Spinor Bose-Einstein Condensation and Bose Ferromagnetism Qiang Gu† Institut F¨Ur Laser-Physik, Universit¨At Hamburg, Luruper Chaussee 149, 22761 Hamburg

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Spinor Bose-Einstein condensation and Bose ferromagnetism Qiang Gu† Institut f¨ur Laser-Physik, Universit¨at Hamburg, Luruper Chaussee 149, 22761 Hamburg

ABSTRACT We discuss ferromagnetism in spinor Bose gases with eﬀective ferromagnetic (FM) couplings between particles. Some basic problems, including interplay between the FM transition and Bose-Einstein condensation, the low-lying collective excitations as Goldstone modes, and the structure of the ground state, are considered. We show on the mean-ﬁeld level that the FM transition occurs above Bose-Einstein condensation. Under the spin conservation rule, the FM transition corresponds to the domain formation. Our results can be applied to the ultracold 87Rb gas and magnetic dipolar gases.

Ferromagnetism: an overview [1]

Weiss molecular-ﬁeld theory: for classical particles Stoner mean-ﬁeld theory: for fermions Spin waves as Goldstone modes

• The model • M increases the band energy due to the Fermi surface splitting: • The spin-wave in Heisenberg ferromagnets: − · H = I ij Si Sj M2 ω = Dk2,D= ISa2 a−the lattice constant E = N( )−the density of state k – Introducing the molecular-field: M = Si = Sj b 2 F 4µBN(F ) – Decoupling the interaction: • The spin-wave in fermionic ferromagnets: z 2 Si · Sj ≈Si·Sj + Si ·Sj−Si·Sj =2MS − M • M leads to a negative molecular-field energy density: i 2 1 ωk = Dk • E = − IM2 The result m 2 – The molecular-field energy: with the spin-wave stiffness T (z-the coordination number) • The Stoner criterion results,   T T ~ (I - I )1/2 F s 1  2 2 − 2 ≤ 1 D = n ∆E − n (∇E ) Em = H = zNIM 0 T ~ I I>I = k k k k F s 2 6MN MI 2µBN(F ) k k – The FM transition point: • The transition temperature: • Summary: ω ∼ k2 1 I I k ∼ s kBTF = S(S +1)zI I I 2 ∼ − 3 TF I Is gapless at k =0

Ferromagnetism in spinor Bose gases in the thermodynamic limit [2,3]

Phase diagram: a microscopic model [2] Phenomenological analysis [3] Spin waves [3]

• The model Hamiltonian: • Free energy density of isotropic spin-F Bose-Einstein condensate • Considering ﬂuctuations: M =(δMx,δMy,M + δMz) † ∗ ∗ ∗ ∗ † 2 Ψ =(Ψ1 + δψ1,δψ0,δψ−1)

− − · H = t ψiσψjσ I Si Sj † 0 2 β0 4 fb = ∗∇Ψ ∇Ψ + α (T − Tc )|Ψ| + |Ψ| • We get ijσ ij 2m 2 βs ∗ ∗ – The density excitation · + ΨσΨσFσγ Fσ γ Ψγ Ψγ z 2 • Suppose S =(0, 0, S )=(0, 0,M), then we arrive at the ∗ fI(δΨ ,δΨ )= δΨ∗(k) δΨ (−k) × mean-ﬁeld Hamiltonian: • Free energy density of the NORMAL ferromagnetic phase 1 1 1 1 k 2 2 − | |2 | |4 k + βΦ1 βΦ1 δΨ1(k) H = (k σHm)nkσ 2 M M ∗ , fm = c|∇M| + a (T − T ) + b 2 2 δΨ (−k) kσ f 2 4 βΦ1 k + βΦ1 1 2 2 ∗ ∗ • with the spectrum with the kinetic energy k = k /2m . m is the mass of the Coupling between two phases

