## 5 Correlation Functions in the Replicated Formalism
 \tag{83}\]Besides the phase and density correlators, one of the most important correlation functions is the current one, defined as  
\[\mathcal{C}_{ij}(\mathbf{r},\tau_{i}\mathbf{r}^{\prime},\tau^{\prime})=m^{-1} \big{[}\left\langle g_{i}(\mathbf{r},\tau)g_{j}(\mathbf{r}^{\prime},\tau^{\prime })\right\rangle-\left\langle g_{i}(\mathbf{r},\tau)\right\rangle\left\langle g _{j}(\mathbf{r}^{\prime},\tau^{\prime})\right\rangle\big{]}\, \tag{84}\]  
where the notation \(\left\langle\ldots\right\rangle\) represents, for now, a general average procedure. In the phase-density representation, from Equation (57), it is easy to verify that the current density \(\mathbf{g}=\hbar(n_{0}+\pi)\
abla\varphi/m\). Thus, by making use of the Wick theorem [70] to disentagle the four-field correlator in Equation (84), in the Fourier space, one obtains [35]  
\[ \begin{split}\mathcal{C}_{ij}(\mathbf{q},\omega_{n};\mathbf{q}^{ \prime},\omega_{n^{\prime}})&=-\hbar^{2}mn_{0}^{2}q_{i}q_{j} \left\langle\varphi(\mathbf{q},\omega_{n})\varphi(\mathbf{q}^{\prime},\omega_ {n^{\prime}})\right\rangle+\\ &-\frac{1}{(\beta L^{d})^{2}}\sum_{\mathbf{k},\omega_{m}}\sum_{ \mathbf{k}^{\prime},\omega_{n^{\prime}}}k_{i}k_{j}^{\prime}\Big{[}\left\langle \pi(\mathbf{q}-\mathbf{k},\omega_{l-m})\pi(\mathbf{q}^{\prime}-\mathbf{k}^{ \prime},\omega_{l^{\prime}-m^{\prime}})\right\rangle\left\langle\varphi( \mathbf{k},\omega_{m})\varphi(\mathbf{k}^{\prime},\omega_{m^{\prime}}) \right\rangle+\\ &\qquad\qquad+\left\langle\pi(\mathbf{k}-\mathbf{q},\omega_{m-l} )\varphi(\mathbf{k}^{\prime},\omega_{m^{\prime}})\right\rangle\left\langle\pi (\mathbf{k}^{\prime}-\mathbf{q}^{\prime},\omega_{m^{\prime}-l^{\prime}}) \varphi(\mathbf{k},\omega_{m})\right\rangle\Big{]}\.\end{split}  \tag{85}\]  
Thanks to the replica trick, the disorder contribution is taken into account by considering the correlators over the replicated Gaussian action, namely Equations (81) and (83). The current response function in Equation (85) is crucial since it encodes the superfluid character of the system. This point is clarified by noticing that \(\mathcal{C}_{ij}\) is a 2-rank tensor, so it can be decomposed in its longitudinal and transverse components by means of the projectors \(\mathcal{P}_{ij}^{L}=q_{i}q_{j}/q^{2}\) and \(P_{ij}^{T}=\delta_{ij}-\mathcal{P}_{ij}^{L}\). It has been shown (see [71; 72] for a pedagogical review) that the static limit of transverse component of Equation (85) \(\mathcal{C}_{ij}^{T}(\mathbf{q},\omega_{n})\equiv(d-1)^{-1}\sum_{ij}\mathcal{ P}_{ij}^{T}(\mathbf{q},\omega_{n})\mathcal{C}_{ij}(\mathbf{q},\omega_{n})/\hbar\) corresponds to the normal component of the system. By replacing the replicated correlator given by Equations (81) and (83) into Equation (85), after burdensome (but standard) algebra, we find  
\[ \begin{split} n_{n}&=\lim_{q\to 0}\left\langle \mathcal{C}^{T}(\mathbf{q},0)\right\rangle_{\text{th+dis}}\\ &=\frac{1}{d\beta L^{d}}\sum_{\mathbf{k},\omega_{m}}\left[\left( \frac{\hbar^{2}k^{2}}{m}\right)\frac{E_{k}^{2}-\hbar^{2}\omega_{m}^{2}}{(\hbar ^{2}\omega_{m}^{2}+E_{q}^{2})^{2}}+n_{0}\left(\frac{\hbar^{2}k^{2}}{m}\right)^ {2}\frac{E_{k}^{2}-3\hbar^{2}\omega_{m}^{2}}{(\hbar^{2}\omega_{m}^{2}+E_{k}^{2 })^{3}}\gamma^{2}(\mathbf{k},\omega_{m})\right]\,.\end{split}  \tag{86}\]  
When the time scale of the disorder potential is frozen, i.e., for  
\[\gamma^{2}(\mathbf{q},\omega_{n})=\gamma^{2}(\mathbf{q})\beta\delta_{n,0}\, \tag{87}\]  
Equation (86) becomes  
\[n_{n}(n_{0},T)=\frac{\beta}{d}\int\frac{d^{d}\mathbf{q}}{(2\pi)^{d}}\frac{ \hbar^{2}q^{2}}{m}\frac{e^{\beta E_{q}}}{(e^{\beta E_{q}}-1)^{2}}+\frac{4n_{0} }{d}\int\frac{d^{d}\mathbf{q}}{(2\pi)^{d}}\left(\frac{\hbar^{2}q^{2}}{2m} \right)^{2}\frac{\gamma^{2}(\mathbf{q})}{E_{q}^{4}}. \tag{88}\]  
With the further simplification of a constant disorder correlator \(\gamma^{2}\), the equation above equates to Equation (52). Thus, Equation (88) reduces correctly to the quenched disorder regime and agrees with results reported in [34; 47; 35].