Spin susceptibility and effects of a harmonic trap in the BCS-BEC crossover regime of an ultracold Fermi gas

We theoretically investigate magnetic properties of a trapped ultracold Fermi gas. Including pairing fluctuations within the framework of an extended $T$-matrix approximation, as well as effects of a harmonic trap in the local density approximation, we calculate the local spin susceptibility ${\chi }_{\mathrm{t}}(r,T)$ in the Bardeen-Cooper-Schrieffer–Bose-Einstein condensation crossover region. We show that pairing fluctuations cause nonmonotonic temperature dependence of ${\chi }_{\mathrm{t}}(r,T)$. Although this behavior looks similar to the spin-gap phenomenon associated with pairing fluctuations in a uniform Fermi gas, the trapped case is found to also be influenced by the temperature-dependent density profile, in addition to pairing fluctuations. We demonstrate how to remove this extrinsic effect from ${\chi }_{\mathrm{t}}(r,T)$, to study the interesting spin-gap phenomenon purely originating from pairing fluctuations. Since experiments in cold-atom physics are always done in a trap, our results would be useful for the assessment of preformed pair scenario, from the viewpoint of spin-gap phenomenon.

Figure 1

(a) Self-energy correction $\widehat{\mathrm{\Sigma }}$ in combined ETMA with LDA (LDA-ETMA). The $2\times 2$-matrix particle-particle scattering vertex $\widehat{\mathrm{\Gamma }}={\left\{\mathrm{\Gamma }\right\}}^{j,j^{\prime }}$ is given in (b). In this figure, the double and single solid lines denote the LDA-ETMA dressed Green's function $\widehat{G}$ in Eq. (2.6), and the bare one ${\widehat{G}}^0$ in Eq. (2.10), respectively. The filled circle is a pairing interaction $-U$.

Figure 2

(a) Calculated local spin susceptibility ${\chi }_{\mathrm{t}}(r,T)$ at $r=0.4R_{\mathrm{F}}$, as a function of temperature. $R_{\mathrm{F}}=\sqrt{2{\varepsilon }_{\mathrm{F}}^{\mathrm{t}}/\left(m{\mathrm{\Omega }}_{\mathrm{tr}}^2\right)}$ is the Thomas-Fermi radius, where ${\varepsilon }_{\mathrm{F}}^{\mathrm{t}}={k_{\mathrm{F}}^{\mathrm{t}}}^2/\left(2m\right)={\left(3N\right)}^{1/3}{\mathrm{\Omega }}_{\mathrm{tr}}$ is the LDA Fermi energy in a trap (which equals the LDA Fermi temperature $T_{\mathrm{F}}^{\mathrm{t}}$). ${\chi }_{\mathrm{t}}^0\left(r\right)=3mn{(r,T)}^{1/3}/{\left(3{\pi }^2\right)}^{2/3}$ is the expression for the spin susceptibility in a free Fermi gas at $T=0$ where the number density is replaced by the LDA-ETMA local density $n(r,T)=n_{\uparrow }(r,T)+n_{\downarrow }(r,T)$ at $r=0.4R_{\mathrm{F}}$. At each line, the short vertical line shows $T_{\mathrm{c}}^{\mathrm{t}}$, and the open circle represents the peak position of ${\chi }_{\mathrm{t}}(r,T)$ in the normal state. The arrow shows $T_{\mathrm{c}}^{\mathrm{t}}\left(r\right)$, below which the LDA superfluid order parameter $\mathrm{\Delta }(r=0.4R_{\mathrm{F}})$ becomes nonzero. (b) Density profile $n\left(r\right)=n_{\uparrow }\left(r\right)+n_{\downarrow }\left(r\right)$, when ${\left(k_{\mathrm{F}}^{\mathrm{t}}{a}_s\right)}^{-1}=-0.6$.

Figure 3

Calculated density profile $n(r,T)={\sum }_{\sigma }{n}_{\sigma }(r,T)$ in LDA-ETMA. (a) ${\left(k_{\mathrm{F}}^{\mathrm{t}}{a}_s\right)}^{-1}=-0.6$. (b) ${\left(k_{\mathrm{F}}^{\mathrm{t}}{a}_s\right)}^{-1}=0$ (unitarity limit). (c) ${\left(k_{\mathrm{F}}^{\mathrm{t}}{a}_s\right)}^{-1}=0.4$. The filled circles “a”–“i” correspond to those in Fig. 4.

Figure 4

Mapping of local spin susceptibility ${\chi }_{\mathrm{t}}(r,T)$ in a trapped Fermi gas onto spin susceptibility ${\chi }_{\mathrm{u}}\left(T\right)$ in a uniform Fermi gas. (a) Each solid line with a filled circle (“a”-“i”) is the scaled temperature $T/T_{\mathrm{F}}^{\mathrm{u}}\left(r\right)$ as a function of the scaled interaction strength ${\left(k_{\mathrm{F}}^{\mathrm{u}}{a}_s\right)}^{-1}$, which is obtained from the local density $n(r,T)$ in Fig. 3 at the same label (“a”-“i”). [However, since the solid lines in the cases of “d”-“f” are the same vertical line at ${\left(k_{\mathrm{F}}^{\mathrm{u}}{a}_s\right)}^{-1}=0$, we do not draw them in the figure.] $k_{\mathrm{F}}^{\mathrm{u}}, T_{\mathrm{F}}^{\mathrm{u}}, T_{\mathrm{c}}^{\mathrm{u}}$, and $T_{\mathrm{SG}}^{\mathrm{u}}$, are the Fermi momentum, Fermi temperature, the superfluid phase transition temperature, and the spin-gap temperature, in a uniform Fermi gas, respectively. (b) Spin susceptibility ${\chi }_{\mathrm{u}}\left(T\right)$ in a uniform Fermi gas [39]. The filled circles “a”-“i” are the values of the local spin susceptibility ${\chi }_{\mathrm{t}}(r,T)$ at the same labels in Fig. 3. ${\chi }_{\mathrm{u}}^0\left(0\right)=m{k}_{\mathrm{F}}^{\mathrm{u}}/{\pi }^2$ is the uniform spin susceptibility in a free Fermi gas at $T=0$.

