## 3 Perturbative Approach to Quenched Disorder
### Thermodynamic Picture and Disorder-Driven Condensate Depletion
\begin{split}\mathcal{Z}(U_{\text{D}})&=e^{-S_{\text{ mf}}[\psi_{0},\mu]/\hbar}\int\mathcal{D}[\chi]\exp\bigg{\{}-\frac{1}{\hbar}\int_{0}^{ \beta\hbar}d\tau\int d^{d}\mathbf{r}\left[\mathcal{L}_{S}^{\text{(pure)}}+ \mathcal{L}_{\text{dis}}\right]\bigg{\}}\\ &=e^{\beta L^{d}\mu^{2}/(2g)}\int\mathcal{D}[\chi]\exp\bigg{\{}- \frac{1}{\hbar}\int d^{d+1}\mathbf{s}\bigg{[}\frac{1}{2}\chi^{\dagger} \mathcal{G}^{-1}(\partial_{\tau},\
abla)\chi-U_{\text{D}}(\mathbf{r})\Upsilon ^{T}\chi\bigg{]}\bigg{\}}\,\end{split}  \tag{23}\]  
with \(\mathbf{s}=(\mathbf{r},\tau)\). In the equation above, the integral in \(\chi\) is Gaussian and, consequently, can be computed exactly, leading us to the following result  
\[\mathcal{Z}(U_{\text{D}})=e^{-\beta\Omega_{\text{mf}}}\ \big{(}\det\mathcal{G}^{-1} \big{)}^{-1}\exp\bigg{\{}\frac{1}{2}\int d^{d+1}\mathbf{s}\int d^{d+1} \mathbf{s}^{\prime}U_{\text{D}}(\mathbf{r})\Upsilon^{T}\mathcal{G}(\mathbf{s} -\mathbf{s}^{\prime})\Upsilon U_{\text{D}}(\mathbf{r}^{\prime})\bigg{\}}. \tag{24}\]  
Let us also recall that, from Equation (18), it is easy to realize that the mean-field grand potential is simply \(\Omega_{\text{mf}}/L^{d}=-\mu^{2}/2g\). The remaining steps of the calculation are easier in the Fourier space, where the (inverse) Gaussian propagator is the following \(2\times 2\) matrix  
\[\mathcal{G}^{-1}(\mathbf{q},\omega)= \begin{pmatrix}-i\hbar\omega_{n}+\epsilon _{q}-\mu+2g\psi_{0}^{2}&g\psi_{0}^{2}\\ g\psi_{0}^{2}&+i\hbar\omega_{n}+\epsilon_{q}-\mu+2g\psi_{0}^{2}\end{pmatrix}  \tag{25}\]  
with \(\omega_{n}=2\pi n/\beta\) the bosonic Matsubara frequencies and \(\epsilon_{q}=\hbar q^{2}/(2m)\). The (functional) determinant in Equation (24) can be handled more easily thanks to the identities  
\[\big{(}\det\mathcal{G}^{-1}\big{)}^{-1}=\exp\big{(}-\text{tr}\log\mathcal{G}^{ -1}\big{)}=\exp\big{(}-\log\det\mathcal{G}^{-1}\big{)}. \tag{26}\]By putting all these pieces together, Equation (24) is built by three blocks, namely  
\[\mathcal{Z}(U_{\text{D}})=e^{-\beta\Omega_{\text{mf}}}\exp\left\{\underbrace{- \frac{1}{2}\sum_{\mathbf{q}}\sum_{n}\log\det\mathcal{G}^{-1}(\mathbf{q},\omega_ {n})}_{\text{Pure Gaussian term}}\exp\left\{\underbrace{\frac{1}{2}\sum_{ \mathbf{q}}\left|U_{\text{D}}(\mathbf{q})\right|^{2}\text{Y}^{T}\mathcal{G}( \mathbf{q},0)\text{Y}}_{\text{disorder Gaussian term}}\right\}\right.. \tag{27}\]  
The disorder contribution in the equation above does not depend on the frequency, since the time scales of the quenched disordered are frozen, implying that the corresponding frequencies goes to zero. Concerning the pure Gaussian contribution, we have [32]  
\[\frac{1}{2\beta}\sum_{n}\log\det\mathcal{G}^{-1}(\mathbf{q},\omega_{n})=\frac {1}{2\beta}\sum_{n}\log\left[\beta^{2}\big{(}\hbar^{2}\omega_{n}^{2}+E_{q}^{2} \big{)}\right]=\frac{1}{2}\,E_{q}+\frac{1}{\beta}\log\left(1-e^{-\beta E_{q}} \right)\,, \tag{28}\]  
where a crucial role is played by the excitation spectrum of collective excitations above the uniform saddle-point configuration, i.e.,  
\[E_{q}(\mu,\psi_{0})=\sqrt{\left(\epsilon_{q}-\mu+2g\psi_{0}^{2}\right)^{2}-g^{ 2}\psi_{0}^{4}}\,. \tag{29}\]  
Focusing on the zero-temperature case, we find  
\[\mathcal{Z}(U_{\text{D}})=e^{-\beta\left(\Omega_{\text{mf}}+\Omega_{S}^{(0)} \right)}\ \exp\left\{\frac{1}{2}\sum_{\mathbf{q}}\left|U_{\text{D}}(\mathbf{q})\right|^ {2}\text{Y}^{T}\mathcal{G}(\mathbf{q},0)\text{Y}\right\} \tag{30}\]  
with  
\[\Omega_{S}^{(0)}=\frac{1}{2}\sum_{\mathbf{q}}E_{q}(\mu,\psi_{0}) \tag{31}\]  
the zero-temperature Gaussian grand potential. At this point, by recalling the relation \(\Omega=\beta^{-1}\log\mathcal{Z}\), we are ready compute the quenched disorder average average of the grand potential. It is immediate to realize the \(\Omega_{\text{mf}}\) and \(\Omega_{S}^{(0)}\) are transparent to this operation.