP bosons. + +\begin_inset Formula +\begin{eqnarray} +\tilde{\bm{S}}_{1} & = & \left(\sqrt{\frac{S_{1}}{2}}(b_{1}^{\dagger}+b_{1}^{\;}),i\sqrt{\frac{S_{1}}{2}}(b_{1}^{\dagger}-b_{1}^{\;}),S_{1}-b_{1}^{\dagger}b_{1}^{\;}\right)+\mathcal{O}(b_{1}^{3}),\nonumber \\ +\tilde{\bm{S}}_{2} & = & \left(\sqrt{\frac{S_{2}}{2}}(b_{2}^{\dagger}+b_{2}^{\;}),i\sqrt{\frac{S_{2}}{2}}(b_{2}^{\dagger}-b_{2}^{\;}),S_{2}-b_{2}^{\dagger}b_{2}^{\;}\right)+\mathcal{O}(b_{2}^{3}),\label{eq:HP1} +\end{eqnarray} + +\end_inset + +After introducing the definition +\begin_inset Formula $R^{r}=R_{1}^{T}R_{2}$ +\end_inset + +, we get: +\begin_inset Formula +\begin{equation} +(\tilde{\bm{S}}_{1}\cdot R^{r}\tilde{\bm{S}}_{2})=C_{0}+C_{\mu}b_{\mu}^{\dagger}+C_{\mu}^{*}b_{\mu}^{\;}+[A_{\mu\nu}b_{\mu}^{\dagger}b_{\nu}^{\;}+B_{\mu\nu}b_{\mu}^{\dagger}b_{\nu}^{\dagger}+B_{\mu\nu}^{*}b_{\nu}^{\;}b_{\mu}^{\;}]\label{eq:bilinear} +\end{equation} + +\end_inset + +with +\begin_inset Formula $\mu,\nu=1,2$ +\end_inset + + (we are using the convention of summation over repeated Greek indices) + and +\begin_inset Formula +\begin{eqnarray} +C_{0} & = & R_{33}^{r}S_{1}S_{2},\nonumber \\ +C_{1} & = & S_{2}\sqrt{\frac{S_{1}}{2}}(R_{13}^{r}+iR_{23}^{r}),\nonumber \\ +C_{2} & = & S_{1}\sqrt{\frac{S_{2}}{2}}(R_{31}^{r}+iR_{32}^{r}), +\end{eqnarray} + +\end_inset + + +\begin_inset Formula +\begin{eqnarray} +A_{11} & = & -R_{33}^{r}S_{2}\nonumber \\ +A_{22} & = & -R_{33}^{r}S_{1}\nonumber \\ +A_{21} & = & \frac{\sqrt{S_{1}S_{2}}}{2}(R_{11}^{r}-iR_{12}^{r}-iR_{21}^{r}+R_{22}^{r})\nonumber \\ +A_{12} & = & \frac{\sqrt{S_{1}S_{2}}}{2}(R_{11}^{r}+iR_{12}^{r}+iR_{21}^{r}+R_{22}^{r}) +\end{eqnarray} + +\end_inset + +and +\begin_inset Formula +\begin{eqnarray} +B_{11} & = & 0\nonumber \\ +B_{22} & = & 0\nonumber \\ +B_{21} & = & \frac{\sqrt{S_{1}S_{2}}}{4}(R_{11}^{r}+iR_{12}^{r}+iR_{21}^{r}-R_{22}^{r})\nonumber \\ +B_{12} & = & \frac{\sqrt{S_{1}S_{2}}}{4}(R_{11}^{r}+iR_{12}^{r}+iR_{21}^{r}-R_{22}^{r}) +\end{eqnarray} + +\end_inset + + +\end_layout + +\begin_layout Standard +By taking the square of Eq. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:bilinear" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, we obtain the contribution of the biquadratic term +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:biq" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + to the spin wave Hamiltonian: +\begin_inset Formula +\begin{equation} +\mathcal{H}_{{\rm B}}^{{\rm SW}}=K_{12}[\tilde{C}_{0}+\tilde{C}_{\mu}b_{\mu}^{\dagger}+\tilde{C}_{\mu}^{*}b_{\mu}^{\;}+\tilde{A}_{\mu\nu}b_{\mu}^{\dagger}b_{\nu}^{\;}+\tilde{B}_{\mu\nu}b_{\mu}^{\dagger}b_{\nu}^{\dagger}+\tilde{B}_{\mu\nu}^{*}b_{\nu}^{\;}b_{\mu}^{\;}]\label{eq:biquad_swt} +\end{equation} + +\end_inset + +with +\begin_inset Formula +\begin{eqnarray} +\tilde{C}_{0} & = & C_{0}^{2}+|C_{1}|^{2}+|C_{2}|^{2}\nonumber \\ +\tilde{C}_{\mu} & = & 2C_{0}C_{\mu}\nonumber \\ +\tilde{A}_{\mu\nu} & = & 2C_{0}{A}_{\mu\nu}+2C_{\mu}C_{\nu}^{*}\nonumber \\ +\tilde{B}_{\mu\nu} & = & 2C_{0}{B}_{\mu\nu}+C_{\mu}C_{\nu} +\end{eqnarray} + +\end_inset + + +\end_layout + +\begin_layout Standard +We note that to capture the effects from the +\begin_inset Formula $1/\lambda_{1}$ +\end_inset + + expansion, we need to multiply Eq. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:biquad_swt" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + by the re-scaling factor +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset citation +LatexCommand cite +key "Dahlbom23" +literal "false" + +\end_inset + + +\begin_inset Formula +\begin{equation} +1-\frac{1}{S}+\frac{1}{4S^{2}}. +\end{equation} + +\end_inset + +Moreover, we need to consider the additional bilinear interaction +\begin_inset Formula +\begin{equation} +-\frac{J}{2}\bm{S}_{1}\cdot\bm{S}_{2}. +\end{equation} + +\end_inset + +The LSWT Hamiltonian from this additional interaction can be easily read + off from the expressions given in Sec. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand ref +reference "sec:blinear" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. +\end_layout + +\begin_layout Section +Single-ion anisotropy +\end_layout + +\begin_layout Standard +A generic contribution to the single-ion anisotropy term is given by: +\begin_inset Formula +\begin{equation} +{\cal H}_{{\rm SI}}=c_{qm}\hat{O}_{qm}({\bm{S}}) +\end{equation} + +\end_inset + +where +\begin_inset Formula $\hat{O}_{qm}$ +\end_inset + + is a so-called Steven's operator (see Appendix +\begin_inset CommandInset ref +LatexCommand ref +reference "SO" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +) and +\begin_inset Formula $c_{qm}$ +\end_inset + + is the value of the corresponding anisotropy (are assuming implicit summation + over repeated indices +\begin_inset Formula $qm$ +\end_inset + + to consider the most general single-ion anisotropy term ). + As we did in the previous section, we need to perform a rotation +\begin_inset Formula $R$ +\end_inset + + into the local reference frame whose quantization +\begin_inset Formula $\tilde{\bm{z}}$ +\end_inset + + is parallel to the direction of the ordered moment: +\begin_inset Formula +\begin{equation} +\bm{z}=R\tilde{\bm{z}}. +\end{equation} + +\end_inset + +Once again, we need to rotate the spin operator into the local reference + frame into a new frame where +\begin_inset Formula $\tilde{\bm{z}}$ +\end_inset + + is the local quantization axis: +\begin_inset Formula +\begin{equation} +\bm{S}=R\tilde{\bm{S}}, +\end{equation} + +\end_inset + +and rewrite +\begin_inset Formula $\mathcal{H}_{{\rm SI}}$ +\end_inset + + in terms of +\begin_inset Formula $\tilde{\bm{S}}$ +\end_inset + +: +\begin_inset Formula +\begin{equation} +{\cal H}_{{\rm SI}}=c_{qm}\hat{O}_{qm}(R\tilde{\bm{S}})=c_{qm}\sum_{m'=-q}^{q}\alpha_{m'}^{qm}\hat{O}_{qm'}(\tilde{\bm{S}})=c_{qm'}\hat{O}_{qm'}(\tilde{\bm{S}}) +\end{equation} + +\end_inset + +where we have used the fact that the Stevens operators for a fixed value + of +\begin_inset Formula $q$ +\end_inset + + form an invariant subspace of SO(3). + To obtain +\begin_inset Formula $c_{qm'}$ +\end_inset + + from +\begin_inset Formula $c_{qm}$ +\end_inset + + in Sunny, we simply call +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + + +\backslash +inlinecode +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +{ +\end_layout + +\end_inset + +rotate_stevens_coeffcients +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +} +\end_layout + +\end_inset + +. + The next step is to expand each Steven's operator +\begin_inset Formula $\hat{O}_{qm}(\tilde{\bm{S}})$ +\end_inset + + to quadratic order in the HP bosons. + As we mentioned before: +\begin_inset Formula +\begin{eqnarray} +\tilde{S}^{+} & \simeq & \sqrt{2S}b,\nonumber \\ +\tilde{S}^{-} & \simeq & \sqrt{2S}b^{\dagger},\nonumber \\ +\tilde{S}^{z} & \simeq & S-b^{\dagger}b. +\end{eqnarray} + +\end_inset + +It is clear that only the operators +\begin_inset Formula $\hat{O}_{q2}(\tilde{\bm{S}})$ +\end_inset + +, +\begin_inset Formula $\hat{O}_{q0}(\tilde{\bm{S}})$ +\end_inset + + and +\begin_inset Formula $\hat{O}_{q\bar{2}}(\tilde{\bm{S}})$ +\end_inset + + contribute to the LSW Hamiltonian (see Appendix +\begin_inset CommandInset ref +LatexCommand ref +reference "SO" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +): +\begin_inset Formula +\begin{equation} +{\cal H}_{{\rm SI}}^{{\rm sw}}=A_{qm}\left[\alpha_{-2}^{qm}\hat{\tilde{O}}_{q\bar{2}}(b^{\dagger},b)+\alpha_{0}^{qm}\hat{\tilde{O}}_{q0}(b^{\dagger},b)+\alpha_{2}^{qm}\hat{\tilde{O}}_{q2}(b^{\dagger},b)\right] +\end{equation} + +\end_inset + +where the expressions +\begin_inset Formula $\hat{\tilde{O}}_{qm}(b^{\dagger},b)$ +\end_inset + + are the expansions of Steven's operators up to quadratic order in +\begin_inset Formula $b^{(\dagger)}$ +\end_inset + + given in Appendix +\begin_inset CommandInset ref +LatexCommand ref +reference "expSO" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. + The linear contributions (not included here) arise from the operators +\begin_inset Formula $\hat{O}_{q\bar{1}}(\tilde{\bm{S}})$ +\end_inset + + and +\begin_inset Formula $\hat{O}_{q1}(\tilde{\bm{S}})$ +\end_inset + +: +\begin_inset Formula +\begin{equation} +{\cal H}_{{\rm SI}}^{{\rm Linear}}=A_{qm}\left[\alpha_{-1}^{qm}\hat{\tilde{O}}_{q\bar{1}}(b^{\dagger},b)+\alpha_{1}^{qm}\hat{\tilde{O}}_{q1}(b^{\dagger},b)\right] +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Section +Appendix +\begin_inset CommandInset label +LatexCommand label +name "app" + +\end_inset + + +\end_layout + +\begin_layout Subsection +Steven's operators +\begin_inset CommandInset label +LatexCommand label +name "SO" + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +X=J(J+1) +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\hat{O}_{00}=1 +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\begin{aligned}\hat{O}_{2\bar{2}} & =\frac{-i}{2}\left[J_{+}^{2}-J_{-}^{2}\right]=J_{x}J_{y}+J_{y}J_{x}=2P_{xy}=I_{4}\\ +\hat{O}_{2\bar{1}} & =\frac{-i}{4}\left[J_{z}\left(J_{+}-J_{-}\right)+\left(J_{+}-J_{-}\right)J_{z}\right]=\frac{1}{2}\left[J_{y}J_{z}+J_{z}J_{y}\right]=P_{yz}=I_{5}\\ +\hat{O}_{20} & =\left[3J_{z}^{2}-X\right]=I_{6}\\ +\hat{O}_{21} & =\frac{1}{4}\left[J_{z}\left(J_{+}+J_{-}\right)+\left(J_{+}+J_{-}\right)J_{z}\right]=\frac{1}{2}\left[J_{x}J_{z}+J_{z}J_{x}\right]=P_{xz}=I_{7}\\ +\hat{O}_{22} & =\frac{1}{2}\left[J_{+}^{2}+J_{-}^{2}\right]=J_{x}^{2}-J_{y}^{2}=I_{8} +\end{aligned} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\begin{aligned}O_{4\bar{4}} & =\frac{-i}{2}\left[\left(J_{+}^{4}-J_{-}^{4}\right]=I_{16}\right.\\ +O_{4\bar{3}} & =\frac{-i}{4}\left[\left(J_{+}^{3}-J_{-}^{3}\right)J_{z}+J_{z}\left(J_{+}^{3}-J_{-}^{3}\right)\right]=I_{17}\\ +O_{4\bar{2}} & =\frac{-i}{4}\left[\left(J_{+}^{2}-J_{-}^{2}\right)\left(7J_{z}^{2}-X-5\right)+\left(7J_{z}^{2}-X-5\right)\left(J_{+}^{2}-J_{-}^{2}\right)\right]=I_{18}\\ +O_{4\bar{1}} & =\frac{-i}{4}\left[\left(J_{+}-J_{-}\right)\left(7J_{z}^{3}-(3X+1)J_{z}\right)+\left(7J_{z}^{3}-(3X+1)J_{z}\right)\left(J_{+}-J_{-}\right)\right]=I_{19}\\ +O_{40} & =\left[35J_{z}^{4}-(30X-25)J_{z}^{2}+3X^{2}-6X\right]=I_{20}\\ +O_{41} & =\frac{1}{4}\left[\left(J_{+}+J_{-}\right)\left(7J_{z}^{3}-(3X+1)J_{z}\right)+\left(7J_{z}^{3}-(3X+1)J_{z}\right)\left(J_{+}+J_{-}\right)\right]=I_{21}\\ +O_{42} & =\frac{1}{4}\left[\left(J_{+}^{2}+J_{-}^{2}\right)\left(7J_{z}^{2}-X-5\right)+\left(7J_{z}^{2}-X-5\right)\left(J_{+}^{2}+J_{-}^{2}\right)\right]=I_{22}\\ +O_{43} & =\frac{1}{4}\left[\left(J_{+}^{3}+J_{-}^{3}\right)J_{z}+J_{z}\left(J_{+}^{3}+J_{-}^{3}\right)\right]=I_{23}\\ +O_{44} & =\frac{1}{2}\left[\left(J_{+}^{4}+J_{-}^{4}\right]=I_{24}\right. +\end{aligned} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\begin{aligned} & \hat{O}_{6\bar{6}}=\frac{-i}{2}\left[J_{+}^{6}-J_{-}^{6}\right]=I_{36}\\ + & \hat{O}_{6\bar{5}}=\frac{-i}{4}\left[\left(J_{+}^{5}-J_{-}^{5}\right)J_{z}+J_{z}\left(J_{+}^{5}-J_{-}^{5}\right)\right]=I_{37}\\ + & \hat{O}_{6\bar{4}}=\frac{-i}{4}\left[\left(J_{+}^{4}-J_{-}^{4}\right)\left(11J_{z}^{2}-X-38\right)+\left(11J_{z}^{2}-X-38\right)\left(J_{+}^{4}-J_{-}^{4}\right)\right]=I_{38}\\ + & \hat{O}_{6\bar{3}}=\frac{-i}{4}\left[\left(J_{+}^{3}-J_{-}^{3}\right)\left(11J_{z}^{3}-(3X+59)J_{z}\right)+\left(11J_{z}^{3}-(3X+59)J_{z}\right)\left(J_{+}^{3}-J_{-}^{3}\right)\right]=I_{39}\\ + & \hat{O}_{6\bar{2}}=\frac{-i}{4}\left[\left(J_{+}^{2}-J_{-}^{2}\right)\left\{ 33J_{z}^{4}-(18X+123)J_{z}^{2}+X^{2}+10X+102\right\} +\{\ldots\}\left(J_{+}^{2}-J_{-}^{2}\right)\right]=I_{40}\\ + & \hat{O}_{6\bar{1}}=\frac{-i}{4}\left[\left(J_{+}-J_{-}\right)\left\{ 33J_{z}^{5}-(30X-15)J_{z}^{3}+\left(5X^{2}-10X+12\right)J_{z}\right\} +\{\ldots\}\left(J_{+}-J_{-}\right)\right]=I_{41}\\ + & \hat{O}_{60}=\left[231J_{z}^{6}-(315X-735)J_{z}^{4}+\left(105X^{2}-525X+294\right)J_{z}^{2}-5X^{3}+40X^{2}-60X\right]=I_{42}\\ + & \hat{O}_{61}=\frac{1}{4}\left[\left(J_{+}+J_{-}\right)\left\{ 33J_{z}^{5}-(30X-15)J_{z}^{3}+\left(5X^{2}-10X+12\right)J_{z}\right\} +\{\ldots\}\left(J_{+}+J_{-}\right)\right]=I_{43}\\ + & \hat{O}_{62}=\frac{1}{4}\left[\left(J_{+}^{2}+J_{-}^{2}\right)\left\{ 33J_{z}^{4}-(18X+123)J_{z}^{2}+X^{2}+10X+102\right\} +\{\ldots\}\left(J_{+}^{2}+J_{-}^{2}\right)\right]=I_{44}\\ + & \hat{O}_{63}=\frac{1}{4}\left[\left(J_{+}^{3}+J_{-}^{3}\right)\left(11J_{z}^{3}-(3X+59)J_{z}\right)+\left(11J_{z}^{3}-(3X+59)J_{z}\right)\left(J_{+}^{3}+J_{-}^{3}\right)\right]=I_{45}\\ + & \hat{O}_{64}=\frac{1}{4}\left[\left(J_{+}^{4}+J_{-}^{4}\right)\left(11J_{z}^{2}-X-38\right)+\left(11J_{z}^{2}-X-38\right)\left(J_{+}^{4}+J_{-}^{4}\right)\right]=I_{46}\\ + & \hat{O}_{65}=\frac{1}{4}\left[\left(J_{+}^{5}+J_{-}^{5}\right)J_{z}+J_{z}\left(J_{+}^{5}+J_{-}^{5}\right)\right]=I_{47}\\ + & \hat{O}_{66}=\frac{1}{2}\left[J_{+}^{6}+J_{-}^{6}\right]=I_{48} +\end{aligned} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsection +Expansion of Steven's operators in HP bosons +\begin_inset CommandInset label +LatexCommand label +name "expSO" + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{alignat*}{2} +\hat{\tilde{O}}_{2\bar{2}} & =-iS\left(bb-b^{\dagger}b^{\dagger}\right)\quad\hat{\tilde{O}}_{4\bar{2}} & =-6iS^{3}\left(bb-b^{\dagger}b^{\dagger}\right)\quad\hat{\tilde{O}}_{6\bar{2}} & =-16iS^{5}\left(bb-b^{\dagger}b^{\dagger}\right)\\ +\hat{\tilde{O}}_{20} & =6(S^{2}-Sb^{\dagger}b)\quad\quad\hat{\tilde{O}}_{40}\;\; & =8S^{4}-80S^{3}b^{\dagger}b\quad\quad\hat{\tilde{O}}_{60} & =16S^{6}-336S^{5}b^{\dagger}b\\ +\hat{\tilde{O}}_{22} & =S\left(bb+b^{\dagger}b^{\dagger}\right)\quad\hat{\tilde{O}}_{42} & =6S^{3}\left(bb+b^{\dagger}b^{\dagger}\right)\quad\hat{\tilde{O}}_{62} & =16S^{5}\left(bb+b^{\dagger}b^{\dagger}\right) +\end{alignat*} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{alignat*}{2} +\hat{\tilde{O}}_{2\bar{1}} & =-i\sqrt{\frac{S}{2}}S(b-b^{\dagger})\quad\hat{\tilde{O}}_{4\bar{1}} & =-i2\sqrt{2S}S^{3}(b-b^{\dagger})\quad\hat{\tilde{O}}_{6\bar{1}} & =-i4\sqrt{2S}S^{5}(b-b^{\dagger})\\ +\hat{\tilde{O}}_{21} & =\sqrt{\frac{S}{2}}S(b+b^{\dagger})\quad\hat{\tilde{O}}_{41} & =2\sqrt{2S}S^{3}(b+b^{\dagger})\quad\hat{\tilde{O}}_{61} & =4\sqrt{2S}S^{5}(b+b^{\dagger}) +\end{alignat*} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + + +\backslash +bibliographystyle{apsrev4-2} +\end_layout 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+arXiv:2304.03874 [cond-mat.str-el] +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + + +\backslash +BibitemShut{ +\end_layout + +\end_inset + +NoStop +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +} +\end_layout + +\end_inset + + +\end_layout + +\end_body +\end_document diff --git a/docs/lyx_notes/SunnySWT.lyx b/docs/lyx_notes/SunnySWT.lyx new file mode 100644 index 000000000..784b9ff55 --- /dev/null +++ b/docs/lyx_notes/SunnySWT.lyx @@ -0,0 +1,1743 @@ +#LyX 2.3 created this file. For more info see http://www.lyx.org/ +\lyxformat 544 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass revtex4-2 +\begin_preamble +\usepackage[T1]{fontenc} +\usepackage{beramono} +\usepackage{listings} +\usepackage[usenames,dvipsnames]{xcolor} + +%% +%% Julia definition (c) 2014 Jubobs +%% +\lstdefinelanguage{Julia}% + {morekeywords={abstract,break,case,catch,const,continue,do,else,elseif,% + end,export,false,for,function,immutable,import,importall,if,in,% + macro,module,otherwise,quote,return,switch,true,try,type,typealias,% + using,while},% + sensitive=true,% + alsoother={$},% + morecomment=[l]\#,% + morecomment=[n]{\#=}{=\#},% + morestring=[s]{"}{"},% + morestring=[m]{'}{'},% +}[keywords,comments,strings]% + +\lstset{% + language = Julia, + basicstyle = \ttfamily, + keywordstyle = \bfseries\color{blue}, + stringstyle = \color{magenta}, + commentstyle = \color{ForestGreen}, + showstringspaces = false, +} +\end_preamble +\use_default_options true +\maintain_unincluded_children false +\language english +\language_package default +\inputencoding auto +\fontencoding global +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\spacing single +\use_hyperref false +\papersize default +\use_geometry false +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine natbib +\cite_engine_type authoryear +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 1 +\use_minted 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\is_math_indent 0 +\math_numbering_side default +\quotes_style english +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Title +SU( +\begin_inset Formula $N$ +\end_inset + +) Linear Spin-Wave Theory Interface to Sunny.jl +\end_layout + +\begin_layout Author +Hao Zhang +\end_layout + +\begin_layout Section +Overview +\end_layout + +\begin_layout Standard +In this note, we try to lay out the implementation of the SU( +\begin_inset Formula $N$ +\end_inset + +) +\emph on +linear +\emph default + spin-wave theory interface to Sunny.jl. + The implementation will be based on the following assumptions: I) We can + inherit from Sunny.jl the crystal symmetry information and the functionalities + of the automatic propagation of the interaction to all other symmetry equivalen +t bonds or sites; II) The ground state (magnetic ordering), +\begin_inset Formula $|Z_{\text{gs}}\rangle=\otimes_{j=1}^{N_{\text{matom}}}|Z_{j}\rangle$ +\end_inset + +, that is a direct product state of +\begin_inset Formula $N_{\text{matom}}$ +\end_inset + + (number of magnetic atoms in the magnetic unit cell) SU( +\begin_inset Formula $N$ +\end_inset + +) coherent states, is known, i.e. + the user needs to provide +\begin_inset Formula $N_{\text{matom}}N$ +\end_inset + + complex numbers (and the positions of magnetic atoms) to specify +\begin_inset Formula $|Z_{\text{gs}}\rangle$ +\end_inset + +, where +\begin_inset Formula +\begin{equation} +|Z_{j}\rangle=\sum_{m=-(N-1)/2}^{(N-1)/2}Z_{j}^{m}|m\rangle,\label{eq:coefs} +\end{equation} + +\end_inset + +with +\begin_inset Formula $\sum_{m}|Z_{j}^{m}|^{2}=1$ +\end_inset + +. + In addition, the user needs to specify the basis vectors +\begin_inset Formula $\tilde{\bm{a}}_{i}$ +\end_inset + + (in units of the chemical unit cell +\begin_inset Formula $\bm{a}_{i}$ +\end_inset + +) of the magnetic unit cell . +\end_layout + +\begin_layout Section +Schwinger boson representation +\end_layout + +\begin_layout Standard +Let us work with the fundamental representation of SU( +\begin_inset Formula $N$ +\end_inset + +). + To make connection with realistic spin Hamiltonians, we will work with + a hermitian basis (physical basis) of SU( +\begin_inset Formula $N$ +\end_inset + +) generators (that are in general polynomial functions of spin operators + +\begin_inset Formula $\hat{\bm{S}}$ +\end_inset + + (can be obtained from Sunny by calling +\color blue +Sunny.gen_spin_ops( +\begin_inset Formula $N$ +\end_inset + +) +\color inherit +). +\begin_inset Formula +\begin{equation} +\hat{T}_{\bm{r}}^{\mu}=\bm{b}_{\bm{r}}^{\dagger}\mathcal{T}_{\bm{r}}^{\mu}\bm{b}_{\bm{r}},\quad\mu=1,2,\ldots,N^{2}-1, +\end{equation} + +\end_inset + +where +\begin_inset Formula $\bm{b}_{\bm{r}}=(b_{\bm{r},(N-1)/2},\ldots,b_{\bm{r}-(N-1)/2})^{T}$ +\end_inset + + is a column vector of Schwinger boson operators that satisfy an additonal + local constaint +\begin_inset Formula +\begin{equation} +\sum_{m=-(N-1)/2}^{(N-1)/2}b_{\bm{r}m}^{\dagger}b_{\bm{r}m}=M,\quad(M=1\ \text{for fundamental rep.)}\label{eq:constraint} +\end{equation} + +\end_inset + +and +\begin_inset Formula $\mathcal{T}_{\bm{r}}^{\mu}$ +\end_inset + + is an +\begin_inset Formula $N\times N$ +\end_inset + + hermitian matrix. + Let us consider a general spin Hamiltonian +\begin_inset Formula +\begin{equation} +\hat{\mathcal{H}}=\frac{1}{2}\sum_{\bm{r},\bm{\delta}}\sum_{\mu,\nu=1}^{N^{2}-1}\mathcal{J}_{\bm{\delta}}^{\mu\nu}\hat{T}_{\bm{r}}^{\mu}\hat{T}_{\bm{r}+\bm{\delta}}^{\nu}+\sum_{\bm{r}}\sum_{\mu=1}^{N^{2}-1}\mathcal{D}_{\mu}\hat{T}^{\mu}. +\end{equation} + +\end_inset + +The first term of +\begin_inset Formula $\hat{\mathcal{H}}$ +\end_inset + + corresponds to an anisotropic exchange interaction between two SU( +\begin_inset Formula $N$ +\end_inset + +) spins connected by the bond +\begin_inset Formula $\bm{\delta}$ +\end_inset + + (at this moment Sunny handles the symmetry analysis for dipolar exchange + interactions only. + Here we consider the general exchange interactions for the sake of completeness +). + The second term of +\begin_inset Formula $\hat{\mathcal{H}}$ +\end_inset + + represents any arbitrary on-site term that can be interpreted as generalized + Zeeman coupling between the SU( +\begin_inset Formula $N$ +\end_inset + +) spin and an external SU( +\begin_inset Formula $N$ +\end_inset + +) field. +\end_layout + +\begin_layout Standard +Now we associate the local coherent state +\begin_inset Formula $|Z_{\bm{r}}\rangle$ +\end_inset + + with a single-particle state of a local Schwinger boson +\begin_inset Formula $\tilde{b}_{\bm{r}(N-1)/2}^{\dagger}$ +\end_inset + +: +\begin_inset Formula +\begin{equation} +|Z_{\bm{r}}\rangle=\tilde{b}_{\bm{r}(N-1)/2}^{\dagger}|\varnothing\rangle,\label{eq:chs} +\end{equation} + +\end_inset + +where +\begin_inset Formula $|\varnothing\rangle$ +\end_inset + + is the common vacuum of the local Schwinger bosons (SBs) +\begin_inset Formula $\tilde{b}_{\bm{r}m}^{\dagger},\ m=(N-1)/2,\ldots,-(N-1)/2$ +\end_inset + +. + We can introduce an SU( +\begin_inset Formula $N$ +\end_inset + +) transformation +\begin_inset Formula $\tilde{\bm{b}}_{\bm{r}}^{\dagger}=\bm{b}_{\bm{r}}^{\dagger}U_{\bm{r}}^{\dagger}$ +\end_inset + + that maps the original Schwinger bosons +\begin_inset Formula $\bm{b}_{\bm{r}}^{\dagger}$ +\end_inset + + into a new set of Schwinger bosons +\begin_inset Formula $\tilde{\bm{b}}_{\bm{r}}^{\dagger}$ +\end_inset + + containing the boson +\begin_inset Formula $\tilde{b}_{\bm{r}(N-1)/2}^{\dagger}$ +\end_inset + + that creates the local coherent state +\begin_inset Formula $|Z_{\bm{r}}\rangle$ +\end_inset + +. + We note that +\begin_inset Formula ${\color{blue}U_{\bm{r}}[1,:]=(Z_{\bm{r}(N-1)/2},\ldots,Z_{\bm{r}-(N-1)/2})^{T}}$ +\end_inset + + and the remaining columns are given by +\begin_inset Formula ${\color{blue}U_{\bm{r}}[2:N,:]=\text{nullspace(U_{\bm{r}}[:,\ 1]')}}$ +\end_inset + +. + In terms of the local SBs, the Hamiltonian takes the form +\begin_inset Formula +\begin{equation} +\mathcal{\hat{H}}=\frac{1}{2}\sum_{\bm{r},\bm{\delta}}\sum_{\mu,\nu=1}^{N^{2}-1}\mathcal{J}_{\bm{\delta}}^{\mu\nu}\tilde{\bm{b}}_{\bm{r}}^{\dagger}\tilde{\mathcal{T}}_{\bm{r}}^{\mu}\tilde{\bm{b}}_{\bm{r}}\tilde{\bm{b}}_{\bm{r}+\bm{\delta}}^{\dagger}\tilde{\mathcal{T}}_{\bm{r}+\bm{\delta}}^{\nu}\tilde{\bm{b}}_{\bm{r}+\bm{\delta}}+\sum_{\bm{r}}\sum_{\mu=1}^{N^{2}-1}\mathcal{D}_{\mu}\tilde{\bm{b}}_{\bm{r}}^{\dagger}\tilde{\mathcal{T}}_{\bm{r}}^{\mu}\tilde{\bm{b}}_{\bm{r}},\label{eq:sunham} +\end{equation} + +\end_inset + +where +\begin_inset Formula +\begin{equation} +\tilde{\mathcal{T}}_{\bm{r}}^{\mu}=U_{\bm{r}}\mathcal{T}_{\bm{r}}^{\mu}U_{\bm{r}}^{\dagger}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Section +Condensation of Schwinger bosons +\end_layout + +\begin_layout Standard +For simplicity, we rename the flavor index +\begin_inset Formula $m$ +\end_inset + + of the SBs from +\begin_inset Formula $(N-1)/2,\ldots-(N-1)/2$ +\end_inset + + to +\begin_inset Formula $1,\ldots N$ +\end_inset + +. + We note that the local coherent (ground) state can be expressed as a condensati +on of the boson +\begin_inset Formula $\tilde{b}_{\bm{r}1}^{\dagger}$ +\end_inset + +: +\begin_inset Formula +\begin{equation} +\tilde{b}_{\bm{r}1}^{\dagger}=\tilde{b}_{\bm{r}1}=\sqrt{M}\sqrt{1-\frac{1}{M}\sum_{m=2}^{N}\tilde{b}_{\bm{r}m}^{\dagger}\tilde{b}_{\bm{r}m}}. +\end{equation} + +\end_inset + +We have +\begin_inset Formula +\begin{align} +\hat{T}_{\bm{r}}^{\mu} & =[\tilde{\mathcal{T}}_{\bm{r}}^{\mu}]_{11}\left(M-\sum_{m=2}^{N}\tilde{b}_{\bm{r}m}^{\dagger}\tilde{b}_{\bm{r}m}\right)+\sqrt{M}\sum_{m=2}^{N}\left(\tilde{b}_{\bm{r}m}^{\dagger}[\tilde{\mathcal{T}}_{\bm{r}}^{\mu}]_{m1}\sqrt{1-\frac{1}{M}\sum_{m=2}^{N}\tilde{b}_{\bm{r}m}^{\dagger}\tilde{b}_{\bm{r}m}}+h.c.