# Clusterization of weakly-interacting bosons in one dimension: an analytic study at zero temperature

## TL;DR

This paper analytically investigates the clustering transition of weakly-interacting bosons in one dimension at zero temperature, comparing GP and variational approaches, and analyzing superfluid properties.

## Contribution

It provides an analytic comparison of GP and variational methods for boson clusterization and characterizes the phase transition and superfluid fraction.

## Key findings

- Clusterization occurs at a specific density with phase transition predicted by all models.
- GP theory links clusterization with roton softening.
- Analytic expressions for high-density behavior and superfluid fraction are derived.

## Abstract

We study a system of penetrable bosons on a line, focusing on the high-density/weak-interaction regime, where the ground state is, to a good approximation, a condensate. Under compression, the system clusterizes at zero temperature, i.e., particles gather together in separate, equally populated bunches. We compare predictions from the Gross-Pitaevskii (GP) equation with those of two distinct variational approximations of the single-particle state, written as either a sum of Gaussians or the square root of it. Not only the wave functions in the three theories are similar, but also the phase-transition density is the same for all. In particular, clusterization occurs together with the softening of roton excitations in GP theory. Compared to the latter theory, Gaussian variational theory has the advantage that the mean-field energy functional is written in (almost) closed form, which enables us to extract the phase-transition and high-density behaviors in fully analytic terms. We also compute the superfluid fraction of the clustered system, uncovering its exact behavior close, as well as very far away from, the transition.