Ferromagnetic phase transition in spinor Bose gases

TL;DR

This paper reviews the ferromagnetic phase transition in spinor Bose gases, highlighting how ferromagnetic order emerges above Bose-Einstein condensation and analyzing collective excitations and their spectra.

Contribution

It provides a theoretical analysis of the phase diagram, transition conditions, and collective excitations in ferromagnetically interacting spinor Bose gases.

Findings

Ferromagnetic transition occurs above Bose-Einstein condensation.

The Bose condensate is fully polarized in the ferromagnetic phase.

Density modes follow a Bogliubov spectrum, and spin waves have a quadratic dispersion at long wavelengths.

Abstract

The achievement in cooling alkali atomic gases, such as $^{87}$Rb, $^{23}$Na and $^{7}$Li, to quantum degeneracy opens up a way to study magnetism in spinor bosons, because these constituent atoms usually have a hyperfine spin degree of freedom. This article reviews several basic problems related to the ferromagnetic phase transition in spinor atomic Bose gases from a theoretical perspective. After a brief discussion on various possible origins of the ferromagnetic interaction, the phase diagram of the ferromagnetically coupled spinor bosons is investigated. It is found that the ferromagnetic transition occurs always above Bose-Einstein condensation and the Bose condensate is fully polarized. The low-lying collective excitations of the spinor condensate, including spin and density modes, are discussed. The spectrum of the density mode is of the Bogliubov form and the spin wave spectrum has a $k^2$-formed dispersion relation at long wavelengths. The spin-wave stiffness coefficient contains contribut