Ferromagnetic phase transition and Bose-Einstein condensation in spinor Bose gases

Abstract

Phase transitions in spinor Bose gases with ferromagnetic (FM) couplings are studied via meanfield theory. We show that an infinitesimal value of the coupling can induce a FM phase transition at a finite temperature always above the critical temperature of Bose-Einstein condensation. This contrasts sharply with the case of Fermi gases, in which the Stoner coupling Is can not lead to a FM phase transition unless it is larger than a threshold value I0. The FM coupling also increases the critical temperatures of both the ferromagnetic transition and the Bose-Einstein condensation.PACS numbers: 05.30. Jp, 03.75.Mn, 75.10.Lp, 75.30.Kz The magnetism of Fermi (electron) gases has long been a research topic in solid state physics. Although many open questions remain, the magnetic properties of Fermi gases have been well understood [1]. The Fermi surface plays an important role in determining their magnetism. For example, a magnetization density M in Fermi gases increases the band energy by splitting the Fermi surfaces for spin-up and spin-down particles. As a result, Fermi gases mainly behave as Pauli paramagnets in the absence of an exchange interaction. If an effective ferromagnetic (FM) exchange I s is present, electron gases can exhibit ferromagnetism. Within the framework of the Stoner theory, I s results in a negative molecular field energy when M is finite. If I s > I 0 , the Stoner threshold, the value of the molecular field energy becomes larger than that of the increase of the band energy induced by M , and then a FM ground state is energetically favored.The magnetism of Bose gases has been less studied. But since the realization of Bose-Einstein condensation (BEC) in ultracold atomic gases[2], more and more attention has been attracted to this topic, because the constituent atoms, such as 87 Rb, 23 Na, and 7 Li usually have (hyperfine) spin degrees of freedom and thus a magnetic moment m. m can be large in atoms such as 52 Cr[3], for which m = 6µ B , where µ B is the Bohr magneton. Since 1998, atomic gases can be confined and cooled in purely optical traps in which their spin degrees of freedom remain active, and therefore investigating their magnetic properties becomes experimentally possible [4].Ferromagnetism in spinor bosons without any spindependent interactions has already been theoretically studied. As opposed to the case of fermions, the ground states of spinor bosons are degenerate and a ferromagnetic state is among the ground states [5,6]. Furthermore, the spinor Bose gas is rather apt to be magnetized by an external magnetic field even at a finite temperature T , as long as T < T c , the BEC critical temperature [7,8,9,10].In ultracold atomic gases, a Heisenberg-like exchange interaction is usually present, arising from spin-flip scattering processes. For example, the effective two-body interaction in spin-1 atoms was given by [11],where n(r)with S α (α = x, y, z) being 3 × 3 spin matrices, ψ σ (r) is a field annihilation operator for an atom in the hyperfine state |F = 1, σ at point r, and σ = ...

Spin waves in ferromagnetically coupled spinor Bose gases

“…These results agree with the prediction of Ho [8] and Ohmi and Machida [9], and also the microscopic theory [7]. Obviously, the BEC critical temperature is enhanced by M 0 , the magnetization in the normal phase,…”

“…It only enhances the condensate fraction, but does not result in another FM transition. This point is confirmed by the microscopic theory [7], which shows that both the magnetization M and the derivatives (∂M/∂T ) seem smooth at T c .…”

“…It is observed that these materials first undergo a FM transition, then go into the superconducting (SC) state. Their temperature-pressure phase diagram seems quite similar to the temperature-FM coupling phase diagram of the Bose ferromagnet [7], as shown in Figure 1. In such SC ferromagnets, both ferromagnetism and superconductivity are believed due to the same band electrons and the Cooper pairs are most likely in the triplet configuration [12].…”

“…One may question whether the ferromagnetic (FM) transition induced by such a weak FM coupling could be observed in experiments. Recently, it was already shown that the FM transition in Bose gases takes place always above Bose-Einstein condensation (BEC), regardless of the value of the FM coupling [7]. It means that Bose gases can exhibit ferromagnetism at relatively high temperatures in comparison to the energy scale of the FM coupling.…”

“…[7]. Neglecting all the fluctuations and minimizing the total free energy density f t with respect to Φ † , one gets the stable solution for the condensate.…”