Random external potential

As a simple model for disorder we use the potential produced by a uniform random distribution of quenched impurities. The random external potential is then given by

$$V({\bf r}) = \sum\limits_{i=1}^{N_{imp}} v_{imp}({\bf r-r_i}),$$ (2.1)

where $N_{imp}$ is the number of impurities present in the volume $V$ located at the fixed positions ${\bf r_i},~i = \overline{1,N_{imp}}$ and $v_{imp}({\bf r})$ is a two-body potential which describes the particle-impurity interaction. This model of disorder is particularly convenient for two reasons:

• it can be easily treated analytically within the Bogoliubov theory of a dilute Bose gas
• can be easily implemented in a numerical simulation.

If the gas of impurities is dilute, as it is the case in the ``weak'' disorder regime which is of interest here, the particle-impurity interaction potential can be replaced by a pseudopotential

$$V({\bf r}) = \sum\limits_{i=1}^{N_{imp}} g_{imp} \delta ({\bf r-r_i}),$$ (2.2)

The coupling constant $g_{imp}$ is defined by the $s$-wave scattering length $b$ of the particle-impurity collision process

$$g_{imp} = \frac{2\pi \hbar^2 b}{m},$$ (2.3)

here $m$ is the mass of the scattering particle, since the mass of the impurity is taken to be infinite (quenched impurities). In this case the particle-impurity reduced mass is twice as large as the corresponding particle-particle reduced mass. This explains the factor two difference between (1.2) and (2.3). The important quantities which describe the statistical properties of the random external potential are the mean value

$$\overline V_0 = \left\langle\frac{1}{V} \int V({\bf r})\,{\bf dr}\right\rangle,$$ (2.4)

and the correlation function $\langle V_{\bf p}V_{\bf -p}\rangle$, where $V_{\bf p}$ denotes the Fourier component

$$V_{\bf p} = \frac{1}{V} \int e^{-i{\bf pr}/\hbar}V({\bf r})\,{\bf dr}$$ (2.5)

Here $\langle ...\rangle$ means average over disorder configurations. For our random external potential (2.2) the correlation function can be rewritten as

$$\langle V_{\bf p}V_{\bf -p}\rangle = \left<\frac{1}{V^2}\int\!\!\... ... \sum\limits_j \langle e^{-i{\bf p(r_1 - r_i})/\hbar}\rangle \qquad\qquad\qquad$$

By assuming that the impurities have a uniform distribution the mean value of the random potential is given by

$$\overline V_0 = n_{imp} g_{imp} = \frac{1}{2} mc^2 \chi\left(\frac{b}{a}\right),$$ (2.6)

while $\langle V_{\bf p}\rangle = 0$ for ${\bf p\ne 0}$. It can be easily shown that the correlation function becomes

$$\langle V_{\bf p}V_{\bf -p}\rangle = \frac{N_{imp} g_{imp}^2}{V^2} = \frac{1}{4} \frac{(mc^2)^2}{nV} \chi\left(\frac{b}{a}\right)^2,$$ (2.7)

where $n_{imp} = N_{imp}/V$ is the density of impurities and $\chi = N_{imp}/N = n_{imp}/n$ is the concentration of impurities. Eq. (2.7) implies that the external potential is treated as a short correlated white noise in momentum space with amplitude proportional to $g_{imp}$. The independent parameters that describe the properties of the system are the following

$$\begin{tabular}{\vert p{2.5 cm}\vert p{7cm}\vert} \hline $na^3$\ ... ...e $\chi = n^{imp}/n$\ & concentration of the impurities\\ \hline \end{tabular}$$ (2.8)