# Spin-1 Bose Hubbard model with nearest neighbour extended interaction

## TL;DR

This paper investigates the phase diagram of a spin-1 Bose gas with extended interactions, revealing the emergence of charge density wave phases and their relation to Mott insulators and supersolids, supported by mean field and perturbation theory.

## Contribution

It provides a comprehensive analysis of the phase diagram of spin-1 Bose gases with extended interactions, including the effects of harmonic confinement, using mean field and perturbation methods.

## Key findings

- Charge density wave phase appears between Mott insulator and supersolid phases.
- CDW phase fully occupies the MI lobe at high extended interaction strength.
- Phase diagrams are consistent across AF and ferromagnetic cases and agree with spin-0 Bose gas results.

## Abstract

We have studied a spinor (F = 1) Bose gas in presence of the density-density interaction through the mean field approach and the perturbation theory for either sign of the spin dependent interaction, namely the antiferromagnetic (AF) and the ferromagnetic cases. In the AF case, the charge density wave (CDW) phase appears to be sandwiched between the Mott insulating (MI) and the supersolid phases for small values of the extended interaction strength. But the CDW phase completely occupies the MI lobe when the extended interaction strength is larger than a certain critical value related to the width of the MI lobes and hence opens up the possibilities of spin singlet and nematic CDW insulating phases. In the ferromagnetic case, the phase diagram shows similar features as that of the AF case and are in complete agreement with a spin-0 Bose gas. The perturbation expansion calculations nicely corroborate the mean field phase results in both these cases. Further, we extend our calculations in presence of a harmonic confinement and obtained the momentum distribution profile that is related to the absorption spectra in order to distinguish between different phases.