# Phase transitions and spin excitations of spin-1 bosons in optical lattice

## TL;DR

This paper analyzes the phase transitions and spin excitations of spin-1 bosons in optical lattices, providing analytical and numerical insights into the MI-SF transition and various magnetic phases, with results aligning with Monte-Carlo simulations.

## Contribution

It introduces an extended SBO method to study spin-1 bosons, deriving phase diagrams and spin phase boundaries considering quantum and thermal fluctuations.

## Key findings

- Quantitative phase diagram of MI-SF transition agrees with Monte-Carlo results.
- Finite-temperature effects are underestimated by mean-field approximation.
- Boundaries of various spin phases in Mott lobes are analytically and numerically determined.

## Abstract

We investigate ground state properties of spin-1 bosonic system trapped in optical lattice with extended standard basis operator (SBO) method. For both ferromagnetic ($U_2<0$) and antiferromagnetic ($U_2>0$) systems, we analytically figure out the symmetry properties in Mott-insulator and superfluid phases, which would provide a deeper insight into the MI-SF phase transition process. Then by applying self-consistent approach to the method, we include the effect of quantum and thermal fluctuations and derive the MI-SF transition phase diagram, which is in quantitative agreement with recent Monte-Carlo simulation at zero temperature, and at finite temperature, we find the underestimation of finite-temperature-effect in the mean-field approximation method. If we further consider the spin excitations in the insulating states of spin-1 system in external field, distinct spin phases are expected. Therefore, in the Mott lobes with $n=1$ and $n=2$ atoms per site, we give analytical and numerical boundaries of the singlet, nematic, partially magnetic and ferromagnetic phases in the magnetic phase diagrams.