# Berezinskii-Kosterlitz-Thouless transition of spin-1 spinor Bose gases in the presence of the quadratic Zeeman effect

## TL;DR

This paper numerically investigates the Berezinskii-Kosterlitz-Thouless transition in spin-1 Bose gases under quadratic Zeeman effects, revealing vortex classifications, transition temperature differences, and new universal relations.

## Contribution

It provides new insights into vortex contributions and superfluid density relations in spinor Bose gases with quadratic Zeeman effects, highlighting conditions for different BKT transition behaviors.

## Key findings

- BKT transition occurs only with integer vortices, not $ ext{Z}_2$ vortices.
- Different BKT transition temperatures for mass and spin superfluid densities under positive quadratic Zeeman effect.
- Spin-singlet pairs exhibit quasi-off-diagonal-long-range order below the BKT transition temperature.

## Abstract

We numerically study the Berezinskii-Kosterlitz-Thouless (BKT) transition of a spin-1 spinor Bose gas under the quadratic Zeeman effect. A calculation of the mass and spin superfluid densities shows that (i) the BKT transition occurs only when vortices are classified by the integer group $\mathbb{Z}$, and $\mathbb{Z}_2$ vortices do not contribute to the BKT transition, (ii) the two BKT transition temperatures for mass and spin superfluid densities are different for a positive quadratic Zeeman effect and equal for a negative quadratic Zeeman effect, and (iii) the universal relation of the superfluid densities at the BKT transition temperature is changed when multiple kinds of vortices contribute to the transition. We have further found that (iv) spin-singlet pairs in non-magnetic states show the quasi-off-diagonal-long-range order at the different temperature lower than the BKT transition temperature, giving the new universal relation of the superfluid density.