Ground state energy of a weakly interacting Bose gas in a 1D crystal with vacancies*

For a weakly interacting Bose gas inside an imperfect one-dimensional crystal, we study the effect of vacancies on its ground state properties. To do this, we solve the corresponding Gross-Pitaevskii equation using the ``Gradient Flow with Discrete Normalization" method.

 The imperfect crystal is modeled as a Dirac Comb potential with a fraction of randomly removed deltas coexisting with the weakly interacting Bose gas. We present the ground state energy (GSE), the many particle wave function and the chemical potential as functions of the deleted delta fraction. The GSE decreases linerarly as the vacancy fraction increases, down to the GSE value of a free weakly interacting Bose gas. Contrary to what we expected, we did not find a vacancy fraction that would locally minimize the system energy, which would be a signal of greater system stability.  These calculations were made for different values of the interaction between bosons and system size. Ground state energies are extrapolated  to their value when the system size goes to infinite.