Exactly Solvable System of One-Dimensional Trapped Bosons with Short- and Long-Range Interactions

We introduce a model of trapped bosons with contact interactions as well as Coulomb repulsion or gravitational attraction in one spatial dimension. We find the exact ground-state energy and many-body wave function. The density profile and the pair-correlation function are sampled using Monte Carlo method and show a rich variety of regimes with crossovers between them. Strong attraction leads to a trapped McGuire quantum soliton. Weak repulsion results in an incompressible Laughlin-like fluid with flat density, well reproduced by a Gross-Pitaevskii equation with long-range interactions. Stronger repulsion induces Friedel oscillations and the eventual formation of a Wigner crystal.

Figure 1

Density profile $n(x)$ (upper row), pair distribution $g_2(x)$ (bottom row) for $N=5$ (left-hand column) and $N=100$ (right-hand column) particles and for different values of interaction strength. Solid lines are results of Monte Carlo sampling, while dashed lines show the mean-field profile, Eq. (19). Harmonic oscillator units are used.