## 2 Disorder in Field Theories
It follows then that  
\[\left\langle\ldots\right\rangle_{\mathrm{dis}}\equiv\int\mathcal{D}[U_{ \mathrm{D}}]\mathsf{P}[U_{\mathrm{D}}](\ldots)\;. \tag{6}\]  
For instance, we can surely imagine a Gaussian distributed disorder, i.e.,  
\[\mathsf{P}[U_{\mathrm{D}}]=\exp\bigg{\{}-\frac{1}{2\gamma^{2}}\int d^{d+1} \mathbf{s}\int d^{d+1}\mathbf{s}^{\prime}\ U_{\mathrm{D}}(\mathbf{s})K^{-1}( \mathbf{s}-\mathbf{s}^{\prime})U_{\mathrm{D}}(\mathbf{s}^{\prime})\bigg{\}}, \tag{7}\]  
where \(\mathbf{s}=(\mathbf{r},\tau)\) and \(d^{d+1}\mathbf{s}=d\tau\,d^{d}\mathbf{r}\). The expectation value of a central Gaussian distributed variable is zero, i.e.,  
\[\left\langle U_{\mathrm{D}}(\mathbf{s})\right\rangle_{\mathrm{dis}}=0, \tag{8}\]  
while its second (central) momentum corresponds to  
\[\left\langle U_{\mathrm{D}}(\mathbf{s})U_{\mathrm{D}}(\mathbf{s}^{\prime}) \right\rangle_{\mathrm{dis}}=\gamma^{2}\,K(\mathbf{s}-\mathbf{s}^{\prime})\;. \tag{9}\]  
For a \(\delta\)-correlated disorder, where \(K(\mathbf{s}-\mathbf{s}^{\prime})=\delta^{(d+1)}(\mathbf{s}-\mathbf{s}^{ \prime})\) with \(\delta^{(d+1)}(\mathbf{s})\) the Dirac delta function in \(d+1\) dimensions, the above equation is further simplified in  
\[\mathsf{P}[U_{\mathrm{D}}]=\exp\bigg{\{}-\frac{1}{2\gamma^{2}}\int_{0}^{\beta \hbar}d\tau\int d^{d}\mathbf{r}\ U_{\mathrm{D}}^{2}(\mathbf{r},\tau)\bigg{\}}\;. \tag{10}\]  
If we deal with atomic bosons in the superfluid phase, the atomic wavelength is much larger than the scattering range of impurities generating the disorder potential \(U_{\mathrm{D}}(\mathbf{s})\). This physical situation clarifies the simplifying assumption of a \(\delta\)-correlated disorder as in Equation (10). Moreover, in this section, we explicitly consider the case of a quenched disorder, where the time scale of \(U_{\mathrm{D}}\) is much longer than the thermodynamic one. From a technical point of view, this implies that \(U_{\mathrm{D}}\) retains only a spatial dependence on \(\mathbf{r}\) and that the disorder average in Equation (6) has to be performed after the thermal one [37]. Thus, the average over a quenched disorder has to be intended as  
\[\left\langle\mathcal{O}[\psi,\psi^{*}]\right\rangle_{\mathrm{dis+th}}=\int \mathcal{D}[U_{\mathrm{D}}]\mathsf{P}[U_{\mathrm{D}}]\left\langle\mathcal{O} [\psi,\psi^{*}]\right\rangle_{\mathrm{th},U_{\mathrm{D}}}, \tag{11}\]  
where  
\[\left\langle\mathcal{O}[\psi,\psi^{*}]\right\rangle_{\mathrm{th},U_{\mathrm{D} }}=\int\mathcal{D}[\psi,\psi^{*}]\frac{1}{\mathcal{Z}(U_{\mathrm{D}})} \mathcal{O}[\psi,\psi^{*}]e^{-S[\psi,\psi^{*}]/\hbar} \tag{12}\]  
is the usual thermal average over the grand canonical ensemble for a given impurity configuration. Within the scheme outlined by Equations (11) and (12), \(U_{\mathrm{D}}(\mathbf{r})\) and \(\psi(\mathbf{r},\tau)\) are not being treated on the same footing, since the random field \(U_{\mathrm{D}}\) is fixed when we average over the bosonic field \(\psi\). More precisely, Equation (12) is telling us to compute the partition function over a precise microscopic realization of the system (i.e., with a given \(U_{\mathrm{D}}\)). Then, according to Equation (11), we have to average all the partition functions obtained in this way over the disorder field. The quenched average presented above is certainly a viable scheme to adopt when the disorder can be treated in a perturbative way. However, this is not always the case. It can occur a physical situation where one has to perform the disorder average early in the calculation and/or the random field \(U_{\mathrm{D}}\) has to be considered at the same level of \(\psi\). In order to highlight the eventual technical obstacle, we make use of the source method to compute expectation values of physical observable. According to this field-theory technique, the expectation value of a given observable \(\mathcal{O}\) can be extracted by means of the source method [44; 50].