Three-dimensional Bose gas near a Feshbach resonance

In this paper, we explore the nature of three-dimensional Bose gases at large positive scattering lengths via resummation of dominating processes involving a minimum number of virtual atoms. We focus on the energetics of the nearly fermionized Bose gases beyond the usual dilute limit. We also find that an onset instability sets in at a critical scattering length, beyond which the near-resonance Bose gases become strongly coupled to molecules and lose the metastability. Near the point of instability, the chemical potential reaches a maximum, and the effect of the three-body forces can be estimated to be around a few percent.

Figure 1

Diagrams showing contributions to the total energy $E(n_0,\mu )$. The dashed lines are for $k=0$ condensed atoms, thick solid internal lines in (a) and (b) are for interacting Green's functions $G^{-1}(\epsilon ,\mathbf{k})=\epsilon -{\epsilon }_{\mathbf{k}}-{\Sigma }_{11}+\mu +i\delta$, and thin solid lines in (c) and (d) are for noninteracting Green's function $G_0^{-1}(\epsilon ,\mathbf{k})=\epsilon -{\epsilon }_{\mathbf{k}}+\mu +i\delta$. (a) The blue circle is for $g_2(n_0,\mu )$; vertices here represent the bare interaction $U_0$ in Eq. (1). (b) ($N=1,2,\cdots$)-loop diagrams that lead to the integral equation for $G_3(-3\eta ,p)$ in Eq. (7). Note that the usual tree-level diagram violates the momentum conservation and does not exist; the one-loop diagram has already been included in $g_2(n_0,\mu )$ and therefore needs to be subtracted when calculating $g_3(n_0,\mu )$. Arrowed dashed lines here as well as in (c) and (d) stand for outgoing condensed atoms, and the remaining dashed lines stand for incoming ones. (c) and (d) The tree level and examples of one-loop diagrams that yield the usual Lee-Huang-Yang corrections in the limit of small $n{a}^3$. The self-consistent approach contains contributions from (c)-type diagrams but not (d)-type ones (see further discussion in the text). Patterned green circles also represent the sum of diagrams in (a), but with thin internal lines, or the noninteracting Green's function $G_0$ lines. All vertices are time ordered from left to right.

Figure 2

(a) Chemical potential $\mu$ in units of the Fermi energy ${\epsilon }_F$ and (b) condensation fraction as a function of $n^{1/3}a$. Beyond a critical value of $0.18$ (shown as circles), the solutions become complex, and only the real part of $\mu$ is plotted; the imaginary part of $\mu$ scales like ${\epsilon }_F{(a/a_{cr}-1)}^{1/2}$ near $a_{cr}$. (However, the sharp transition would be smeared out if the small imaginary part of $G_3$ is included.) Dashed lines are the result of the Lee-Huang-Yang theory, thin solid blue lines are the solution without three-body effects (i.e., $g_3=0$). Thick solid red lines are the solution with $g_3$ included; the momentum cutoff is $\Lambda =100n^{1/3}$. The inset is the relative weight of three-body effects in the chemical potential as a function of $\Lambda n^{-1/3}$ at the critical point.

Figure 3

$\text{Re}{G}_3(-3\eta ,0)\eta$ as a function of $\eta ={\Sigma }_{11}\left(n_0\right)-\mu$. $E_{\Lambda }={\Lambda }^2/2m$ and $\Lambda$ is the momentum cutoff. The imaginary part of $G_3$ (not shown) is zero once $3\eta >1/\left(m{a}^2\right)$.