Goldstino spectrum in an ultracold Bose-Fermi mixture with explicitly broken supersymmetry

We theoretically investigate a supersymmetric collective mode called the Goldstino in a Bose-Fermi mixture. The explicit supersymmetry breaking, which is unavoidable in cold-atom experiments, is considered. We derive the Gell-Mann–Oakes–Renner (GOR) relation for the Goldstino, which gives the relation between the energy gap at zero momentum and the explicit breaking term. We also numerically evaluate the gap of the Goldstino above the Bose-Einstein condensation temperature within the random phase approximation (RPA). While the gap obtained from the GOR relation coincides with that in the RPA for the mass-balanced system, there is a deviation from the GOR relation in the mass-imbalanced system. We point out that the deviation becomes large when the Goldstino pole is close to the branch point, although it is parametrically a higher order with respect to the mass-imbalanced parameter. To examine the existence of the Goldstino pole in realistic cold atomic systems, we show how the mass-imbalance effect appears in ${}^6\mathrm{Li}\text{−}{}^7\mathrm{Li}$, ${}^{40}\mathrm{K}\text{−}{}^{41}\mathrm{K}$, and ${}^{173}\mathrm{Yb}\text{−}{}^{174}\mathrm{Yb}$ mixtures. Furthermore, we analyze the Goldstino spectral weight in a ${}^{173}\mathrm{Yb}\text{−}{}^{174}\mathrm{Yb}$ mixture with realistic interactions and show a clear peak due to the Goldstino pole. As a possibility to observe the Goldstino spectrum in cold-atom experiments, we discuss the effects of the Goldstino pole on fermionic single-particle excitation as well as the relationship between the GOR relation and Tan's contact.

Figure 1

Feynman diagrams for the Goldstino propagator $\mathrm{\Gamma }$ consisting of RPA series of boson-fermion bubble $\mathrm{\Pi }$. The solid (dashed) line and the black dot represent the fermion (boson) propagator and the boson-fermion interaction $U_{bf}$, respectively.

Figure 2

Ratio between fermionic and bosonic number densities $N_f/N_b$ with different interactions as functions of the temperature $T$ in (a1) ${}^6\mathrm{Li}\text{−}{}^7\mathrm{Li}$, (b1) ${}^{40}\mathrm{K}\text{−}{}^{41}\mathrm{K}$, and (c1) ${}^{173}\mathrm{Yb}\text{−}{}^{174}\mathrm{Yb}$ Bose-Fermi mixtures with $\mathrm{\Delta }\mu =\mathrm{\Delta }U=0$. (a2), (b2), and (c2) show the bosonic chemical potential ${\mu }_b$ in systems corresponding to (a1), (b1), and (c1), respectively. In these plots, ${\mu }_b$ is divided by the energy scale ${\varepsilon }_b$ characterizing $N_b$ as ${\varepsilon }_b=k_b^2/\left(2m_b\right)$, where $k_b={\left(6{\pi }^2{N}_b\right)}^{\frac{1}{3}}$. $T_{\mathrm{BEC}}$ is the BEC temperature.

Figure 3

The Goldstino gap ${\omega }_G$ calculated by the GOR (solid lines) and RPA (dashed lines) with $\mathrm{\Delta }U=\mathrm{\Delta }\mu =0$. For ${}^6\mathrm{Li}\text{−}{}^7\mathrm{Li}$ mixtures, we plot ${\omega }_G$ at different interaction strengths, $k_b{a}_{bb}=0.2$, 0.4, and 0.8. We also show ${\omega }_G$ in ${}^{40}\mathrm{K}\text{−}{}^{41}\mathrm{K}$ and ${}^{173}\mathrm{Yb}\text{−}{}^{174}\mathrm{Yb}$ mixtures at $k_b{a}_{bb}=0.2$.

