Finite-temperature phase diagram of a spin-1 Bose gas

We formulate a self-consistent Hartree-Fock theory for a spin-1 Bose gas at finite temperature and apply it to characterization of the phase diagram. We find that spin coherence between thermal atoms in different magnetic sublevels develops via coherent collisions with the condensed atoms, and is a crucial factor in determining the phase diagram. We develop analytical expressions to characterize the interaction- and temperature-dependent shifts of the phase boundaries.

Figure 1

The $T=0$ phase diagram of a spin-1 Bose gas for cases where the spin-dependent interaction is (a) antiferromagnetic ($c_1>0$) and (b) ferromagnetic ($c_1<0$). The vertical and horizontal axes are the linear and quadratic Zeeman energies (see text) in units of $|c_1|n$, where $n$ is the total number density (which is identical to the condensate number density at $T=0$). The phases shown are (FM) ferromagnetic, (P) polar, (AFM) antiferromagnetic, and (BA) broken-axisymmetry phases (see Sec. 4a and Refs. [1, 4]). The rotational symmetry about the direction of the applied field is spontaneously broken in the AFM and BA phases.

Figure 2

Results of the HF calculation for antiferromagnetic interactions with $c_1/c_0=0.05$. (a) Temperature dependence of the phase diagram in $(q,p)$ space, where the FM-P and AFM-P phase boundaries are independent of temperature. The region of the AFM phase shrinks as the temperature increases. The longitudinal magnetization per atom of (b) the condensate and (c) the noncondensate at $T/T_0=0.1$. The transverse magnetizations are always zero for both condensed and noncondensed atoms.

Figure 3

Temperature dependence of the AFM-FM phase boundary $p_{\mathrm{b}}$ at $q=-3c_{1}n$ and $c_1/c_0=0.05$ obtained by (I) the full HF calculation and (II) the HF calculation but neglecting the off-diagonal elements of $n_{ij}^{\mathrm{nc}}$, together with the curves indicating $n^{\mathrm{c}}/n$, Eq. (35), and Eq. (36).

Figure 4

Results of the HF calculation for ferromagnetic interactions with $c_1/c_0=-0.05$. (a) Temperature dependence of the phase diagram in $(q,p)$ space, where the FM-BA phase boundary is independent of temperature. The region of the BA phase shrinks as the temperature increases. The longitudinal and transverse magnetization per atom at $T/T_0=0.1$ for (b),(c) the condensate and (d),(e) the noncondensate. In (e), $F_{\perp }^{\mathrm{nc}}<0$ means that the transverse magnetization of the noncondensate is antiparallel to that of the condensate.

Figure 5

Temperature dependence of the BA-P phase boundary $q_{\mathrm{b}}$ at $p=0$ and $c_1/c_0=-0.05$ obtained by (I) the full HF calculation and (II) the HF calculation but neglecting the off-diagonal elements of $n_{ij}^{\mathrm{nc}}$, together with the curves indicating $2n^{\mathrm{c}}/n$, Eq. (38), and Eq. (40).