## 5 Correlation Functions in the Replicated Formalism  
As detailed in Appendix A, the matrix \(\mathbb{M}\) in Equation (75) is diagonalized by a well-known unitary transformation, reading  
\[\tilde{\mathbb{M}}^{-1}(Q)=\text{diag}\{\tilde{\mathcal{M}}_{R}^{-1}(Q), \underbrace{\tilde{\mathcal{M}}^{-1}(Q),\ldots,\tilde{\mathcal{M}}^{-1}(Q)}_{ R-1\text{ times}}\} \tag{78}\]  
with  
\[\mathbb{M}_{R}^{-1}(Q)= \begin{pmatrix}n_{0}\frac{\hbar^{2}q^{2}}{m}&-\hbar \omega_{n}\\ +\hbar\omega_{n}&\frac{\hbar^{2}q^{2}}{4 mn_{0}}+\bar{V}(\mathbf{q})-R\gamma^{ 2}(\mathbf{q},\omega_{n})\end{pmatrix} . \tag{79}\]  
Up to this level of approximation, correlation functions are specified by the entries of the fluctuation propagator. For the pure system, by assuming translational invariance both in time and space, we have  
\[\left\langle\chi_{j}(\mathbf{q},\omega_{n})\chi_{k}(\mathbf{q}^{\prime},\omega _{n^{\prime}})\right\rangle_{\text{th}}=\mathcal{M}_{jk}(\mathbf{q},\omega_{ n})\delta(\mathbf{q}+\mathbf{q}^{\prime})\beta\hbar\delta_{n,-n^{\prime}}. \tag{80}\]  
On the other hand, according to Equation (68), the replicated formalism provides both the disorder and the thermal average at the same time. For instance, concerning the density-density correlation function, we have  
\[ \begin{split}\left\langle\pi^{*}(\mathbf{q},\omega_{n})\pi( \mathbf{q}^{\prime},\omega_{n^{\prime}})\right\rangle_{\text{dis+th}}& =\lim_{R\to 0}\frac{1}{R}\sum_{\alpha=1}^{R}\left\langle\pi_{\alpha}( \mathbf{q},\omega_{n})\pi_{\alpha}(\mathbf{q}^{\prime},\omega_{n^{\prime}}) \right\rangle_{S^{\text{imp}}_{x}}\\ &=\lim_{R\to 0}\frac{1}{R}\bigg{[}(R-1)\frac{n_{0}\hbar^{2}q^{2}/m}{ \hbar^{2}\omega_{n}^{2}+E_{q}^{2}}+\frac{n_{0}\hbar^{2}q^{2}/m}{\hbar^{2} \omega_{n}^{2}+E_{q}^{2}-R\gamma^{2}(\mathbf{q},\omega_{n})n_{0}\frac{\hbar^ {2}q^{2}}{m}}\bigg{]}\delta(\mathbf{q}+\mathbf{q}^{\prime})\beta\hbar\delta_{n,-n^{\prime}}\\ &=\bigg{[}\frac{n_{0}\hbar^{3}q^{2}/m}{\hbar^{2}\omega_{n}^{2}+E_ {q}^{2}}+\gamma^{2}(\mathbf{q},\omega_{n})\frac{n_{0}\hbar^{2}q^{2}q^{2}/m}{( \hbar^{2}\omega_{n}^{2}+E_{q}^{2})(\hbar^{2}\omega_{n^{\prime}}^{2}+E_{q^{ \prime}})}\bigg{]}\delta(\mathbf{q}+\mathbf{q}^{\prime})\beta\hbar\delta_{n,- n^{\prime}},\end{split}  \tag{81}\]  
where the final result is obtained by expanding the second line up to the first order in \(R\). Moreover, we have  
\[E_{q}=\sqrt{\epsilon_{q}\left[\epsilon_{q}+2n_{0}\bar{V}(\mathbf{q})\right]}\, \tag{82}\]  
nothing else than Equation (29) with the saddle-point replacement \(\mu=\bar{V}(0)\,n_{0}\). The remaining correlations are computed in the same way, leading to  
\[ \begin{split}\left\langle\varphi^{*}(\mathbf{q},\omega_{n}) \varphi(\mathbf{q}^{\prime},\omega_{n^{\prime}})\right\rangle_{\text{dis+th}}& =\bigg{[}\frac{mE_{q}^{2}/(n_{0}\hbar q^{2})}{\hbar^{2}\omega_{ n}^{2}+E_{q}^{2}}+\frac{\hbar^{2}\omega_{n}\omega_{n^{\prime}}}{(\hbar^{2} \omega_{n}^{2}+E_{q}^{2})(\hbar^{2}\omega_{n^{\prime}}^{2}+E_{q^{\prime}})} \gamma^{2}(\mathbf{q},\omega_{n})\bigg{]}\delta(\mathbf{q}+\mathbf{q}^{ \prime})\beta\hbar\delta_{n,-n^{\prime}},\\ \left\langle\varphi^{*}(\mathbf{q},\omega_{n})\pi(\mathbf{q}^{ \prime},\omega_{n^{\prime}})\right\rangle_{\text{dis+th}}&=\bigg{[} \frac{\hbar^{2}\omega_{n}}{\hbar^{2}\omega_{n}^{2}+E_{q}^{2}}+\frac{n_{0}\hbar^ {3}q^{2}\omega_{n^{\prime}}/m}{(\hbar^{2}\omega_{n}^{2}+E_{q}^{2})(\hbar^{2} \omega_{n^{\prime}}^{2}+E_{q^{\prime}})}\gamma^{2}(\mathbf{q},\omega_{n}) \bigg{]}\delta(\mathbf{q}+\mathbf{q}^{\prime})\beta\hbar\delta_{n,-n^{\prime}}. \end{split}