\right)\nonumber \\ + & +\sum_{m,m'=2}^{N}[\tilde{\mathcal{T}}_{\bm{r}}^{\mu}]_{mm'}\tilde{b}_{\bm{r}m}^{\dagger}\tilde{b}_{\bm{r}m'}\label{eq:ghpt} +\end{align} + +\end_inset + +where +\begin_inset Formula $\tilde{\mathcal{T}}_{\bm{r}}^{\mu}=U_{\bm{r}}\mathcal{T}_{\bm{r}}^{\mu}U_{\bm{r}}^{\dagger}$ +\end_inset + +. + As in the case of the +\begin_inset Formula $1/S$ +\end_inset + +-expansion, we implement a Taylor expansion of the square-root that appears + in the above equation. + This is justified by assuming +\begin_inset Formula $\sum_{m=2}^{N}\langle\tilde{b}_{\bm{r}m}^{\dagger}\tilde{b}_{\bm{r}m}\rangle\ll M$ +\end_inset + +. + After plugging Eq. +\begin_inset ERT +status open + +\begin_layout Plain Layout + +~ +\end_layout + +\end_inset + + +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:ghpt" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + into the general Hamiltonian Eq. +\begin_inset ERT +status open + +\begin_layout Plain Layout + +~ +\end_layout + +\end_inset + + +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:sunham" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + and expanding the square root, we obtain +\begin_inset Formula +\begin{equation} +\hat{\mathcal{H}}=M^{2}\mathcal{H}^{(0)}+M\hat{\mathcal{H}}^{(2)}+M^{1/2}\hat{\mathcal{H}}^{(3)}+M^{0}\hat{\mathcal{H}}^{(4)}+\mathcal{O}(M^{-1}).\label{eq:hamlargeM} +\end{equation} + +\end_inset + +where +\begin_inset Formula $\hat{\mathcal{H}}^{(n)}$ +\end_inset + + denotes the terms of the +\begin_inset Formula $n$ +\end_inset + +-th power in the +\begin_inset Formula $N-1$ +\end_inset + + uncondensed SB operators +\begin_inset Formula $\tilde{b}_{\bm{r}m}^{\dagger}$ +\end_inset + + and +\begin_inset Formula $\tilde{b}_{\bm{r}m}$ +\end_inset + + with +\begin_inset Formula $m\neq1$ +\end_inset + +. + +\begin_inset Formula $M^{2}\mathcal{H}^{(0)}$ +\end_inset + + is the classical Hamiltonian, and the quadratic bosonic Hamiltonian +\begin_inset Formula $\hat{\mathcal{H}}^{(2)}$ +\end_inset + + is known as the +\emph on +linear +\emph default + spin-wave Hamiltonian, that corresponds to a non-interacting theory of + bosons. + The higher order corrections in the +\begin_inset Formula $1/M$ +\end_inset + + expansion ( +\begin_inset Formula $n\geq3$ +\end_inset + +-particle terms) correspond to interactions between quasiparticles of the + linear spin wave theory. +\end_layout + +\begin_layout Section +Linear spin-wave Hamiltonian and dispersion relations +\end_layout + +\begin_layout Standard +Let us focus on the linear spin-wave Hamiltonian +\begin_inset Formula $\hat{\mathcal{H}}^{(2)}$ +\end_inset + +: +\begin_inset Formula +\begin{align} +\hat{\mathcal{H}}^{(2)} & =M\sum_{\bm{r}}\bigg\{\sum_{\delta=\bm{\delta}_{\text{culled}}}\sum_{m,n=2}^{N}\sum_{\mu,\nu=1}^{N^{2}-1}\mathcal{J}_{\bm{\delta}}^{\mu\nu}\bigg[\big([\tilde{\mathcal{T}}_{\bm{r}}^{\mu}]_{mn}-\delta_{mn}[\tilde{\mathcal{T}}_{\bm{r}}^{\mu}]_{11}\big)[\tilde{\mathcal{T}}_{\bm{r}+\bm{\delta}}^{\nu}]_{11}\tilde{b}_{\bm{r}m}^{\dagger}\tilde{b}_{\bm{r}n}\nonumber \\ + & +[\tilde{\mathcal{T}}_{\bm{r}}^{\mu}]_{mn}\big([\tilde{\mathcal{T}}_{\bm{r}+\bm{\delta}}^{\nu}]_{mn}-\delta_{mn}[\tilde{\mathcal{T}}_{\bm{r}+\bm{\delta}}^{\nu}]_{11}\big)\tilde{b}_{\bm{r}+\bm{\delta}m}^{\dagger}\tilde{b}_{\bm{r}+\bm{\delta}n}\nonumber \\ + & +[\tilde{\mathcal{T}}_{\bm{r}}^{\mu}]_{m1}[\tilde{\mathcal{T}}_{\bm{r}+\bm{\delta}}^{\nu}]_{1n}\tilde{b}_{\bm{r}m}^{\dagger}\tilde{b}_{\bm{r}+\bm{\delta}n}+[\tilde{\mathcal{T}}_{\bm{r}}^{\mu}]_{1m}[\tilde{\mathcal{T}}_{\bm{r}+\bm{\delta}}^{\nu}]_{n1}\tilde{b}_{\bm{r}m}\tilde{b}_{\bm{r}+\bm{\delta}n}^{\dagger}\nonumber \\ + & +[\tilde{\mathcal{T}}_{\bm{r}}^{\mu}]_{m1}[\tilde{\mathcal{T}}_{\bm{r}+\bm{\delta}}^{\nu}]_{n1}\tilde{b}_{\bm{r}m}^{\dagger}\tilde{b}_{\bm{r}+\bm{\delta}n}^{\dagger}+[\tilde{\mathcal{T}}_{\bm{r}}^{\mu}]_{1m}[\tilde{\mathcal{T}}_{\bm{r}+\bm{\delta}}^{\nu}]_{1n}\tilde{b}_{\bm{r}m}\tilde{b}_{\bm{r}+\bm{\delta}n}\bigg]\nonumber \\ + & +\sum_{\mu=1}^{N^{2}-1}\mathcal{D}_{\mu}\sum_{mm'=2}^{N}\big([\tilde{\mathcal{T}}_{\bm{r}}^{\mu}]_{mm'}-\delta_{mm'}[\tilde{\mathcal{T}}_{\bm{r}}^{\mu}]_{11}\big)\tilde{b}_{\bm{r}m}^{\dagger}\tilde{b}_{\bm{r}m'}\bigg\}, +\end{align} + +\end_inset + +where +\begin_inset Formula $\bm{\delta}_{\text{culled}}$ +\end_inset + + indicates the culled bonds defined in Sunny.jl to avoid double counting. + The next step is to implement the Fourier tranform. + We use the letter +\begin_inset Formula $\bm{k}$ +\end_inset + + without a tilde to indicate a Bloch wave vector of the chemical lattice + and +\begin_inset Formula $\tilde{\bm{k}}$ +\end_inset + + to indicate a Bloch wave vector of the magnetic lattice. + Note that the good quantum numbers are the +\begin_inset Formula $\tilde{\bm{k}}$ +\end_inset + +s instead of +\begin_inset Formula $\bm{k}$ +\end_inset + +s because of the reduced translation symmetry of the magnetic lattice. + As a result, we introduce the following Fourier transform on the boson + operator +\begin_inset Formula +\begin{equation} +\tilde{b}_{\bm{r}m}=\tilde{b}_{\bm{R}+\bm{d}m}=\sqrt{\frac{N_{\text{matom}}}{N_{s}}}\sum_{\tilde{\bm{k}}}e^{i\tilde{\bm{k}}\cdot\bm{R}}\tilde{b}_{\tilde{\bm{k}},\bm{d}m}\label{eq:fourier1} +\end{equation} + +\end_inset + + +\begin_inset Formula +\begin{equation} +\tilde{b}_{\bm{r}m}^{\dagger}=\tilde{b}_{\bm{R}+\bm{d}m}^{\dagger}=\sqrt{\frac{N_{\text{matom}}}{N_{s}}}\sum_{\tilde{\bm{k}}}e^{-i\tilde{\bm{k}}\cdot\bm{R}}\tilde{b}_{\tilde{\bm{k}},\bm{d}m}^{\dagger}\ ,\label{eq:fourier2} +\end{equation} + +\end_inset + +where +\begin_inset Formula $\bm{R}$ +\end_inset + + indicates the position of the origin of a magnetic cell and +\begin_inset Formula $\bm{d}$ +\end_inset + + is the relative displacement of the magnetic atom (there are +\begin_inset Formula $N_{\text{matom}}$ +\end_inset + + of them) with respect to the origin of the magnetic cell +\begin_inset ERT +status open + +\begin_layout Plain Layout + +~ +\backslash +footnote{There is a U(1) gauge freedom to choose the phase factor for the + Fourier transform. + As it will become clear later, our choice here simplifies the calculation.} +\end_layout + +\end_inset + +. + +\begin_inset Formula $N_{s}$ +\end_inset + + is the total number of sites in the above equations. + After Fourier tranforming the linear spin-wave Hamiltonian, we have +\begin_inset Formula +\begin{align} +M\hat{\mathcal{H}}^{(2)} & =M\sum_{\tilde{\bm{k}}}\sum_{\bm{d}}\bigg\{\sum_{\delta=\bm{\delta}_{\text{culled}}}\sum_{mn=2}^{N}\sum_{\mu,\nu=1}^{N^{2}-1}\mathcal{J}_{\bm{\delta}}^{\mu\nu}\bigg[\big([\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{mn}-\delta_{mn}[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{11}\big)[\tilde{\mathcal{T}}_{\bm{d}+\bm{\delta}}^{\nu}]_{11}\tilde{b}_{\tilde{\bm{k}},\bm{d}m}^{\dagger}\tilde{b}_{\tilde{\bm{k}},\bm{d}n}\nonumber \\ + & +[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{11}\big([\tilde{\mathcal{T}}_{\bm{d}+\bm{\delta}}^{\nu}]_{mn}-\delta_{mn}[\tilde{\mathcal{T}}_{\bm{d}+\bm{\delta}}^{\nu}]_{11}\big)\tilde{b}_{\tilde{\bm{k}},\bm{d}+\bm{\delta}m}^{\dagger}\tilde{b}_{\tilde{\bm{k}},\bm{d}+\bm{\delta}n}\nonumber \\ + & [\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{m1}[\tilde{\mathcal{T}}_{\bm{d}+\bm{\delta}}^{\nu}]_{1n}\tilde{b}_{\tilde{\bm{k}},\bm{d}m}^{\dagger}\tilde{b}_{\tilde{\bm{k}},\bm{d}+\bm{\delta}n}e^{i\tilde{\bm{k}}\cdot\Delta\bm{R}_{\bm{\delta}}}+[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{1m}[\tilde{\mathcal{T}}_{\bm{d}+\bm{\delta}}^{\nu}]_{n1}\tilde{b}_{\tilde{\bm{k}},dm}\tilde{b}_{\tilde{\bm{k}},\bm{d}+\bm{\delta}n}^{\dagger}e^{-i\tilde{\bm{k}}\cdot\Delta\bm{R}_{\bm{\delta}}}\nonumber \\ + & +[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{m1}[\tilde{\mathcal{T}}_{\bm{d}+\bm{\delta}}^{\nu}]_{n1}\tilde{b}_{\tilde{\bm{k}},\bm{d}m}^{\dagger}\tilde{b}_{-\tilde{\bm{k}},\bm{d}+\bm{\delta}n}^{\dagger}e^{i\tilde{\bm{k}}\cdot\Delta\bm{R}_{\bm{\delta}}}+[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{1m}[\tilde{\mathcal{T}}_{\bm{d}+\bm{\delta}}^{\nu}]_{1n}\tilde{b}_{-\tilde{\bm{k}},dm}\tilde{b}_{\tilde{\bm{k}},\bm{d}+\bm{\delta}n}e^{i\tilde{\bm{k}}\cdot\Delta\bm{R}_{\bm{\delta}}}\bigg]\label{eq:hamfourier}\\ + & +\sum_{\mu=1}^{N^{2}-1}\mathcal{D}_{\mu}\sum_{mm'=2}^{N}\big([\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{mm'}-\delta_{mm'}[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{11}\big)\tilde{b}_{\tilde{\bm{k}},\bm{d}m}^{\dagger}\tilde{b}_{\tilde{\bm{k}},\bm{d}m'}\bigg\}, +\end{align} + +\end_inset + +where the additional phase factor +\begin_inset Formula $e^{i\tilde{\bm{k}}\cdot\Delta\bm{R}_{\bm{\delta}}}$ +\end_inset + + comes from the Bloch theorem and +\begin_inset Formula $\Delta\bm{R}_{\bm{\delta}}$ +\end_inset + + is the displacement of two magnetic unit cells that contain the two sites + connected by the bond +\begin_inset Formula $\bm{\delta}$ +\end_inset + +. + In a more compact way, +\begin_inset Formula +\begin{equation} +M\hat{\mathcal{H}}^{(2)}=M\bigg[\sum_{\tilde{\bm{k}}}\Psi_{\tilde{\bm{k}}}^{\dagger}\mathcal{H}^{(2)}(\tilde{\bm{k}})\Psi_{\tilde{\bm{k}}}+\xi\bigg], +\end{equation} + +\end_inset + +where +\begin_inset Formula $\Psi_{\bm{k}}=(\tilde{b}_{\tilde{\bm{k}},\bm{d}_{1}2},\ldots\tilde{b}_{\tilde{\bm{k}},\bm{d}_{1}N},\ldots\tilde{b}_{\tilde{\bm{k}},\bm{d}_{N_{\text{matom}}}2},\ldots\tilde{b}_{\tilde{\bm{k}},\bm{d}_{N_{\text{matom}}}N},\tilde{b}_{-\tilde{\bm{k}},\bm{d}_{1}2}^{\dagger},\ldots\tilde{b}_{-\tilde{\bm{k}},\bm{d}_{1}N}^{\dagger},\ldots\tilde{b}_{-\tilde{\bm{k}},\bm{d}_{N_{\text{matom}}}2}^{\dagger},\ldots\tilde{b}_{-\tilde{\bm{k}},\bm{d}_{N_{\text{matom}}}N}^{\dagger})^{T}$ +\end_inset + + is a +\begin_inset Formula $2(N-1)N_{\text{matom}}$ +\end_inset + +-dimensional Nambu spinor and +\begin_inset Formula $\xi$ +\end_inset + + is a number that renormalizes the classical energy. + The matrix element of +\begin_inset Formula $\mathcal{H}^{(2)}$ +\end_inset + + can be read from Eq. +\begin_inset ERT +status open + +\begin_layout Plain Layout + +~ +\end_layout + +\end_inset + + +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:hamfourier" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. +\end_layout + +\begin_layout Standard +The next goal is to diagonalize +\begin_inset Formula $\tilde{\mathcal{H}}^{(2)}$ +\end_inset + +. + Here we follow +\begin_inset ERT +status open + +\begin_layout Plain Layout + +~ +\end_layout + +\end_inset + + +\begin_inset CommandInset citation +LatexCommand citep +key "Colpa1978" +literal "false" + +\end_inset + + by performming the so-called Bogoliubov tranform: +\begin_inset Formula +\begin{equation} +\tilde{\Psi}_{\tilde{\bm{k}}}=V_{\tilde{\bm{k}}}^{-1}\Psi_{\tilde{\bm{k}}}, +\end{equation} + +\end_inset + +we have +\begin_inset Formula +\begin{equation} +\sum_{\tilde{\bm{k}}}\Psi_{\tilde{\bm{k}}}^{\dagger}\mathcal{H}^{(2)}(\tilde{\bm{k}})\Psi_{\tilde{\bm{k}}}=\sum_{\tilde{\bm{k}}}\tilde{\Psi}_{\tilde{\bm{k}}}^{\dagger}\tilde{\mathcal{H}}^{(2)}(\tilde{\bm{k}})\tilde{\Psi}_{\tilde{\bm{k}}}, +\end{equation} + +\end_inset + +where +\begin_inset Formula +\begin{equation} +\tilde{\mathcal{H}}^{(2)}(\tilde{\bm{k}})=\text{diag}(\text{\tilde{\omega}_{\tilde{\bm{k}}1},\ldots}\tilde{\omega}_{\tilde{\bm{k}}(N-1)N_{\text{matom}}},\text{\tilde{\omega}_{-\tilde{\bm{k}}1},\ldots}\tilde{\omega}_{-\tilde{\bm{k}}(N-1)N_{\text{matom}}}) +\end{equation} + +\end_inset + +and +\begin_inset Formula +\begin{equation} +\tilde{\Psi}_{\tilde{\bm{k}}}=(\beta_{\tilde{\bm{k}}1},\ldots,\beta_{\tilde{\bm{k}}(N-1)N_{\text{matom}}},\beta_{-\tilde{\bm{k}}1}^{\dagger},\ldots,\beta_{\tilde{-\bm{k}}(N-1)N_{\text{matom}}}^{\dagger})^{T}. +\end{equation} + +\end_inset + +Putting the above result in normal ordering, we have the Hamiltonian of + +\begin_inset Formula $(N-1)N_{\text{matom}}$ +\end_inset + + decoupled quantum harmonic oscillators for each +\begin_inset Formula $\tilde{\bm{k}}$ +\end_inset + +: +\begin_inset Formula +\begin{equation} +\sum_{\tilde{\bm{k}}}\tilde{\Psi}_{\tilde{\bm{k}}}^{\dagger}\tilde{\mathcal{H}}^{(2)}(\tilde{\bm{k}})\tilde{\Psi}_{\tilde{\bm{k}}}=\sum_{n=1}^{(N-1)N_{\text{matom}}}\sum_{\tilde{\bm{k}}}\omega_{\tilde{\bm{k}},n}\bigg(\beta_{\tilde{\bm{k}}n}^{\dagger}\beta_{\tilde{\bm{k}}n}+\frac{1}{2}\bigg), +\end{equation} + +\end_inset + +where +\begin_inset Formula $\omega_{\tilde{\bm{k}},n}=2\tilde{\omega}_{\tilde{\bm{k}},n}$ +\end_inset + + gives the linear spin-wave dispersion of band +\begin_inset Formula $n$ +\end_inset + +. + The details of finding +\begin_inset Formula $\tilde{\omega}_{\tilde{\bm{k}}}$ +\end_inset + + and +\begin_inset Formula $V_{\tilde{\bm{k}}}$ +\end_inset + + can be found in +\begin_inset ERT +status open + +\begin_layout Plain Layout + +~ +\end_layout + +\end_inset + + +\begin_inset CommandInset citation +LatexCommand citep +key "Colpa1978" +literal "false" + +\end_inset + + and will be easily implemented in the code (see documentation for the code + later). +\end_layout + +\begin_layout Section +Linear spin-wave theory–Spectral weight +\end_layout + +\begin_layout Standard +Let us write down the Bogoliubov transform explicitly for later convinence + (note that +\begin_inset Formula $N_{m}=N_{\text{matoms}}$ +\end_inset + + for shorter notations), the combined index +\begin_inset Formula $d_{\alpha}=(d-1)(N-1)+\alpha-1$ +\end_inset + +, where +\begin_inset Formula $d=1,\ldots N_{m},\alpha=2,\ldots N$ +\end_inset + +: +\begin_inset Formula +\begin{align} +\tilde{b}_{\tilde{\bm{k}}d_{\alpha}} & =\sum_{n=1}^{(N-1)N_{m}}\big([V_{\tilde{\bm{k}}}]_{d_{\alpha},n}\beta_{\tilde{\bm{k}},n}+[V_{\tilde{\bm{k}}}]_{d_{\alpha},n+(N-1)N_{m}}\beta_{-\tilde{\bm{k}},n}^{\dagger}\big)\label{eq:bk1}\\ +\tilde{b}_{-\tilde{\bm{k}}d_{\alpha}}^{\dagger} & =\sum_{n=1}^{(N-1)N_{m}}\big([V_{\tilde{\bm{k}}}]_{d_{\alpha}+(N-1)N_{m},n}\beta_{\tilde{\bm{k}},n}+[V_{\tilde{\bm{k}}}]_{d_{\alpha}+(N-1)N_{m},n+(N-1)N_{m}}\beta_{-\tilde{\bm{k}},n}^{\dagger}\big)\\ +\tilde{b}_{-\tilde{\bm{k}}d_{\alpha}} & =\sum_{n=1}^{(N-1)N_{m}}\big([V_{\tilde{\bm{k}}}]_{d_{\alpha}+(N-1)N_{m},n}^{*}\beta_{\tilde{\bm{k}},n}^{\dagger}+[V_{\tilde{\bm{k}}}]_{d_{\alpha}+(N-1)N_{m},n+(N-1)N_{m}}^{*}\beta_{-\tilde{\bm{k}},n}\big)\\ +\tilde{b}_{\tilde{\bm{k}}d_{\alpha}}^{\dagger} & =\sum_{n=1}^{(N-1)N_{m}}\big([V_{\tilde{\bm{k}}}]_{d_{\alpha},n}^{*}\beta_{\tilde{\bm{k}},n}^{\dagger}+[V_{\tilde{\bm{k}}}]_{d_{\alpha},n+(N-1)N_{m}}^{*}\beta_{-\tilde{\bm{k}},n}\big),\label{eq:bk4} +\end{align} + +\end_inset + +where the third and fourth line is obtained from Hermitian conjugating the + second and the first line, respectively. + The +\begin_inset Formula $T=0$ +\end_inset + + (generalized) spin structure factor is related to the dynamical spin susceptibi +lity through the fluctuation dissipation theorem: +\begin_inset Formula +\begin{align} +\mathcal{S}^{\mu\nu}(\bm{q},\omega) & =-2\text{Im}\chi^{\mu\nu}(\bm{q},\omega)\nonumber \\ + & =\sum_{\nu}\langle0|\hat{T}_{\bm{q}}^{\mu}|\nu\rangle\langle\nu|\hat{T}_{-\bm{q}}^{\nu}|0\rangle\delta(\omega+E_{0}-E_{\nu})\nonumber \\ + & =\mathcal{S}_{\text{ssf}}^{\mu\nu}(\bm{q},\omega)+\mathcal{S}_{\text{dssf}}^{\mu\nu}(\bm{q},\omega), +\end{align} + +\end_inset + +where +\begin_inset Formula +\begin{align} +\mathcal{S}_{\text{ssf}}^{\mu\nu} & =\langle0|\hat{T}_{\bm{q}}^{\mu}|0\rangle\langle0|\hat{T}_{-\bm{q}}^{\nu}|0\rangle\delta(\omega),\\ +\mathcal{S}_{\text{dssf}}^{\mu\nu}(\bm{q},\omega) & =\sum_{\nu\neq0}\langle0|\hat{T}_{\bm{q}}^{\mu}|\nu\rangle\langle\nu|\hat{T}_{-\bm{q}}^{\nu}|0\rangle\delta(\omega+E_{0}-E_{\nu}). +\end{align} + +\end_inset + +In the above expressions, +\begin_inset Formula $\bm{q}$ +\end_inset + + belongs to the wave vector in the reciprocal space of the chemical lattice. + We have +\begin_inset Formula $\bm{q}=\bm{K}+\tilde{\bm{q}}$ +\end_inset + +, where +\begin_inset Formula $\bm{K}$ +\end_inset + + is the reciprocal lattice vector of the magnetic lattice, and +\begin_inset Formula $\tilde{\bm{q}}$ +\end_inset + + is the wave vector in the first BZ of the magnetic lattice. + An useful relation: +\begin_inset Formula +\begin{equation} +e^{i\bm{K}\cdot\bm{R}}=1, +\end{equation} + +\end_inset + +where +\begin_inset Formula $\bm{R}$ +\end_inset + + sits on the magnetic lattice. + Consider the Fourier transform of the SU( +\begin_inset Formula $N$ +\end_inset + +) generator +\begin_inset Formula +\begin{align*} +\hat{T}_{\bm{q}}^{\mu} & =\frac{1}{\sqrt{N_{s}}}\sum_{\bm{r}}e^{-i\bm{q}\cdot\bm{r}}\hat{T}_{\bm{r}}^{\mu}=\frac{1}{\sqrt{N_{s}}}\sum_{\bm{r}}e^{-i\bm{q}\cdot\bm{r}}\bm{b}_{\bm{r}}^{\dagger}\mathcal{T}_{\bm{r}}^{\mu}\bm{b}_{\bm{r}}\\ + & =\frac{1}{\sqrt{N_{s}}}\sum_{\bm{r}}e^{-i\bm{q}\cdot\bm{r}}\bm{b}_{\bm{r}}^{\dagger}U_{\bm{r}}^{\dagger}U_{\bm{r}}\mathcal{T}_{\bm{r}}^{\mu}U_{\bm{r}}^{\dagger}U_{\bm{r}}\bm{b}_{\bm{r}}=\frac{1}{\sqrt{N_{s}}}\sum_{\bm{r}}e^{-i\bm{q}\cdot\bm{r}}\tilde{\bm{b}}_{\bm{r}}^{\dagger}\mathcal{\tilde{T}}_{\bm{r}}^{\mu}\tilde{\bm{b}}_{\bm{r}}\\ + & \approx\frac{1}{\sqrt{N_{s}}}\sum_{\bm{r}}e^{-i\bm{q}\cdot\bm{r}}\bigg[M[\tilde{\mathcal{T}}_{\bm{r}}^{\mu}]_{11}+\sqrt{M}\sum_{\alpha=2}^{N}\big([\tilde{\mathcal{T}}_{\bm{r}}^{\mu}]_{\alpha1}\tilde{b}_{\bm{r}\alpha}^{\dagger}+[\tilde{\mathcal{T}}_{\bm{r}}^{\mu}]_{1\alpha}\tilde{b}_{\bm{r}\alpha}\big)\\ + & +M^{0}\sum_{\alpha,\beta=2}^{N}\big([\tilde{\mathcal{T}}_{\bm{r}}^{\mu}]_{\alpha\beta}-[\tilde{\mathcal{T}}_{\bm{r}}^{\mu}]_{11}\delta_{\alpha\beta}\big)\tilde{b}_{\bm{r}\alpha}^{\dagger}\tilde{b}_{\bm{r}\beta}\bigg]\\ + & =\frac{1}{\sqrt{N_{s}}}\sum_{\bm{R}}\sum_{\bm{d}}e^{-i(\bm{K}+\tilde{\bm{q}})\cdot(\bm{R}+\bm{d})}\bigg[M[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{11}\\ + & +\sqrt{\frac{MN_{m}}{N_{s}}}\sum_{\tilde{\bm{k}}}\sum_{\alpha=2}^{N}\big([\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{\alpha1}\tilde{b}_{\tilde{\bm{k}},\bm{d}\alpha}^{\dagger}e^{-i\tilde{\bm{k}}\cdot\bm{R}}+[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{1\alpha}\tilde{b}_{\tilde{\bm{k}},\bm{d}\alpha}e^{i\tilde{\bm{k}}\cdot\bm{R}}\big)\\ + & +\frac{M^{0}N_{m}}{N_{s}}\sum_{\tilde{\bm{k}}_{1},\tilde{\bm{k}}_{2}}\sum_{\alpha,\beta=2}^{N}\big([\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{\alpha\beta}-[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{11}\delta_{\alpha\beta}\big)\tilde{b}_{\tilde{\bm{k}}_{1}\bm{d}\alpha}^{\dagger}\tilde{b}_{\tilde{\bm{k}}_{2},\bm{d}\beta}e^{-i(\tilde{\bm{k}}_{1}-\tilde{\bm{k}}_{2})\cdot\bm{R}}\bigg]\\ + & =\frac{1}{\sqrt{N_{s}}}\sum_{\bm{d}}e^{-i(\bm{K}+\tilde{\bm{q}})\cdot\bm{d}}\bigg[\frac{MN_{s}}{N_{m}}[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{11}\delta_{\tilde{\bm{q}},0}+\sqrt{\frac{MN_{s}}{N_{m}}}\sum_{\alpha=2}^{N}\big([\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{\alpha1}\tilde{b}_{-\tilde{\bm{q}},\bm{d}\alpha}^{\dagger}+[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{1\alpha}\tilde{b}_{\tilde{\bm{q}},\bm{d}\alpha}\big)\\ + & +\sum_{\tilde{\bm{k}}}\sum_{\alpha,\beta=2}^{N}\big([\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{\alpha\beta}-[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{11}\delta_{\alpha\beta}\big)\tilde{b}_{\tilde{\bm{k}}-\tilde{\bm{q}}\bm{d}\alpha}^{\dagger}\tilde{b}_{\tilde{\bm{k}},\bm{d}\beta}\bigg] +\end{align*} + +\end_inset + +As a result, after the Bogoliubov transform, we have +\begin_inset Formula +\begin{align} +\langle0|\hat{T}_{\bm{q}}^{\mu} & \approx\frac{1}{\sqrt{N_{s}}}\sum_{\bm{d}}e^{-i\bm{q}\cdot\bm{d}}\bigg[\frac{MN_{s}}{N_{m}}[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{11}\delta_{\tilde{\bm{q}},0}\nonumber \\ + & +\sqrt{\frac{MN_{s}}{N_{m}}}\sum_{\alpha=2}^{N}\sum_{n=1}^{(N-1)N_{m}}\big([\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{\alpha1}[V_{\tilde{\bm{q}}}]_{d_{\alpha}+(N-1)N_{m},n}\beta_{\tilde{\bm{k}},n}+[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{1\alpha}[V_{\tilde{\bm{q}}}]_{d_{\alpha},n}\beta_{\tilde{\bm{q}},n}\big)\nonumber \\ + & +\langle0|\sum_{\tilde{\bm{k}}}\sum_{\alpha,\beta=2}^{N}\big([\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{\alpha\beta}-[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{11}\delta_{\alpha\beta}\big)\tilde{b}_{\tilde{\bm{k}}-\tilde{\bm{q}}\bm{d}\alpha}^{\dagger}\tilde{b}_{\tilde{\bm{k}},\bm{d}\beta}\bigg] +\end{align} + +\end_inset + + +\begin_inset Formula +\begin{align} +\hat{T}_{-\bm{q}}^{\nu}|0\rangle & \approx\frac{1}{\sqrt{N_{s}}}\sum_{\bm{d}}e^{i\bm{q}\cdot\bm{d}}\bigg[\frac{MN_{s}}{N_{m}}[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{11}\delta_{\tilde{\bm{q}},0}\nonumber \\ + & +\sqrt{\frac{MN_{s}}{N_{m}}}\sum_{\alpha=2}^{N}\sum_{n=1}^{(N-1)N_{m}}\big([\tilde{\mathcal{T}}_{\bm{d}}^{\nu}]_{\alpha1}[V_{\tilde{\bm{q}}}]_{d_{\alpha},n}^{*}\beta_{\tilde{\bm{q}},n}^{\dagger}+[\tilde{\mathcal{T}}_{\bm{d}}^{\nu}]_{1\alpha}[V_{\tilde{\bm{q}}}]_{d_{\alpha}+(N-1)N_{m},n}^{*}\beta_{\tilde{\bm{q}},n}^{\dagger}\big)\nonumber \\ + & +\sum_{\tilde{\bm{k}}}\sum_{\alpha,\beta=2}^{N}\big([\tilde{\mathcal{T}}_{\bm{d}}^{\nu}]_{\alpha\beta}-[\tilde{\mathcal{T}}_{\bm{d}}^{\nu}]_{11}\delta_{\alpha\beta}\big)\tilde{b}_{\tilde{\bm{k}}+\tilde{\bm{q}}\bm{d}\alpha}^{\dagger}\tilde{b}_{\tilde{\bm{k}},\bm{d}\beta} +\end{align} + +\end_inset + +Furthermore (keeping contributions up to the +\begin_inset Formula $1/M$ +\end_inset + + order), +\begin_inset Formula +\begin{equation} +\langle0|\hat{T}_{\bm{q}}^{\mu}|n\neq0\rangle\approx\sqrt{\frac{M}{N_{m}}}\sum_{\bm{d}}e^{-i\bm{q}\cdot\bm{d}}\sum_{\alpha=2}^{N}\sum_{n=1}^{(N-1)N_{m}}\big([\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{\alpha1}[V_{\tilde{\bm{q}}}]_{d_{\alpha}+(N-1)N_{m},n}+[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{1\alpha}[V_{\tilde{\bm{q}}}]_{d_{\alpha},n}\big) +\end{equation} + +\end_inset + + +\begin_inset Formula +\begin{equation} +\langle0|\hat{T}_{\bm{q}}^{\mu}|0\rangle\approx\sqrt{N_{s}}\sum_{\bm{d}}e^{-i\bm{q}\cdot\bm{d}}\bigg[\frac{M}{N_{m}}[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{11}\delta_{\tilde{\bm{q}},0}+\frac{1}{N_{m}}\frac{1}{N_{u}}\sum_{\tilde{\bm{k}}}\sum_{\alpha,\beta=2}^{N}\big([\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{\alpha\beta}-[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{11}\delta_{\alpha\beta}\big)\tilde{b}_{\tilde{\bm{k}}-\tilde{\bm{q}}\bm{d}\alpha}^{\dagger}\tilde{b}_{\tilde{\bm{k}},\bm{d}\beta}\bigg], +\end{equation} + +\end_inset + +where +\begin_inset Formula $N_{u}=N_{s}/N_{m}$ +\end_inset + +. + In summary, +\begin_inset Formula +\begin{align} +\mathcal{S}_{\text{ssf}}^{\mu\nu} & (\bm{q},\omega)=\langle0|\hat{T}_{\bm{q}}^{\mu}|0\rangle\langle0|\hat{T}_{\bm{-q}}^{\nu}|0\rangle\delta(\omega),\nonumber \\ + & \approx N_{s}\delta_{\tilde{\bm{q}},0}\delta(\omega)\bigg(\frac{1}{N_{m}}\sum_{\bm{d}}e^{-i\bm{q}\cdot\bm{d}}\bigg[M[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{11}+\delta\tilde{T}_{\bm{d}}^{\mu}\bigg]\bigg)\nonumber \\ + & \times\bigg(\frac{1}{N_{m}}\sum_{\bm{d}}e^{i\bm{q}\cdot\bm{d}}\bigg[M[\tilde{\mathcal{T}}_{\bm{d}}^{\nu}]_{11}+\delta\tilde{T}_{\bm{d}}^{\nu}\bigg]\bigg) +\end{align} + +\end_inset + +where +\begin_inset Formula +\begin{align} +\delta T_{\bm{d}}^{\mu} & =\frac{1}{N_{u}}\sum_{\tilde{\bm{k}}}\sum_{\alpha,\beta=2}^{N}\big([\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{\alpha\beta}-[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{11}\delta_{\alpha\beta}\big)\langle0|\tilde{b}_{\tilde{\bm{k}},\bm{d}\alpha}^{\dagger}\tilde{b}_{\tilde{\bm{k}},\bm{d}\beta}|0\rangle\bigg],\nonumber \\ + & =\frac{1}{N_{u}}\sum_{\tilde{\bm{k}}}\sum_{\alpha,\beta=2}^{N}\big([\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{\alpha\beta}-[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{11}\delta_{\alpha\beta}\big)\sum_{n=1}^{(N-1)N_{m}}[V_{\tilde{\bm{k}}}]_{d_{\alpha},n+(N-1)N_{m}}^{*}[V_{\tilde{\bm{k}}}]_{d_{\beta},n+(N-1)N_{m}}, +\end{align} + +\end_inset + +and +\begin_inset Formula +\begin{align} +\mathcal{S}_{\text{dssf}}^{\mu\nu}(\bm{q},\omega) & =\frac{M}{N_{m}}\bigg(\sum_{\bm{d}}\sum_{\alpha=2}^{N}\sum_{n=1}^{(N-1)N_{m}}e^{-i\bm{q}\cdot\bm{d}}\big[[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{\alpha1}[V_{\tilde{\bm{q}}}]_{d_{\alpha}+(N-1)N_{m},n}+[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{1\alpha}[V_{\tilde{\bm{q}}}]_{d_{\alpha},n}\big]\bigg)\nonumber \\ + & \times\bigg(\sum_{\bm{d}}\sum_{\alpha=2}^{N}\sum_{n=1}^{(N-1)N_{m}}e^{i\bm{q}\cdot\bm{d}}\big[[\tilde{\mathcal{T}}_{\bm{d}}^{\nu}]_{\alpha1}[V_{\tilde{\bm{q}}}]_{d_{\alpha},n}^{*}+[\tilde{\mathcal{T}}_{\bm{d}}^{\nu}]_{1\alpha}[V_{\tilde{\bm{q}}}]_{d_{\alpha}+(N-1)N_{m},n}^{*}\big]\bigg)\delta(\omega-\omega_{\tilde{\bm{q}},n}) +\end{align} + +\end_inset + +We note that the diagonal part +\begin_inset Formula $\mu\mu$ +\end_inset + + is pure real, the sum of the off-diagonal part +\begin_inset Formula $\mu\nu+\nu\mu$ +\end_inset + + is also pure real, but the difference of the off-diagonal part +\begin_inset Formula $\mu\nu-\nu\mu$ +\end_inset + + is pure imaginary. + Moreover, +\begin_inset Formula $\delta T_{\bm{d}}^{\mu}$ +\end_inset + + is pure real +\begin_inset Formula +\begin{align} +(\delta T_{\bm{d}}^{\mu})^{*} & =\frac{1}{N_{u}}\sum_{\tilde{\bm{k}}}\sum_{n=1}^{(N-1)N_{m}}\sum_{\alpha,\beta=2}^{N}\big([\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{\alpha\beta}^{*}-[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{11}^{*}\delta_{\alpha\beta}\big)[V_{\tilde{\bm{k}}}]_{d_{\beta},n+(N-1)N_{m}}^{*}[V_{\tilde{\bm{k}}}]_{d_{\alpha},n+(N-1)N_{m}}\nonumber \\ + & =\frac{1}{N_{u}}\sum_{\tilde{\bm{k}}}\sum_{n=1}^{(N-1)N_{m}}\sum_{\alpha,\beta=2}^{N}\big([\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{\beta\alpha}-[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{11}\delta_{\beta\alpha}\big)[V_{\tilde{\bm{k}}}]_{d_{\beta},n+(N-1)N_{m}}^{*}[V_{\tilde{\bm{k}}}]_{d_{\alpha},n+(N-1)N_{m}}\nonumber \\ + & =\delta T_{\bm{d}}^{\mu}, +\end{align} + +\end_inset + +where we have used the fact that +\begin_inset Formula $\hat{T}^{\mu}$ +\end_inset + + is hermitian ( +\begin_inset Formula $[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{11}$ +\end_inset + + is pure real) that +\begin_inset Formula $\alpha,\beta$ +\end_inset + + are dummy indices. +\end_layout + +\begin_layout Section +Cubic Hamiltonian +\end_layout + +\begin_layout Standard +The cubic Hamiltonian is written as +\begin_inset Formula +\begin{align} +\hat{\mathcal{H}}^{(3)} & =\sqrt{M}\sum_{\langle i,j\rangle}\bigg\{\sum_{m,n=2}^{N}\bigg(V_{(3,1)}^{m}(i,j)\tilde{b}_{j,n}^{\dagger}\tilde{b}_{j,n}\tilde{b}_{j,m}+V_{(3,1)}^{m}(j,i)\tilde{b}_{i,n}^{\dagger}\tilde{b}_{i,n}\tilde{b}_{i,m}+h.c.\bigg)\nonumber \\ + & +\sum_{m,n,l=2}^{N}\bigg(V_{(3,2)}^{lmn}(i,j)\tilde{b}_{j,m}^{\dagger}\tilde{b}_{j,n}\tilde{b}_{i,l}+V_{(3,2)}^{lmn}(j,i)\tilde{b}_{i,m}^{\dagger}\tilde{b}_{i,n}\tilde{b}_{j,l}+h.c.\bigg)\bigg\}\nonumber \\ + & -\frac{1}{2\sqrt{M}}\sum_{i}\sum_{\mu=1}^{N^{2}-2}\mathcal{D}_{\mu}\sum_{m,n=2}^{N}\bigg([\tilde{\mathcal{T}}_{i}^{\mu}]_{1m}\tilde{b}_{i,n}^{\dagger}\tilde{b}_{i,n}\tilde{b}_{i,m}+h.c.\bigg), +\end{align} + +\end_inset + +where +\begin_inset Formula +\begin{equation} +V_{(3,1)}^{m}(i,j)=-\frac{1}{2}[\tilde{\mathcal{T}}_{i}^{\mu}]_{11}\mathcal{J}_{ij}^{\mu\nu}[\tilde{\mathcal{T}}_{j}^{\nu}]_{1m},\quad V_{(3,2)}^{lmn}=[\tilde{\mathcal{T}}_{i}]_{1l}\mathcal{J}_{ij}^{\mu\nu}\big([\tilde{\mathcal{T}}_{j}^{\nu}]_{mn}-\delta_{mn}[\tilde{\mathcal{T}}_{j}^{\nu}]_{11}\big). +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Again (assuming translation symmetry and) applying Fourier transforms Eq. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:fourier1" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + and +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:fourier2" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, we have (In this section, we drop the tilde for wave vectors in the magnetic + Brillouin zone for notation simplicity and introduce the notation +\begin_inset Formula $\bar{\bm{q}}=-\bm{q}$ +\end_inset + +) +\begin_inset Formula +\begin{equation} +\mathcal{H}^{(3)}=\frac{1}{\sqrt{N_{u}}}\sum_{\bm{q}_{a}\in MBZ}\sum_{\alpha_{a}}\sum_{\sigma_{a}\neq1}\delta\big(\sum_{a=1}^{3}\bm{q}_{a}\big)V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3})\tilde{b}_{\bar{\bm{q}}_{1}\alpha_{1}\sigma_{1}}^{\dagger}\tilde{b}_{\bm{q}_{2}\alpha_{2}\sigma_{2}}\tilde{b}_{\bm{q}_{3}\alpha_{3}\sigma_{3}}+h.c., +\end{equation} + +\end_inset + +with +\begin_inset Formula +\begin{equation} +V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3})=\sum_{\langle i,j\rangle'}\tilde{V}_{\langle i,j\rangle'}(1,2,3)+\sum_{\alpha}\tilde{V}_{\alpha}(1,2,3), +\end{equation} + +\end_inset + +where +\begin_inset Formula $\sum_{\langle i,j\rangle'}$ +\end_inset + + sums over translationally inequivalent bonds (culled bonds given by Sunny), + +\begin_inset Formula $\alpha$ +\end_inset + + is the sublattice index (in the magnetic unit cell), and +\begin_inset Formula $\sigma$ +\end_inset + + is the Schwinger boson flavor index (Recall that +\begin_inset Formula $\sigma\neq1$ +\end_inset + + because the +\begin_inset Quotes eld +\end_inset + + +\begin_inset Formula $1$ +\end_inset + + +\begin_inset Quotes erd +\end_inset + + is chosen to be condensed). + The first term includes the bond contributions to the cubic vertex that + arises from the exchange interactions. + The corresponding vertex function is +\begin_inset Formula +\begin{align} +\tilde{V}_{\langle i,j\rangle'}(1,2,3) & =\sqrt{M}\bigg[V_{(3,1)}^{\sigma_{3}}(\alpha_{i},\bm{\delta}_{\langle i,j\rangle})\delta_{\alpha_{1}\alpha_{j}}\delta_{\alpha_{2}\alpha_{j}}\delta_{\alpha_{3}\alpha_{j}}\delta_{\sigma_{1}\sigma_{2}}\nonumber \\ + & +V_{(3,1)}^{\sigma_{3}}(\alpha_{i},\bar{\bm{\delta}}_{\langle i,j\rangle})\delta_{\alpha_{1}\alpha_{i}}\delta_{\alpha_{2}\alpha_{i}}\delta_{\alpha_{3}\alpha_{i}}\delta_{\sigma_{1}\sigma_{2}}\nonumber \\ + & +V_{(3,2)}^{\sigma_{3}\sigma_{1}\sigma_{2}}(\alpha_{i},\bm{\delta}_{\langle i,j\rangle})e^{-i\bm{q}_{3}\cdot\bm{\delta}_{ij}}\delta_{\alpha_{1}\alpha_{j}}\delta_{\alpha_{2}\alpha_{j}}\delta_{\alpha_{3}\alpha_{i}}\nonumber \\ + & +V_{(3,2)}^{\sigma_{3}\sigma_{1}\sigma_{2}}(\alpha_{i},\bar{\bm{\delta}}_{\langle i,j\rangle})e^{i\bm{q}_{3}\cdot\bm{\delta}_{ij}}\delta_{\alpha_{1}\alpha_{i}}\delta_{\alpha_{2}\alpha_{i}}\delta_{\alpha_{3}\alpha_{j}}, +\end{align} + +\end_inset + +and +\begin_inset Formula +\begin{equation} +\tilde{V}_{\alpha}(1,2,3)=\frac{1}{2\sqrt{M}}\sum_{\mu}\mathcal{D}_{\mu}[\tilde{\mathcal{T}}_{i}^{\mu}]_{1\sigma_{3}}\delta_{\alpha_{1}\alpha}\delta_{\alpha_{2}\alpha}\delta_{\alpha_{3}\alpha}\delta_{\sigma_{1}\sigma_{2}}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Then we write the vertex functions in the quasi-particle basis by applying + the Bogoliubov transforms Eq. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:bk1" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +- +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:bk4" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. + [The useful formula to transform the combined index of sublattice and flavor + to a linear index: +\begin_inset Formula $(\alpha,\sigma)=(\alpha-1)(N-1)+\sigma-1$ +\end_inset + +]. + +\begin_inset Formula +\begin{align} +\tilde{b}_{\bar{\bm{q}}_{1}\alpha_{1}\sigma_{1}}^{\dagger}\tilde{b}_{\bm{q}_{2}\alpha_{2}\sigma_{2}}\tilde{b}_{\bm{q}_{3}\alpha_{3}\sigma_{3}} & =\sum_{n_{1}n_{2}n_{3}=1}^{(N-1)N_{m}}\big[V_{\bm{q}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\bm{q}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\bm{q}_{3}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\bm{q}_{1},n_{1}}\beta_{\bm{q}_{2},n_{2}}\beta_{\bm{q}_{3},n_{3}}\nonumber \\ + & +\big[\big[V_{\bm{q}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\bm{q}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\bm{q}_{3}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\bm{q}_{1},n_{1}}\beta_{\bar{\bm{q}}_{2},n_{2}}^{\dagger}\beta_{\bm{q}_{3},n_{3}}\nonumber \\ + & +\big[V_{\bm{q}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\bm{q}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\bm{q}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\bm{q}_{1},n_{1}}\beta_{\bm{q}_{2},n_{2}}\beta_{\bar{\bm{q}}_{3},n_{3}}^{\dagger}\nonumber \\ + & +\big[V_{\bm{q}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\bm{q}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\bm{q}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\bm{q}_{1},n_{1}}\beta_{\bar{\bm{q}}_{2},n_{2}}^{\dagger}\beta_{\bar{\bm{q}}_{3},n_{3}}^{\dagger}\nonumber \\ + & +\big[V_{\bm{q}_{1}}^{22}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\bm{q}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\bm{q}_{3}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\bar{\bm{q}}_{1},n_{1}}^{\dagger}\beta_{\bm{q}_{2},n_{2}}\beta_{\bm{q}_{3},n_{3}}\nonumber \\ + & +\big[V_{\bm{q}_{1}}^{22}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\bm{q}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\bm{q}_{3}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\bar{\bm{q}}_{1},n_{1}}^{\dagger}\beta_{\bar{\bm{q}}_{2},n_{2}}^{\dagger}\beta_{\bm{q}_{3},n_{3}}\nonumber \\ + & +\big[V_{\bm{q}_{1}}^{22}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\bm{q}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\bm{q}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\bar{\bm{q}}_{1},n_{1}}^{\dagger}\beta_{\bm{q}_{2},n_{2}}\beta_{\bar{\bm{q}}_{3},n_{3}}^{\dagger}\nonumber \\ + & +\big[V_{\bm{q}_{1}}^{22}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\bm{q}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\bm{q}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\bar{\bm{q}}_{1},n_{1}}^{\dagger}\beta_{\bar{\bm{q}}_{2},n_{2}}^{\dagger}\beta_{\bar{\bm{q}}_{3},n_{3}}^{\dagger}. +\end{align} + +\end_inset + +Here we write the matrix of eigenvectors +\begin_inset Formula $V_{\bm{k}}$ +\end_inset + + into +\begin_inset Formula $2\times2$ +\end_inset + + block form. + After putting the +\begin_inset Formula $\beta$ +\end_inset + + operators in normal ordering, we have the cubic vertex the additional cubic-lin +ear vertex (see appendix for details of the calculation) +\begin_inset Formula +\begin{align} + & V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3})\tilde{b}_{\bar{\bm{q}}_{1}\alpha_{1}\sigma_{1}}^{\dagger}\tilde{b}_{\bm{q}_{2}\alpha_{2}\sigma_{2}}\tilde{b}_{\bm{q}_{3}\alpha_{3}\sigma_{3}}+h.c.\nonumber \\ + & =\sum_{\{n_{i}\}}\tilde{V}_{n_{1}n_{2}n_{3}}^{(3,1)}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3})\beta_{\bar{\bm{q}}_{1},n_{1}}^{\dagger}\beta_{\bm{q}_{2},n_{2}}\beta_{\bm{q}_{3},n_{3}}+\sum_{\{n_{i}\}}\tilde{V}_{n_{1}n_{2}n_{3}}^{(3,2)}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3})\beta_{\bar{\bm{q}}_{1},n_{1}}^{\dagger}\beta_{\bar{\bm{q}}_{2},n_{2}}^{\dagger}\beta_{\bar{\bm{q}}_{3},n_{3}}^{\dagger}+h.c.