particle. Hm = IM is the molecular ﬁeld. − | | ∗ 2 2 fc = g M ΨσFσγΨγ ω1 = +2βΦ1 k • The mean-ﬁeld equations are given by k • The total Ginzburg-Landau free energy is ft = fb + fm + fc. – The spin excitation N 1 ∗ n = = nk,1 + nk,0 + nk,−1 (1) Minimizing ft with respect to Ψ and M,onegets V V I ∗ ∗ f (δM ,δM−; δΨ ,δΨ )= δM (k) δΨ (k) × k + 0 0 + 0 1 g g a − 0 0 −  k √  Hm = I nk,1 nk,−1 (2) Tc = Tc + M = Tc + [Tf Tc] 2 V α α b 2 gΦ1 2 k ck + − gΦ δM−(k)  √ 2M0 2 1  • At I → 0, both δT = T − T and δT = T − T tend to zero. − 2 δΨ0(k) • The relation between the chemical potential a and the coupling c c 0 f f 0 2 gΦ1 k + gM0 So we get Im for small Im δT 4π − f which suggest the spin waves in normal and condensed compo- a ≈ I2 (δTc)=δTf 1 ∗ 3 2 m T nents are coupled, with the mixed spin-wave spectrum (3ζ[ ])3 2 ∗ 2 with T =(g/α ) a /b 2 2

Φ where a and Im are rescaled as ≈ 2 1 1 • ω0 csk , with cs = c + Suppose δTf = CI when I<<1, 2M2 2m − 1/3 ∗ 2 0 a = (µ + Hm)/(kBT ),Im = In m /(2π ) CI δTc = CI 1 − It means that an infinitesimal FM coupling can induce T ∗ 1. The spectrum has the same momentum-dependance as a FM phase transition at a finite temperature above the in usual ferromagnets. 2. The spin-wave stiffness contains BEC critical point Thus the phase diagram is reproduced. two parts, implying “two fluids” feature of the system.

Discussions and outlooks Phase diagram of ferromagnets • Magnetic dipolar Bose gases [5] – FM transition corresponding to domain formation – Domains formed before BEC µ0 · − · · Umd = 3[m1 m2 3(m1 rˆ)(m2 rˆ)] – Fz=1 and -1 domains spatially separated bosons 4πr T - T ~ I T F 0 – Fz=0 domains much smaller T - T ~ I c 0 The dipolar interaction causes ferromagnetism: • If not, [6,7] T 0 the dipolar ferromagnetism T ~ (I - I )1/2 F s – Fz=1,0 and -1 particles miscible classical particles Though the FM interaction is very week, we expect that T ~ I fermions F ± the FM transition or dipolar ferromagnetism can be ob- – Fz=0 bosons condensed earlier than 1 bosons – Fz=0 component dominating over others I I served at a relatively high temperature. s Outlooks Spinor Bose gases under experimental conditions Experimental background How does the ground state manifest in experiments: Our results are relevant to current experiments on quantum degen- In experiments, the system is away from the thermodynamic limit. the 3 components miscible or spatially separated? erate atomic bosons. • Total spin conservation To examine this, one can • Atomic bosons with FM scattering: 87Rb [4] • Small particle number • investigate spin dynamics [8]

c0 cs Can the spin-rotational symmetry be broken spontaneously? • determine fractions of all the components V = n · n + S · S,cs < 0 2 2 • If yes, [2,3,6] † with K. Bongs, K. Sengstock and R.A. Klemm

References: [5] Q. Gu, Phys. Rev. A 68, 025601 (2003); S. Yi, L. You and H. Pu, Phys. Rev. Lett. 93, 040403 (2004) [1]P.Mohn,Magnetism in the Solid: An Introduction (Springer-Verlag, Berlin, 2003) [6]T.Isoshima,T.OhmiandK.Machida,J.Phys.Soc.Jpn.69, 3864 (2000) [2] Q. Gu and R.A. Klemm, Phys. Rev. A 68, 031604(R) (2003); S. Ashhab, cond-mat/0403720 [7] W. Zhang, S. Yi and L. You, cond-mat/0409680, to appear in PRA [3] Q. Gu, K. Bongs and K. Sengstock, cond-mat/0405094, to appear in PRA 70 [8] H. Schmaljohann et al., Phys. Rev. Lett. 92, 040402 (2004); M. Erhard et al., Phys. Rev. A 70, [4] T.L. Ho, Phys. Rev. Lett. 81, 742 (1998); T. Ohmi & K. Machida, J. Phys. Soc. Jpn. 67, 1822 (1998) 031602(R) (2004); cond-mat/0406485