## 4 The Replicated Action for Superfluid Bosons
Let us consider in the following the generalization of Equation (10), namely given by  
\[\hat{\rho}[U_{\text{D}}]=\exp\bigg{\{}-\frac{1}{2}\int_{0}^{\beta\hbar}d\tau \int d^{d}\mathbf{r}\ \gamma^{-2}(\mathbf{r},\tau)U_{\text{D}}^{2}(\mathbf{r},\tau) \bigg{\}}, \tag{56}\]  
where, actually, the disorder is assumed translationally invariant in time and space. In order to implement the replica trick within the context of superfluid bosonic systems, as an alternative to Equation (19), we move to the phase-density representation, where  
\[\psi(\mathbf{r},\tau)=\sqrt{n_{0}+\pi(\mathbf{r},\tau)}\ e^{i\varphi(\mathbf{r },\tau)} \tag{57}\]  
with \(n_{0}=\psi_{0}^{2}\), \(\psi_{0}^{2}\) being given by Equation (18). From the equation above, we deduce that \(n(\mathbf{r},\tau)=n_{0}+\pi(\mathbf{r},\tau)\) and a perturbative strategy is easily implemented by assuming the smallness of density fluctuation. Thus, Equation (57) becomes  
\[\psi(\mathbf{r},\tau)\simeq\sqrt{n_{0}}\bigg{[}1+\frac{\pi(\mathbf{r},\tau)}{2 n_{0}}\bigg{]}e^{i\varphi(\mathbf{r},\tau)}. \tag{58}\]  
In order to understand the limitations of our approach, it is crucial to recall that \(\left\langle n(\mathbf{r},\tau)\right\rangle_{\text{th}}=\left\langle\left| \psi(\mathbf{r},\tau)\right|^{2}\right\rangle_{\text{th}}\), while we often set \(n\simeq n_{0}\) with \(n_{0}=\big{|}\left\langle\psi(\mathbf{r},\tau)\right\rangle_{\text{th}}\big{|} ^{2}\). Up to the Gaussian level, this is an acceptable approximation since \(\left\langle\pi(\mathbf{r},\tau)\right\rangle_{\text{th}}\simeq 0\). In the regime of small fluctuations, we can expand the mean-field analysis by retaining only the quadratic terms in \(\pi(\mathbf{r},\tau)\) and \(\varphi(\mathbf{r},\tau)\). The change of variables \(\left\{\psi,\psi^{*}\right\}\to\left\{\pi,\varphi\right\}\) has a constant Jacobian determinant; therefore, Equation (3) reads  
\[\mathcal{Z}\simeq\int\mathcal{D}[\pi,\varphi]\exp\bigg{\{}-\frac{1}{\hbar}S_{g }[\pi,\varphi]\bigg{\}} \tag{59}\]  
with the periodic boundary conditions \(\pi(\mathbf{r},\beta\hbar)=\pi(\mathbf{r},0)\) and \(\varphi(\mathbf{r},\beta\hbar)=\varphi(\mathbf{r},0)\). The smallness of fluctuations allows for extending the integration range of the phase variable from \([-\pi,+\pi]\) to \([-\infty,+\infty]\). This decompactification obviously has to be addressed with particular attention when dealing with \(d\leq 2\), where the enhancement of fluctuations prevents the occurring of a _true_ long-range order and a more refined analysis is required [44; 65; 66; 67; 68]. The Gaussian action in Equation (59) is given by  
\[ \begin{split} S_{g}[\pi,\varphi]&=\int_{0}^{\beta \hbar}d\tau\int d^{d}\mathbf{r}\bigg{[}\hat{\imath}\hbar\frac{\pi\partial \varphi}{\partial\tau}+\frac{n_{0}\hbar^{2}(\
abla\varphi)^{2}}{2m}+\frac{ \hbar^{2}(\
abla\pi)^{2}}{8 mn_{0}}-U_{\text{D}}(\mathbf{r},\tau)(n_{0}+\pi)\\ &\qquad\qquad+\frac{1}{2}\int d^{d}\mathbf{r}^{\prime}\pi(\mathbf{ r}^{\prime},\tau)V(|\mathbf{r}-\mathbf{r}^{\prime}|)\pi(\mathbf{r},\tau)\bigg{]}- \frac{1}{2}\beta\hbar L^{d}\tilde{V}(0)n_{0}^{2},\end{split}  \tag{60}\]where we have imposed Equation (18) to eliminate the dependence on the chemical potential and \(U_{\rm D}\) is distributed according to Equation (56). The pertubative expansion up to the terms contained in Equation (60) is equivalent to a one-loop expansion within a diagrammatic representation. Equivalently, in the second-quantization formalism, this corresponds to the [PERSON] approximation [69]. The implementation of the replica trick requires the calculation of \(\langle\mathcal{Z}^{R}\rangle_{\rm dis}\). The (not-averaged) partition function is simply  
\[\mathcal{Z}^{R}=\int\mathcal{D}[\pi,\varphi]\exp\bigg{\{}-\frac{1}{\hbar}\sum_ {\alpha=1}^{R}S_{g}[\pi_{\alpha},\varphi_{\alpha}]\bigg{\}}\, \tag{61}\]  
where \(S_{g}[\pi_{\alpha},\varphi_{\alpha}]\) is provided by Equation (60) and \(\mathcal{D}[\pi,\varphi]=\prod_{\alpha=1}^{R}\mathcal{D}[\pi_{\alpha},\varphi _{\alpha}]\).