\nonumber \\ + & +\tilde{V}_{n_{1}n_{2}n_{3}}^{(3,3)}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3})\beta_{\tilde{\bm{q}}_{3},n_{3}}+h.c., +\end{align} + +\end_inset + +where +\begin_inset Note Comment +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula +\begin{align} +\tilde{V}_{n_{1}n_{2}n_{3}}^{(3,1)}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3}) & =\sum_{\{\alpha_{i},\sigma_{i}\}}V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3})\big[V_{\bar{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\bm{q}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\bm{q}_{3}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\nonumber \\ + & +V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bm{q}_{3},\bm{q}_{2},\bm{q}_{1})\big[V_{\bm{q}_{3}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{3}}\big[V_{\bm{q}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\bm{q}_{1}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{1}}\nonumber \\ + & +V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bm{q}_{2},\bm{q}_{1},\bm{q}_{3})\big[V_{\bm{q}_{2}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{2}}\big[V_{\bm{q}_{1}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{1}}\big[V_{\bm{q}_{3}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\nonumber \\ + & +\big[V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bar{\bm{q}}_{1},\bar{\bm{q}}_{2},\bar{\bm{q}}_{3})\big]^{*}\big[V_{\bar{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}^{*}\big[V_{\bar{\bm{q}}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}^{*}\big[V_{\bar{\bm{q}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}^{*}\nonumber \\ + & +\big[V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bar{\bm{q}}_{3},\bar{\bm{q}}_{2},\bar{\bm{q}}_{1})\big]^{*}\big[V_{\bar{\bm{q}}_{3}}^{22}\big]_{(\alpha_{1},\sigma_{1}),n_{3}}^{*}\big[V_{\bar{\bm{q}}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}^{*}\big[V_{\bar{\bm{q}}_{1}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{1}}^{*}\nonumber \\ + & +\big[V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bar{\bm{q}}_{2},\bar{\bm{q}}_{1},\bar{\bm{q}}_{3})\big]^{*}\big[V_{\bar{\bm{q}}_{2}}^{22}\big]_{(\alpha_{1},\sigma_{1}),n_{2}}^{*}\big[V_{\bar{\bm{q}}_{1}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{1}}^{*}\big[V_{\bar{\bm{q}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}^{*}, +\end{align} + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align} +\tilde{V}_{n_{1}n_{2}n_{3}}^{(3,1)}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3}) & =\sum_{\{\alpha_{i},\sigma_{i}\}}V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3})\big[V_{\bar{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\bm{q}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\bm{q}_{3}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\nonumber \\ + & +V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bm{q}_{3},\bm{q}_{2},\bm{q}_{1})\big[V_{\bm{q}_{3}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{3}}\big[V_{\bm{q}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\bar{\bm{q}}_{1}}^{21}\big]_{(\alpha_{3},\sigma_{3}),n_{1}}^{*}\nonumber \\ + & +V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bm{q}_{2},\bm{q}_{1},\bm{q}_{3})\big[V_{\bm{q}_{2}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{2}}\big[V_{\bar{\bm{q}}_{1}}^{21}\big]_{(\alpha_{2},\sigma_{2}),n_{1}}^{*}\big[V_{\bm{q}_{3}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\nonumber \\ + & +\big[V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bar{\bm{q}}_{1},\bar{\bm{q}}_{2},\bar{\bm{q}}_{3})\big]^{*}\big[V_{\bar{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}^{*}\big[V_{\bm{q}_{2}}^{21}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\bm{q}_{3}}^{21}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\nonumber \\ + & +\big[V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bar{\bm{q}}_{3},\bar{\bm{q}}_{2},\bar{\bm{q}}_{1})\big]^{*}\big[V_{\bm{q}_{3}}^{11}\big]_{(\alpha_{1},\sigma_{1}),n_{3}}\big[V_{\bm{q}_{2}}^{21}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\bar{\bm{q}}_{1}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{1}}^{*}\nonumber \\ + & +\big[V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bar{\bm{q}}_{2},\bar{\bm{q}}_{1},\bar{\bm{q}}_{3})\big]^{*}\big[V_{\bm{q}_{2}}^{11}\big]_{(\alpha_{1},\sigma_{1}),n_{2}}\big[V_{\bar{\bm{q}}_{1}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{1}}^{*}\big[V_{\bm{q}_{3}}^{21}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}, +\end{align} + +\end_inset + +and +\begin_inset Note Comment +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula +\begin{align} +\tilde{V}_{n_{1}n_{2}n_{3}}^{(3,2)} & =\sum_{\{\alpha_{i},\sigma_{i}\}}V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3})\big[V_{\bm{q}_{1}}^{22}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\bm{q}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\bm{q}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\nonumber \\ + & +\big[V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bar{\bm{q}}_{3},\bar{\bm{q}}_{2},\bar{\bm{q}}_{1})\big]^{*}\big[V_{\bar{\bm{q}}_{3}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{3}}^{*}\big[V_{\bar{\bm{q}}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}^{*}\big[V_{\bar{\bm{q}}_{1}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{1}}^{*}, +\end{align} + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{align} +\tilde{V}_{n_{1}n_{2}n_{3}}^{(3,2)} & =\sum_{\{\alpha_{i},\sigma_{i}\}}V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3})\big[V_{\bar{\bm{q}}_{1}}^{11}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}^{*}\big[V_{\bar{\bm{q}}_{2}}^{21}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}^{*}\big[V_{\bar{\bm{q}}_{3}}^{21}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}^{*}\nonumber \\ + & +\big[V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bar{\bm{q}}_{3},\bar{\bm{q}}_{2},\bar{\bm{q}}_{1})\big]^{*}\big[V_{\bar{\bm{q}}_{3}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{3}}^{*}\big[V_{\bar{\bm{q}}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}^{*}\big[V_{\bar{\bm{q}}_{1}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{1}}^{*}, +\end{align} + +\end_inset + +and +\begin_inset Note Comment +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula +\begin{align} +\tilde{V}_{n_{1}n_{2}n_{3}}^{(3,3)}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3}) & =\sum_{\{\alpha_{i},\sigma_{i}\}}\bigg\{ V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bm{q}_{3},\bm{q}_{2},\bm{q}_{1})\big[V_{\bm{q}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\bm{q}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\bm{q}_{3}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\nonumber \\ + & +V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bm{q}_{3},\bm{q}_{2},\bm{q}_{1})\big[V_{\bm{q}_{3}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{3}}\big[V_{\bm{q}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\bm{q}_{1}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{1}}\nonumber \\ + & +V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bm{q}_{1},\bm{q}_{3},\bm{q}_{2})\big[V_{\bm{q}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\bm{q}_{3}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{3}}\big[V_{\bm{q}_{2}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{2}}\nonumber \\ + & +\big[V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bar{\bm{q}}_{1},\bar{\bm{q}}_{2},\bar{\bm{q}}_{3})\big]^{*}\big[V_{\bar{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}^{*}\big[V_{\bar{\bm{q}}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}^{*}\big[V_{\bar{\bm{q}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}^{*}\nonumber \\ + & +\big[V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bar{\bm{q}}_{1},\bar{\bm{q}}_{3},\bar{\bm{q}}_{2})\big]^{*}\big[V_{\bar{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}^{*}\big[V_{\bar{\bm{q}}_{3}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{3}}^{*}\big[V_{\bar{\bm{q}}_{2}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{2}}^{*}\nonumber \\ + & +\big[V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bar{\bm{q}}_{3},\bar{\bm{q}}_{2},\bar{\bm{q}}_{1})\big]^{*}\big[V_{\bar{\bm{q}}_{3}}^{22}\big]_{(\alpha_{1},\sigma_{1}),n_{3}}^{*}\big[V_{\bar{\bm{q}}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}^{*}\big[V_{\bar{\bm{q}}_{1}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{1}}^{*}\bigg\}\nonumber \\ + & \delta_{\bm{q}_{2}+\bm{q}_{1}}\delta_{n_{2}n_{1}}\beta_{\bm{q}_{3},n_{3}}. +\end{align} + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{align} +\tilde{V}_{n_{1}n_{2}n_{3}}^{(3,3)}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3}) & =\sum_{\{\alpha_{i},\sigma_{i}\}}\bigg\{ V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bm{q}_{3},\bm{q}_{2},\bm{q}_{1})\big[V_{\bm{q}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\bar{\bm{q}}_{2}}^{21}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}^{*}\big[V_{\bm{q}_{3}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\nonumber \\ + & +V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bm{q}_{3},\bm{q}_{2},\bm{q}_{1})\big[V_{\bm{q}_{3}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{3}}\big[V_{\bm{q}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\bar{\bm{q}}_{1}}^{21}\big]_{(\alpha_{3},\sigma_{3}),n_{1}}^{*}\nonumber \\ + & +V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bm{q}_{1},\bm{q}_{3},\bm{q}_{2})\big[V_{\bm{q}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\bm{q}_{3}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{3}}\big[V_{\bar{\bm{q}}_{2}}^{21}\big]_{(\alpha_{3},\sigma_{3}),n_{2}}^{*}\nonumber \\ + & +\big[V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bar{\bm{q}}_{1},\bar{\bm{q}}_{2},\bar{\bm{q}}_{3})\big]^{*}\big[V_{\bar{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}^{*}\big[V_{\bm{q}_{2}}^{21}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\bm{q}_{3}}^{21}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\nonumber \\ + & +\big[V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bar{\bm{q}}_{1},\bar{\bm{q}}_{3},\bar{\bm{q}}_{2})\big]^{*}\big[V_{\bar{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}^{*}\big[V_{\bm{q}_{3}}^{21}\big]_{(\alpha_{2},\sigma_{2}),n_{3}}\big[V_{\bm{q}_{2}}^{21}\big]_{(\alpha_{3},\sigma_{3}),n_{2}}\nonumber \\ + & +\big[V_{\alpha_{1}\alpha_{2}\alpha_{3}}^{\sigma_{1}\sigma_{2}\sigma_{3}}(\bar{\bm{q}}_{3},\bar{\bm{q}}_{2},\bar{\bm{q}}_{1})\big]^{*}\big[V_{\bm{q}_{3}}^{11}\big]_{(\alpha_{1},\sigma_{1}),n_{3}}\big[V_{\bar{\bm{q}}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}^{*}\big[V_{\bm{q}_{1}}^{21}\big]_{(\alpha_{3},\sigma_{3}),n_{1}}\bigg\}\nonumber \\ + & \delta_{\bm{q}_{2}+\bm{q}_{1}}\delta_{n_{2}n_{1}}\beta_{\bm{q}_{3},n_{3}}. +\end{align} + +\end_inset + +The final expression for the cubic interction is obtained after symmetrization + of the vertex: +\begin_inset Formula +\begin{align} +\hat{\mathcal{H}}^{(3)} & =\frac{1}{\sqrt{N_{u}}}\sum_{\bm{q}_{a}\in MBZ}\sum_{\alpha_{a}}\sum_{\sigma_{a}\neq1}\delta(\sum_{\bm{a}}\bm{q}_{a})\bigg[\frac{1}{2!}V_{n_{1}n_{2}n_{3}}^{(3,S1)}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3})\beta_{\bar{\bm{q}}_{1},n_{1}}^{\dagger}\beta_{\bm{q}_{2},n_{2}}\beta_{\bm{q}_{3},n_{3}}\\ + & +\frac{1}{3!