Variational approach to the many-boson problem in one dimension

For one-dimensional many-boson systems there exists a considerable body of analytical treatments of model systems. The existence of some exactly known properties of these systems and the fact that some of these properties differ from those of ⁴He in higher dimensions are a strong impetus to apply microscopic many-body methods to quasi-one-dimensional systems with realistic interactions. The results on the model systems can help as a guide to understanding those systems, where due to the nature of the interaction, exact results are not available. In this paper, we examine three model systems: hard rods, the Morse potential, and the Lennard-Jones 6-12 potential using the nonperturbative variational approach based on Feenberg's method of correlated basis functions. In the hard rod system we can compare our equation of state with the exact result and discuss the low-lying excitations. We examine both finite-size rods and the point rod limit. For the Morse system, the existence of a two-body bound state, dimerization, can be determined analytically. We examine the effect of dimerization on the many-body equation of state and argue that dimerization is manifested in the many-boson system as the appearance of a many-body bound state (i.e., a zero pressure, negative energy system with nonzero density). For the Lennard-Jones system we numerically determine the onset of dimerization and, as with the Morse system, we show that in one dimension the many-body bound state only exists for those systems which dimerize. For each of these systems we track high-density instabilities in the equations of state which we tentatively identify as zero-temperature liquid-solid transitions.