Ground State Properties of an Interacting Bose gas Within an Imperfect 1D Crystal*

For a weakly interacting boson gas within an imperfect one-dimensional crystal, we report the ground state properties as functions of the vacancy number in the crystal. To do this, we solve the corresponding Gross-Pitaevskii equation using the “Gradient Flow with Discrete Normalization”method, also known as the imaginary time method. We model the imperfect one-dimensional crystal as a Dirac Comb potential with several deltas removed at random.

We know that, in the case of a non-interacting Bose gas (IBG) within a 1D perfect crystal, the presence of a single vacancy [1] modifies the Bose gas energy spectrum by lowering only the bottom edge energy, which generates an energy gap between the ground state and the first excited state of the new energy spectrum. The gap magnitude is a function of the intensity of the removed delta. When we turn on the interactions between bosons within a 1D perfect crystal, the ground state energy (GSE) increases proportionally to the magnitude of the interaction. When we introduce the vacancies for a given interaction g, the GSE decreases almost linearly as a function of the vacancy number up to a value g [ ℏ²/2ma²] when we remove all the deltas. We also give the chemical potential as a function of the vacancy number and show some probability density functions for some vacancy number given an interaction g.

[1] E. I. Guerrero et al., March Meeting 2023 (G00.00382).