## 3 Perturbative Approach to Quenched Disorder  
### Thermodynamic Picture and Disorder-Driven Condensate Depletion  
Let us consider the Euclidean action in Equation (4) and apply the saddle-point method by stationarizing it, i.e.,  
\[\bigg{[}\hbar\frac{\partial}{\partial\tau}-\frac{\hbar^{2}\
abla^{2}}{2m}-( \mu+U_{\text{D}}(\mathbf{r},\tau))+\int d^{d}\mathbf{r}^{\prime}\ |\psi(\mathbf{r}^{\prime},\tau)|^{2}V(|\mathbf{r}-\mathbf{r}^{\prime}|) \bigg{]}\psi(\mathbf{r},\tau)=0. \tag{15}\]  
In the case of a two-body contact interaction, we have  
\[V=g\ \delta^{(d)}(\mathbf{r}), \tag{16}\]  
where \(g=\tilde{V}(\mathbf{q}=0)\) and \(\tilde{V}(\mathbf{q})\) is the Fourier transform of the two-body potential. For a uniform configuration of the field, with  
\[\psi(\mathbf{r},\tau)=\psi_{0} \tag{17}\]  
not depending on \(\mathbf{r}\) and \(\tau\), we get  
\[\big{[}-(\mu+\left\langle U_{\text{D}}\right\rangle_{\text{dis}})+g\psi_{0}^{ 2}\big{]}\psi_{0}=0\Longrightarrow \begin{cases}\psi_{0}^{2}=0,\qquad\text{for }\mu+\left\langle U_{\text{D}}\right\rangle_{\text{dis}}\leq 0, \\ \psi_{0}^{2}=(\mu+\left\langle U_{\text{D}}\right\rangle_{\text{dis}})/g, \qquad\text{for }\mu+\left\langle U_{\text{D}}\right\rangle_{\text{dis}}>0\.\end{cases}  \tag{18}\]  
This approximation must not be underrated: actually, in the presence of an external potential, \(\psi_{0}\) should not be uniform. However, at low temperatures, it seems reasonable to assume that the wavelengths of atoms are much larger than the spatial variations caused by the impurity potential. By comparing the equations above with the ones for a pure system [32], it is immediate to realize that, in principle, the disorder may shift the critical point of the superfluid transition. However, if \(U_{\text{D}}\) follows a centered Gaussian distribution like the one in Equation (7), this shift equates to zero. From the saddle-point result, it descends that the total density \(n=\psi_{0}^{2}\). By identifying \(\psi_{0}^{2}\) with the condensate density \(n_{0}\), this signals that, up to this (very rough) level of approximation, all the particles take part in the condensate within the broken-symmetry phase. In this phase, where a \(\mathsf{U}(1)\) symmetry is spontaneously broken, fluctuations can be taken into account by considering the following splitting of the field  
\[\psi(\mathbf{r},\tau)=\psi_{0}+\eta(\mathbf{r},\tau)\, \tag{19}\]  
with \(\eta(\mathbf{r},\tau)\in\mathbb{C}\). Replacing the equation above in the Euclidean action of Equation (4), the disorder contribution reads  
\[\mathcal{L}_{\text{dis}}[\psi,\psi^{*},U_{\text{D}}]=-U_{\text{D}}(\mathbf{r}) \big{|}\psi(\mathbf{r},\tau)\big{|}^{2}=-U_{\text{D}}(\mathbf{r})\big{[}\psi_{ 0}^{2}+\left|\eta(\mathbf{r},\tau)\right|^{2}+v\eta(\mathbf{r},\tau)+v\eta^{*} (\mathbf{r},\tau)\big{]}, \tag{20}\]  
where \(\mathcal{L}\) denotes the (Euclidean) Lagrangian density, i.e., \(S[\psi]=\int d\tau\int d^{d}\mathbf{r}\ \mathcal{L}[\psi]\). The first two terms give rise to an unobservable shift of the chemical potential, so we will neglect them. Moreover, let us remark that we are considering fluctuations over the saddle-point non-trivial configuration \(\psi_{0}^{2}=\mu/g\) with \(\left\langle U_{\text{D}}\right\rangle_{\text{dis}}=0\), but this is not the true stationary point. Consequently, we are not allowed to neglect the linear terms in Equation (20), not even up to the Gaussian order in the fluctuation fields. By introducing the column vectors  
\[\text{Y}=\psi_{0} \begin{pmatrix}1\\ 1\end{pmatrix} \qquad\text{and}\qquad\chi= \begin{pmatrix}\eta\\ \eta^{*}\end{pmatrix} , \tag{21}\]  
Equation (20) is brought to its more compact form  
\[\mathcal{L}_{\text{dis}}[\eta,\eta^{*},U_{\text{D}}]=-U_{\text{D}}(\mathbf{r} )\Upsilon^{T}\chi. \tag{22}\]  
At this point, we can replace the field \(\psi(\mathbf{r},\tau)\) with Equation (19). By retaining terms up to the quadratic order in \(\eta\) and \(\eta^{*}\), the grand canonical partition function reads  
\[