Figure 4

Calculated Goldstino spectral weight $A_G(\mathit{p},\omega ){\varepsilon }_b/N_b$ within the RPA in (a) ${}^6\mathrm{Li}\text{−}{}^7\mathrm{Li}$, (b) ${}^{40}\mathrm{K}\text{−}{}^{41}\mathrm{K}$, and (c) ${}^{173}\mathrm{Yb}\text{−}{}^{174}\mathrm{Yb}$ Bose-Fermi mixtures at $T=T_{\mathrm{BEC}}$. Parameters are set at $k_b{a}_{bb}=0.2$ and $\mathrm{\Delta }U=\mathrm{\Delta }\mu =0$.

Figure 5

(a) Goldstino spectral weight $A_G(\mathit{p}=\mathit{0},\omega )$ at zero momentum in a ${}^6\mathrm{Li}\text{−}{}^7\mathrm{Li}$ mixture at $T=1.16T_{\mathrm{BEC}}$ and $\mathrm{\Delta }\mu =\mathrm{\Delta }U=0$ with $k_b{a}_{bb}=0$, 0.2, 0.4, 0.6, and 0.8. (b) Contour plot of $A_G(\mathit{p},\omega )$ at $k_b{a}_{bb}=0.8$.

Figure 6

(a) Comparison of the Goldstino gap ${\omega }_G$ between the GOR and the RPA and (b) $\tilde{\mathrm{\Phi }}\left({\omega }_G^{\mathrm{GOR}}\right)/{\omega }_G^{\mathrm{GOR}}$ given by Eq. (37) in a ${}^6\mathrm{Li}\text{−}{}^7\mathrm{Li}$ mixture at $T=1.16T_{\mathrm{BEC}}$ and $\mathrm{\Delta }\mu =\mathrm{\Delta }U=0$.

Figure 7

Calculated Goldstino spectral weight $A_G(\mathit{p}=\mathit{0},\omega ){\varepsilon }_b/N_b$ within the RPA in a ${}^{173}\mathrm{Yb}\text{−}{}^{174}\mathrm{Yb}$ Bose-Fermi mixture at $T=T_{\mathrm{BEC}}$ and $\mathrm{\Delta }\mu =0$. Interaction parameters are chosen to reproduce the experimental ratio $a_{bf}/a_{bb}=7.34/5.55$ [60]. The sharp peaks at positive energy are the Goldstino poles.

Figure 8

Self-energy diagram for supersymmetric fluctuations associated with the RPA Goldstino propagator $\mathrm{\Gamma }$. Solid (dashed) lines represent fermion (boson) propagators $G_{b\left(f\right)}^{\mathrm{H}}$ with the Hartree shift ${\mathrm{\Sigma }}_{f\left(b\right)}^{\mathrm{H}}$. Black dots show the boson-fermion coupling $U_{bf}$.

Figure 9

Fermionic single-particle spectral function $A_f(\mathit{p}\to \mathit{0},\omega )$ at the zero-momentum limit in a ${}^{173}\mathrm{Yb}\text{−}{}^{174}\mathrm{Yb}$ mixture at $T=T_{\mathrm{BEC}}$ with $k_b{a}_{bb}=0.3$ and $a_{bf}/a_{bb}=7.34/5.55$ [60]. In the numerical calculation, we take the momentum cutoff $\mathrm{\Lambda }=2k_b$. Dotted and dashed vertical lines represent the fermionic pole position ${\xi }_{\mathit{p}\to \mathit{0}}^f=-({\mu }_f-U_{bf}{N}_b)$ within the mean-field theory and the Goldstino gap ${\omega }_G^{\mathrm{RPA}}$ obtained from the RPA analysis.

Figure 10

Comparison of $A_{G,0}(\mathit{0},\omega )$ in a noninteracting ${}^6\mathrm{Li}\text{−}{}^7\mathrm{Li}$ mixture at $T=1.38T_{\mathrm{BEC}}$ with ${\mu }_b={\mu }_f$ obtained from Eq. (B2) (dashed curve) and the numerical result with $\delta ={10}^{-3}{\varepsilon }_b$ (solid curve).

Figure 11

Fermionic single-particle spectral function $A_f(\mathit{p}\to \mathit{0},\omega )$ at different momentum cutoffs, $\mathrm{\Lambda }/k_b=1.5$, 2, 2.5, and 3. Other parameters are the same as in Fig. 9.