}V_{n_{1}n_{2}n_{3}}^{(3,S2)}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3})\beta_{\bar{\bm{q}}_{1},n_{1}}^{\dagger}\beta_{\bar{\bm{q}}_{2},n_{2}}^{\dagger}\beta_{\bar{\bm{q}}_{3},n_{3}}^{\dagger}+V_{n_{1}n_{2}n_{3}}^{(3,S3)}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3})\beta_{\bm{q}_{3},n_{3}}\bigg]+h.c., +\end{align} + +\end_inset + +where +\begin_inset Formula +\begin{equation} +V_{n_{1}n_{2}n_{3}}^{(3,S1)}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3})=\sum_{P(2,3)}\tilde{V}_{n_{1}n_{2}n_{3}}^{(3,1)}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3}) +\end{equation} + +\end_inset + + +\begin_inset Formula +\begin{equation} +V_{n_{1}n_{2}n_{3}}^{(S2)}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3})=\sum_{P(1,2,3)}\tilde{V}_{n_{1}n_{2}n_{3}}^{(3,2)}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3}) +\end{equation} + +\end_inset + + +\begin_inset Formula +\begin{equation} +V_{n_{1}n_{2}n_{3}}^{(S3)}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3})=\sum_{P(2,3)}\tilde{V}_{n_{1}n_{2}n_{3}}^{(3,3)}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3}), +\end{equation} + +\end_inset + +where +\begin_inset Formula $P(1,2)$ +\end_inset + + denotes the permutation of +\begin_inset Formula $(\bm{q}_{1},n_{1})$ +\end_inset + + and +\begin_inset Formula $(\bm{q}_{2},n_{2})$ +\end_inset + +. +\end_layout + +\begin_layout Section +Quartic Contribution +\end_layout + +\begin_layout Standard +The quartic Hamiltonian is written as +\begin_inset Formula +\begin{align} +\mathcal{H}^{(4)} & =\sum_{\langle i,j\rangle}\big[V_{(4,1)}^{\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}}\tilde{b}_{i\sigma_{1}}^{\dagger}\tilde{b}_{i\sigma_{2}}\tilde{b}_{j\sigma_{3}}^{\dagger}\tilde{b}_{j\sigma_{4}}+(V_{(4,2)}^{\sigma_{1}\sigma_{2}\sigma_{3}}\tilde{b}_{i\sigma_{1}}\tilde{b}_{j\sigma_{2}}^{\dagger}\tilde{b}_{j\sigma_{2}}\tilde{b}_{j\sigma_{3}}+h.c.)\nonumber \\ + & +(V_{(4,3)}^{\sigma_{1}\sigma_{2}\sigma_{3}}\tilde{b}_{i\sigma_{1}}^{\dagger}\tilde{b}_{j\sigma_{2}}^{\dagger}\tilde{b}_{j\sigma_{2}}\tilde{b}_{j\sigma_{3}}+h.c.)\big]\label{eq:quarticHam} +\end{align} + +\end_inset + +where +\begin_inset Formula +\begin{align} +V_{(4,1)}^{\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}} & =\mathcal{J}_{ij}^{\mu\nu}\big([\tilde{\mathcal{T}}_{i}^{\mu}]_{\sigma_{1}\sigma_{2}}-\delta_{\sigma_{1}\sigma_{2}}[\tilde{\mathcal{T}}_{i}^{\mu}]_{11}\big)\big([\tilde{\mathcal{T}}_{j}^{\nu}]_{\sigma_{3}\sigma_{4}}-\delta_{\sigma_{3}\sigma_{4}}[\tilde{\mathcal{T}}_{j}^{\nu}]_{11}\big)\\ +V_{(4,2)}^{\sigma_{1}\sigma_{2}\sigma_{3}} & =\mathcal{J}_{ij}^{\mu\nu}[\tilde{\mathcal{T}}_{i}^{\mu}]_{\sigma_{1}1}[\tilde{\mathcal{T}}_{j}^{\nu}]_{1\sigma_{3}}\\ +V_{(4,3)}^{\sigma_{1}\sigma_{2}\sigma_{3}} & =\mathcal{J}_{ij}^{\mu\nu}[\tilde{\mathcal{T}}_{i}^{\mu}]_{1\sigma_{1}}[\tilde{\mathcal{T}}_{j}^{\nu}]_{1\sigma_{3}}. +\end{align} + +\end_inset + +To deal with these quartic terms, we will perform the +\emph on +Hartree-Fock-like +\emph default + mean-field decouplings +\begin_inset ERT +status open + +\begin_layout Plain Layout + +~ +\backslash +footnote{The decouplings are not self-consistent so they are not rigorously + speaking Hartree-Fock.} +\end_layout + +\end_inset + + by introducing the following mean-field parameters (that are quadratic + in H-P bosons) +\begin_inset Formula +\begin{align} +N_{(\alpha_{1},\sigma_{1}),(\alpha_{2},\sigma_{2})} & (\bm{\delta}_{12})\equiv\frac{1}{N_{u}}\sum_{\langle i,j\rangle}\langle\tilde{b}_{i,(\alpha_{1}\sigma_{1})}^{\dagger}\tilde{b}_{j,(\alpha_{2},\sigma_{2})}\rangle_{0}\nonumber \\ + & =\frac{1}{N_{u}}\sum_{\tilde{\bm{q}}\in BZ}\sum_{n}[V_{\tilde{\bm{q}}}^{21}]_{(\alpha_{1}\sigma_{1}),n}[V_{\tilde{\bm{q}}}^{21}]_{(\alpha_{2}\sigma_{2}),n}^{*}e^{i\bm{q}\cdot\bm{\delta}_{12}} +\end{align} + +\end_inset + + +\begin_inset Formula +\begin{align} +\Delta_{(\alpha_{1},\sigma_{1}),(\alpha_{2},\sigma_{2})} & (\bm{\delta}_{12})\equiv\frac{1}{N_{u}}\sum_{\langle i,j\rangle}\langle\tilde{b}_{i,(\alpha_{1}\sigma_{1})}\tilde{b}_{j,(\alpha_{2},\sigma_{2})}\rangle_{0}\nonumber \\ + & =\frac{1}{N_{u}}\sum_{\tilde{\bm{q}}\in BZ}\sum_{n}[V_{\tilde{\bm{q}}}^{11}]_{(\alpha_{1}\sigma_{1}),n}[V_{\tilde{\bm{q}}}^{21}]_{(\alpha_{2}\sigma_{2}),n}^{*}e^{i\bm{q}\cdot\bm{\delta}_{12}}, +\end{align} + +\end_inset + +where +\begin_inset Formula $\langle\ldots\rangle_{0}$ +\end_inset + + denotes the +\emph on + bare-vacuum +\emph default + of Bogoliubov particles (LSWT). + By applying the Wick's theorem, we obtain the decouplings for the three + quartic terms +\begin_inset Formula +\begin{align} +\tilde{b}_{i\alpha_{1}\sigma_{1}}^{\dagger}\tilde{b}_{i\alpha_{1}\sigma_{2}}\tilde{b}_{j\alpha_{2}\sigma_{3}}^{\dagger}\tilde{b}_{j\alpha_{2}\sigma_{4}} & \simeq N_{(\alpha_{1},\sigma_{1})(\alpha_{1},\sigma_{2})}(\bm{0})\tilde{b}_{j\alpha_{2}\sigma_{3}}^{\dagger}\tilde{b}_{j\alpha_{2}\sigma_{4}}+N_{(\alpha_{2},\sigma_{3})(\alpha_{2},\sigma_{4})}(\bm{0})\tilde{b}_{i\alpha_{1}\sigma_{1}}^{\dagger}\tilde{b}_{i\alpha_{1}\sigma_{2}}\nonumber \\ + & +\Delta_{(\alpha_{1},\sigma_{1})(\alpha_{2},\sigma_{3})}^{*}(\bm{\delta}_{ij})\tilde{b}_{i\alpha_{1}\sigma_{2}}\tilde{b}_{j\alpha_{2}\sigma_{4}}+\Delta_{(\alpha_{1},\sigma_{2})(\alpha_{2},\sigma_{4})}(\bm{\delta}_{ij})\tilde{b}_{i\alpha_{1}\sigma_{1}}^{\dagger}\tilde{b}_{j\alpha_{2}\sigma_{3}}^{\dagger}\nonumber \\ + & +N_{(\alpha_{1},\sigma_{1})(\alpha_{2},\sigma_{4})}(\bm{\delta}_{ij})\tilde{b}_{i\alpha_{1}\sigma_{2}}\tilde{b}_{j\alpha_{2}\sigma_{3}}^{\dagger}+N_{(\alpha_{1},\sigma_{2})(\alpha_{2},\sigma_{3})}^{*}(\bm{\delta}_{ij})\tilde{b}_{i\alpha_{1}\sigma_{1}}^{\dagger}\tilde{b}_{j\alpha_{2}\sigma_{4}},\label{eq:quartic1} +\end{align} + +\end_inset + + +\begin_inset Formula +\begin{align} +\tilde{b}_{i\sigma_{1}}\tilde{b}_{j\sigma_{2}}^{\dagger}\tilde{b}_{j\sigma_{2}}\tilde{b}_{j\sigma_{3}} & \simeq N_{(\alpha_{1},\sigma_{1})(\alpha_{2},\sigma_{2})}^{*}(\bm{\delta}_{ij})\tilde{b}_{j\sigma_{2}}\tilde{b}_{j\sigma_{3}}+\Delta_{(\alpha_{2},\sigma_{2}),(\alpha_{2},\sigma_{3})}(\bm{0})\tilde{b}_{i\sigma_{1}}\tilde{b}_{j\sigma_{2}}^{\dagger}\nonumber \\ + & +\Delta_{(\alpha_{1},\sigma_{1}),(\alpha_{2},\sigma_{2})}(\bm{\delta}_{ij})\tilde{b}_{j\sigma_{2}}^{\dagger}\tilde{b}_{j\sigma_{3}}+N_{(\alpha_{2},\sigma_{2})(\alpha_{2},\sigma_{3})}(\bm{0})\tilde{b}_{i\sigma_{1}}\tilde{b}_{j\sigma_{2}}\nonumber \\ + & +\Delta_{(\alpha_{1},\sigma_{1}),(\alpha_{2},\sigma_{3})}(\bm{\delta}_{ij})\tilde{b}_{j\sigma_{2}}^{\dagger}\tilde{b}_{j\sigma_{2}}+N_{(\alpha_{2},\sigma_{2})(\alpha_{2},\sigma_{2})}(\bm{0})\tilde{b}_{i\sigma_{1}}\tilde{b}_{j\sigma_{3}}, +\end{align} + +\end_inset + +and +\begin_inset Formula +\begin{align} +\tilde{b}_{i\sigma_{1}}^{\dagger}\tilde{b}_{j\sigma_{2}}^{\dagger}\tilde{b}_{j\sigma_{2}}\tilde{b}_{j\sigma_{3}} & \simeq\Delta_{(\alpha_{1},\sigma_{1}),(\alpha_{2},\sigma_{2})}^{*}(\bm{\delta}_{ij})\tilde{b}_{j\sigma_{2}}\tilde{b}_{j\sigma_{3}}+\Delta_{(\alpha_{2},\sigma_{2}),(\alpha_{2},\sigma_{3})}\tilde{b}_{i\sigma_{1}}^{\dagger}\tilde{b}_{j\sigma_{2}}^{\dagger}\nonumber \\ + & +N_{(\alpha_{1},\sigma_{1}),(\alpha_{2},\sigma_{2})}(\bm{\delta}_{ij})\tilde{b}_{j\sigma_{2}}^{\dagger}\tilde{b}_{j\sigma_{3}}+N_{(\alpha_{2},\sigma_{2})(\alpha_{2},\sigma_{3})}(\bm{0})\tilde{b}_{i\sigma_{1}}^{\dagger}\tilde{b}_{j\sigma_{2}}^{\dagger}\nonumber \\ + & +N_{(\alpha_{1},\sigma_{1}),(\alpha_{2},\sigma_{3})}(\bm{\delta}_{ij})\tilde{b}_{j\sigma_{2}}^{\dagger}\tilde{b}_{j\sigma_{2}}+N_{(\alpha_{2},\sigma_{2})(\alpha_{2},\sigma_{2})}(\bm{0})\tilde{b}_{i\sigma_{1}}^{\dagger}\tilde{b}_{j\sigma_{3}}.\label{eq:quartic3} +\end{align} + +\end_inset + +Our decoupling scheme simply leads to a renormalization of the quadratic + Hamiltonian: +\begin_inset Formula +\begin{equation} +\mathcal{H}^{(4)}\simeq\mathcal{H}_{NO}^{(2)}=\sum_{\bm{q}}\sum_{n,n'}\big[V_{nn'}^{(4,N)}(\bm{q})\beta_{\bm{q},n}^{\dagger}\beta_{\bm{q},n'}+(V_{nn'}^{(4,A)}(\bm{q})\beta_{\bar{\bm{q}},n}\beta_{\bm{q},n}+h.c.)\big]. +\end{equation} + +\end_inset + +Note that in presence of quantum fluctuations, the Bogoliubov particles + no-longer correspond to the eigenstates. + As a result, the quartic Hamiltonian after the normal ordering includes + both normal +\begin_inset Formula $V_{nn'}^{(4,N)}(\bm{q})$ +\end_inset + + and anomalous +\begin_inset Formula $V_{nn'}^{(4,A)}(\bm{q})$ +\end_inset + + contributions. + From the above derivation, we see that +\begin_inset Formula $\mathcal{H}_{NO}^{(2)}$ +\end_inset + + is of order +\begin_inset Formula $M^{0}$ +\end_inset + +. + Therefore, only the diagonal normal contribution arising from the normal + vertex +\begin_inset Formula $V_{nn'}^{(4,N)}(\bm{q})\delta_{nn'}$ +\end_inset + + gives a relative correction of +\begin_inset Formula $1/M$ +\end_inset + + to the bare single-particle energy. +\end_layout + +\begin_layout Standard +The next step is to derive +\begin_inset Formula $V_{nn'}^{(4,N)}(\bm{q})\delta_{nn'}$ +\end_inset + + . + Let us use the notation +\begin_inset Formula $f_{N}$ +\end_inset + + to denote such contributions, we have: +\begin_inset Formula +\begin{align} +f_{N}(\tilde{b}_{i\alpha_{1}\sigma_{1}}^{\dagger}\tilde{b}_{j\alpha_{2}\sigma_{2}}) & =\sum_{n}\big[V_{\bm{q}}^{11}\big]_{(\alpha_{1},\sigma_{1}),n}^{*}\big[V_{\bm{q}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n}e^{i\bm{q}\cdot\bm{\delta}_{ij}}+\big[V_{\bar{\bm{q}}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n}^{*}\big[V_{\bar{\bm{q}}}^{21}\big]_{(\alpha_{2},\sigma_{2}),n}e^{-i\bm{q}\cdot\bm{\delta}_{ij}}\\ +f_{N}(\tilde{b}_{i\alpha_{1}\sigma_{1}}\tilde{b}_{j\alpha_{2}\sigma_{2}}) & =\sum_{n}\big[V_{\bar{\bm{q}}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n}\big[V_{\bm{q}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n}e^{i\tilde{\bm{q}}\cdot\bm{\delta}_{ij}}+\big[V_{\bar{\bm{q}}}^{11}\big]_{(\alpha_{1},\sigma_{1}),n}\big[V_{\bar{\bm{q}}}^{21}\big]_{(\alpha_{2},\sigma_{2}),n}^{*}e^{-i\bm{q}\cdot\bm{\delta}_{ij}}. +\end{align} + +\end_inset + +We can then plug in the above results to Eq +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:quarticHam" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + and Eqs +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:quartic1" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +- +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:quartic3" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + to obtain the general expression for +\begin_inset Formula $V_{nn'}^{(4,N)}(\bm{q})\delta_{nn'}$ +\end_inset + + . + We will let computer to perform this simple algebra. +\end_layout + +\begin_layout Section +One-Loop diagrams +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +appendix +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Section +Normal ordering of the cubic vertex +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align} +\tilde{b}_{\bar{\tilde{\bm{q}}}_{1}\alpha_{1}\sigma_{1}}^{\dagger}\tilde{b}_{\tilde{\bm{q}}_{2}\alpha_{2}\sigma_{2}}\tilde{b}_{\tilde{\bm{q}}_{3}\alpha_{3}\sigma_{3}} & =\sum_{n_{1}n_{2}n_{3}=1}^{(N-1)N_{m}}\big[V_{\tilde{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\tilde{\bm{q}}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{3}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\tilde{\bm{q}}_{1},n_{1}}\beta_{\tilde{\bm{q}}_{2},n_{2}}\beta_{\tilde{\bm{q}}_{3},n_{3}}\nonumber \\ + & +\big[V_{\tilde{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\tilde{\bm{q}}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{3}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\tilde{\bm{q}}_{1},n_{1}}\beta_{\tilde{\bar{\bm{q}}}_{2},n_{2}}^{\dagger}\beta_{\tilde{\bm{q}}_{3},n_{3}}\nonumber \\ + & +\big[V_{\tilde{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\tilde{\bm{q}}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\tilde{\bm{q}}_{1},n_{1}}\beta_{\tilde{\bm{q}}_{2},n_{2}}\beta_{\tilde{\bar{\bm{q}}}_{3},n_{3}}^{\dagger}\nonumber \\ + & +\big[V_{\tilde{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\tilde{\bm{q}}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\tilde{\bm{q}}_{1},n_{1}}\beta_{\tilde{\bar{\bm{q}}}_{2},n_{2}}^{\dagger}\beta_{\tilde{\bar{\bm{q}}}_{3},n_{3}}^{\dagger}\nonumber \\ + & +\big[V_{\tilde{\bm{q}}_{1}}^{22}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\tilde{\bm{q}}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{3}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\tilde{\bar{\bm{q}}}_{1},n_{1}}^{\dagger}\beta_{\tilde{\bm{q}}_{2},n_{2}}\beta_{\tilde{\bm{q}}_{3},n_{3}}\nonumber \\ + & +\big[V_{\tilde{\bm{q}}_{1}}^{22}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\tilde{\bm{q}}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{3}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\tilde{\bar{\bm{q}}}_{1},n_{1}}^{\dagger}\beta_{\tilde{\bar{\bm{q}}}_{2},n_{2}}^{\dagger}\beta_{\tilde{\bm{q}}_{3},n_{3}}\nonumber \\ + & +\big[V_{\tilde{\bm{q}}_{1}}^{22}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\tilde{\bm{q}}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\tilde{\bar{\bm{q}}}_{1},n_{1}}^{\dagger}\beta_{\tilde{\bm{q}}_{2},n_{2}}\beta_{\tilde{\bar{\bm{q}}}_{3},n_{3}}^{\dagger}\nonumber \\ + & +\big[V_{\tilde{\bm{q}}_{1}}^{22}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\tilde{\bm{q}}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\tilde{\bar{\bm{q}}}_{1},n_{1}}^{\dagger}\beta_{\tilde{\bar{\bm{q}}}_{2},n_{2}}^{\dagger}\beta_{\tilde{\bar{\bm{q}}}_{3},n_{3}}^{\dagger}. +\end{align} + +\end_inset + +Below we work out the normal ordering +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align} + & \big[V_{\tilde{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\tilde{\bm{q}}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{3}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\tilde{\bm{q}}_{1},n_{1}}\beta_{\tilde{\bar{\bm{q}}}_{2},n_{2}}^{\dagger}\beta_{\tilde{\bm{q}}_{3},n_{3}}\nonumber \\ + & =\big[V_{\tilde{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\tilde{\bm{q}}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{3}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}(\delta_{\tilde{\bm{q}}_{1}+\tilde{\bm{q}}_{2}}\delta_{n_{1}n_{2}}+\beta_{\tilde{\bar{\bm{q}}}_{2},n_{2}}^{\dagger}\beta_{\tilde{\bm{q}}_{1},n_{1}})\beta_{\tilde{\bm{q}}_{3},n_{3}}\nonumber \\ + & =\big[V_{\tilde{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\tilde{\bm{q}}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{3}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\tilde{\bar{\bm{q}}}_{2},n_{2}}^{\dagger}\beta_{\tilde{\bm{q}}_{1},n_{1}}\beta_{\tilde{\bm{q}}_{3},n_{3}}\nonumber \\ + & +\big[V_{\tilde{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\tilde{\bm{q}}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{3}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\delta_{\tilde{\bm{q}}_{1}+\tilde{\bm{q}}_{2}}\delta_{n_{1}n_{2}}\beta_{\tilde{\bm{q}}_{3},n_{3}}\nonumber \\ + & ={\color{blue}\big[V_{\tilde{\bm{q}}_{2}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{2}}\big[V_{\tilde{\bm{q}}_{1}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{1}}\big[V_{\tilde{\bm{q}}_{3}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\tilde{\bar{\bm{q}}}_{1},n_{2}}^{\dagger}\beta_{\tilde{\bm{q}}_{2},n_{1}}\beta_{\tilde{\bm{q}}_{3},n_{3}}}\nonumber \\ + & {\color{blue}+\big[V_{\tilde{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\tilde{\bm{q}}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{3}}^{11}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\delta_{\tilde{\bm{q}}_{1}+\tilde{\bm{q}}_{2}}\delta_{n_{1}n_{2}}\beta_{\tilde{\bm{q}}_{3},n_{3}}} +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align} + & \big[V_{\tilde{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\tilde{\bm{q}}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\tilde{\bm{q}}_{1},n_{1}}\beta_{\tilde{\bm{q}}_{2},n_{2}}\beta_{\tilde{\bar{\bm{q}}}_{3},n_{3}}^{\dagger}\nonumber \\ + & =\big[V_{\tilde{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\tilde{\bm{q}}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\tilde{\bm{q}}_{1},n_{1}}(\delta_{\tilde{\bm{q}}_{2}+\tilde{\bm{q}}_{3}}\delta_{n_{2}n_{3}}+\beta_{\tilde{\bar{\bm{q}}}_{3},n_{3}}^{\dagger}\beta_{\tilde{\bm{q}}_{2},n_{2}})\nonumber \\ + & =\big[V_{\tilde{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\tilde{\bm{q}}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\bigg[\beta_{\tilde{\bm{q}}_{1},n_{1}}\delta_{\tilde{\bm{q}}_{2}+\tilde{\bm{q}}_{3}}\delta_{n_{2}n_{3}}\nonumber \\ + & +(\delta_{\tilde{\bm{q}}_{3}+\tilde{\bm{q}}_{1}}\delta_{n_{3}n_{1}}+\beta_{\tilde{\bar{\bm{q}}}_{3},n_{3}}^{\dagger}\beta_{\tilde{\bm{q}}_{1},n_{1}})\beta_{\tilde{\bm{q}}_{2},n_{2}}\bigg]\nonumber \\ + & =\big[V_{\tilde{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\tilde{\bm{q}}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\tilde{\bar{\bm{q}}}_{3},n_{3}}^{\dagger}\beta_{\tilde{\bm{q}}_{1},n_{1}}\beta_{\tilde{\bm{q}}_{2},n_{2}}\nonumber \\ + & +\big[V_{\tilde{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\tilde{\bm{q}}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\delta_{\tilde{\bm{q}}_{2}+\tilde{\bm{q}}_{3}}\delta_{n_{2}n_{3}}\beta_{\tilde{\bm{q}}_{1},n_{1}}\nonumber \\ + & +\big[V_{\tilde{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\tilde{\bm{q}}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\delta_{\tilde{\bm{q}}_{3}+\tilde{\bm{q}}_{1}}\delta_{n_{3}n_{1}}\beta_{\tilde{\bm{q}}_{2},n_{2}}\nonumber \\ + & {\color{blue}=\big[V_{\tilde{\bm{q}}_{3}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{3}}\big[V_{\tilde{\bm{q}}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{1}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{1}}\beta_{\tilde{\bar{\bm{q}}}_{1},n_{2}}^{\dagger}\beta_{\tilde{\bm{q}}_{2},n_{1}}\beta_{\tilde{\bm{q}}_{3},n_{3}}}\\ + & {\color{blue}+(\big[V_{\tilde{\bm{q}}_{3}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{3}}\big[V_{\tilde{\bm{q}}_{2}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{1}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{1}}+\big[V_{\tilde{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\tilde{\bm{q}}_{3}}^{11}\big]_{(\alpha_{2},\sigma_{2}),n_{3}}\big[V_{\tilde{\bm{q}}_{2}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{2}})\delta_{\tilde{\bm{q}}_{2}+\tilde{\bm{q}}_{1}}\delta_{n_{2}n_{1}}\beta_{\tilde{\bm{q}}_{3},n_{3}}}\nonumber +\end{align} + +\end_inset + + +\begin_inset Formula +\begin{align} + & \bigg(\big[V_{\tilde{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}\big[V_{\tilde{\bm{q}}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}\big[V_{\tilde{\bm{q}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}\beta_{\tilde{\bm{q}}_{1},n_{1}}\beta_{\tilde{\bar{\bm{q}}}_{2},n_{2}}^{\dagger}\beta_{\tilde{\bar{\bm{q}}}_{3},n_{3}}^{\dagger}\bigg)^{\dagger}\nonumber \\ + & =\big[V_{\tilde{\bm{q}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}^{*}\big[V_{\tilde{\bm{q}}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}^{*}\big[V_{\tilde{\bm{q}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}^{*}\beta_{\tilde{\bar{\bm{q}}}_{3},n_{1}}\beta_{\tilde{\bar{\bm{q}}}_{2},n_{2}}\beta_{\tilde{\bm{q}}_{1},n_{1}}^{\dagger}\nonumber \\ + & =\big[V_{\tilde{\bar{\bm{q}}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}^{*}\big[V_{\tilde{\bar{\bm{q}}}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}^{*}\big[V_{\tilde{\bar{\bm{q}}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}^{*}\beta_{\tilde{\bm{q}}_{3},n_{1}}\beta_{\tilde{\bm{q}}_{2},n_{2}}\beta_{\tilde{\bar{\bm{q}}}_{1},n_{1}}^{\dagger}\nonumber \\ + & =\big[V_{\tilde{\bar{\bm{q}}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}^{*}\big[V_{\tilde{\bar{\bm{q}}}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}^{*}\big[V_{\tilde{\bar{\bm{q}}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}^{*}\beta_{\tilde{\bm{q}}_{3},n_{1}}\bigg(\delta_{\tilde{\bm{q}}_{1}+\tilde{\bm{q}}_{2}}\delta_{n_{1}n_{2}}+\beta_{\tilde{\bar{\bm{q}}}_{1},n_{1}}^{\dagger}\beta_{\tilde{\bm{q}}_{2},n_{2}}\bigg)\nonumber \\ + & =\big[V_{\tilde{\bar{\bm{q}}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}^{*}\big[V_{\tilde{\bar{\bm{q}}}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}^{*}\big[V_{\tilde{\bar{\bm{q}}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}^{*}\delta_{\tilde{\bm{q}}_{1}+\tilde{\bm{q}}_{2}}\delta_{n_{1}n_{2}}\beta_{\tilde{\bm{q}}_{3},n_{1}}\nonumber \\ + & +\big[V_{\tilde{\bar{\bm{q}}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}^{*}\big[V_{\tilde{\bar{\bm{q}}}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}^{*}\big[V_{\tilde{\bar{\bm{q}}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}^{*}\delta_{\tilde{\bm{q}}_{1}+\tilde{\bm{q}}_{3}}\delta_{n_{1}n_{3}}\beta_{\tilde{\bm{q}}_{2},n_{2}}\nonumber \\ + & +\big[V_{\tilde{\bar{\bm{q}}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}^{*}\big[V_{\tilde{\bar{\bm{q}}}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}^{*}\big[V_{\tilde{\bar{\bm{q}}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}^{*}\beta_{\tilde{\bar{\bm{q}}}_{1},n_{1}}^{\dagger}\beta_{\tilde{\bm{q}}_{2},n_{2}}\beta_{\tilde{\bm{q}}_{3},n_{1}}\nonumber \\ + & {\color{blue}=\big[V_{\tilde{\bar{\bm{q}}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}^{*}\big[V_{\tilde{\bar{\bm{q}}}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}^{*}\big[V_{\tilde{\bar{\bm{q}}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}^{*}\beta_{\tilde{\bar{\bm{q}}}_{1},n_{1}}^{\dagger}\beta_{\tilde{\bm{q}}_{2},n_{2}}\beta_{\tilde{\bm{q}}_{3},n_{1}}}\\ + & {\color{blue}+\bigg(\big[V_{\tilde{\bar{\bm{q}}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}^{*}\big[V_{\tilde{\bar{\bm{q}}}_{2}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{2}}^{*}\big[V_{\tilde{\bar{\bm{q}}}_{3}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{3}}^{*}+\big[V_{\tilde{\bar{\bm{q}}}_{1}}^{21}\big]_{(\alpha_{1},\sigma_{1}),n_{1}}^{*}\big[V_{\tilde{\bar{\bm{q}}}_{3}}^{12}\big]_{(\alpha_{2},\sigma_{2}),n_{3}}^{*}\big[V_{\tilde{\bar{\bm{q}}}_{2}}^{12}\big]_{(\alpha_{3},\sigma_{3}),n_{2}}^{*}\bigg)\delta_{\tilde{\bm{q}}_{1}+\tilde{\bm{q}}_{2}}\delta_{n_{1}n_{2}}\beta_{\tilde{\bm{q}}_{3},n_{1}}}\nonumber +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +\begin_inset Formula +\begin{align} +\mathcal{S}_{\text{dssf}}^{\mu\nu}(\bm{q},\omega) & =\frac{M}{N_{m}}\bigg(\sum_{\bm{d}}e^{-i\bm{q}\cdot\bm{d}}\sum_{\alpha=2}^{N}\sum_{n=1}^{(N-1)N_{m}}\big([\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{\alpha1}[V_{\tilde{\bm{q}}}]_{d_{\alpha},n}^{*}+[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{1\alpha}[V_{\tilde{\bm{q}}}]_{d_{\alpha},n+(N-1)N_{m}}\big)\bigg)^{*}\nonumber \\ + & \times\bigg(\sum_{\bm{d}}e^{i\bm{q}\cdot\bm{d}}\sum_{\alpha=2}^{N}\sum_{n=1}^{(N-1)N_{m}}\big([\tilde{\mathcal{T}}_{\bm{d}}^{\nu}]_{\alpha1}[V_{-\tilde{\bm{q}}}]_{d_{\alpha},n}^{*}+[\tilde{\mathcal{T}}_{\bm{d}}^{\nu}]_{1\alpha}[V_{-\tilde{\bm{q}}}]_{d_{\alpha},n+(N-1)N_{m}}\big)\bigg)\delta(\omega-\omega_{\tilde{\bm{q}},n})\nonumber \\ + & =\frac{M}{N_{m}}\bigg(\sum_{\bm{d}}e^{-i\bm{q}\cdot\bm{d}}\sum_{\alpha=2}^{N}\sum_{n=1}^{(N-1)N_{m}}\big([\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{\alpha1}[V_{\tilde{\bm{q}}}]_{d_{\alpha},n}^{*}+[\tilde{\mathcal{T}}_{\bm{d}}^{\mu}]_{1\alpha}[V_{\tilde{\bm{q}}}]_{d_{\alpha},n+(N-1)N_{m}}\big)\bigg)^{*}\nonumber \\ + & \times\bigg(\sum_{\bm{d}}e^{i\bm{q}\cdot\bm{d}}\sum_{\alpha=2}^{N}\sum_{n=1}^{(N-1)N_{m}}\big([\tilde{\mathcal{T}}_{\bm{d}}^{\nu}]_{\alpha1}[V_{\bm{q}}]_{d_{\alpha}+(N-1)N_{m},n+(N-1)N_{m}}+[\tilde{\mathcal{T}}_{\bm{d}}^{\nu}]_{1\alpha}[V_{\tilde{\bm{q}}}]_{d_{\alpha}+(N-1)N_{m},n}^{*}\big)\bigg)\delta(\omega-\omega_{\tilde{\bm{q}},n}), +\end{align} + +\end_inset + +where we have used the facts that +\begin_inset Formula +\begin{align} +[V_{-\tilde{\bm{q}}}]_{d_{\alpha},n}^{*} & =[V_{\bm{q}}]_{d_{\alpha}+(N-1)N_{m},n+(N-1)N_{m}}\\{} +[V_{-\tilde{\bm{q}}}]_{d_{\alpha},n+(N-1)N_{m}} & =[V_{\tilde{\bm{q}}}]_{d_{\alpha}+(N-1)N_{m},n}^{*} +\end{align} + +\end_inset + +that can be read from Eq. +\begin_inset ERT +status open + +\begin_layout Plain Layout + +~ +\end_layout + +\end_inset + + +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:bk1" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +- +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:bk4" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. +\end_layout + +\end_inset + + +\end_layout + +\end_body +\end_document