Figure 5

$r-T$ phase diagram of a trapped Fermi gas. (a) ${\left(k_{\mathrm{F}}^{\mathrm{t}}{a}_s\right)}^{-1}=-0.6$ (weak-coupling BCS side). (b) ${\left(k_{\mathrm{F}}^{\mathrm{t}}{a}_s\right)}^{-1}=0$ (unitarity limit). (c) ${\left(k_{\mathrm{F}}^{\mathrm{t}}{a}_s\right)}^{-1}=0.6$ (strong-coupling BEC side). The LDA superfluid order parameter $\mathrm{\Delta }\left(r\right)$ becomes nonzero when $r\le r_{\mathrm{c}}\left(T\right)$ (SF). $T_{\mathrm{p}}^{\mathrm{t}}\left(r\right)$ is the temperature at which ${\chi }_{\mathrm{t}}(r,T)$ takes a maximum value, when $r$ is fixed. $r_{\mathrm{p}}\left(T\right)$ is the spatial position at which ${\chi }_{\mathrm{t}}(r,T)$ takes a maximum value, when $T$ is fixed. ${\chi }_{\mathrm{t}}(r_{\mathrm{SG}}\left(T\right),T)$ is mapped onto ${\chi }_{\mathrm{u}}\left(T_{\mathrm{SG}}^{\mathrm{u}}\right)$ in a uniform Fermi gas. The region $r_{\mathrm{c}}\left(T\right)\le r\le r_{\mathrm{SG}}\left(T\right)$ (SG) is mapped onto the spin-gap regime in a uniform Fermi gas, where ${\chi }_{\mathrm{u}}\left(T\right)$ is suppressed by pairing fluctuations. The region $r>r_{\mathrm{SG}}\left(T\right)$ (NF) is mapped onto the normal Fermi gas regime in the uniform case, where ${\chi }_{\mathrm{u}}\left(T\right)$ monotonically increases with decreasing the temperature. The open and filled circles represent $T_{\mathrm{p}}^{\mathrm{t}}\left(r\right)$ and $r_{\mathrm{p}}\left(T\right)$ obtained from Figs. 2 and 6, respectively. Because of computational problems at low temperatures ($T\lesssim 0.02T_{\mathrm{F}}^{\mathrm{t}}$), we only draw eye guide (thin dashed line) for each line there.

Figure 6

(a) Temperature dependence of local density $n(r,T)$ in the outer region of the gas cloud at $r=0.9R_{\mathrm{F}}$. (b) Scaled local temperature $T/T_{\mathrm{F}}^{\mathrm{t}}(r=0.9R_{\mathrm{F}})$, as a function of $T/T_{\mathrm{F}}^{\mathrm{t}}$.

Figure 7

Local spin susceptibility ${\chi }_{\mathrm{t}}(r,T)$, as a function of the spatial position $r$, measured from the trap center. We also plot the superfluid order parameter $\mathrm{\Delta }\left(r\right)$.

Figure 8

Calculated trap-averaged spin susceptibility $X_{\mathrm{t}}\left(T\right)$ in Eq. (2.13), normalized by the value $X_{\mathrm{t}}^0\left(0\right)=3N/{\varepsilon }_{\mathrm{F}}^{\mathrm{t}}$ in a trapped free Fermi gas at $T=0$. (a) ${\left(k_{\mathrm{F}}^{\mathrm{t}}{a}_s\right)}^{-1}=-0.8$. (b) ${\left(k_{\mathrm{F}}^{\mathrm{t}}{a}_s\right)}^{-1}=0$. (c) ${\left(k_{\mathrm{F}}^{\mathrm{t}}{a}_s\right)}^{-1}=-0.8$. For comparison, we also plot the ETMA spin susceptibility ${\chi }_{\mathrm{u}}\left(T\right)$ in a uniform Fermi gas [40], normalized by the value ${\chi }_{\mathrm{u}}^0\left(0\right)=m{k}_{\mathrm{F}}^{\mathrm{u}}/{\pi }^2$ in a free Fermi gas at $T=0$. In each result, the filled circle shows the temperature at which the spin susceptibility takes a maximum value. In the uniform case, it gives the spin-gap temperature $T_{\mathrm{SG}}^{\mathrm{u}}$. In the trapped case, it gives ${\tilde{T}}_{\mathrm{p}}^{\mathrm{t}}$. The open circles are the recent experimental data on a ${}^6\mathrm{Li}$ Fermi gas [46]. Because of computational problems, our LDA-ETMA results end at $T\simeq 0.02T_{\mathrm{F}}^{\mathrm{t}}$; the thin dashed lines at lower temperatures in panels (a) and (b) are eye guides.

Figure 9

Peak temperature ${\tilde{T}}_{\mathrm{p}}^{\mathrm{t}}$ of $X_{\mathrm{t}}\left(T\right)$. For comparison, we also plot the spin-gap temperature $T_{\mathrm{SG}}^{\mathrm{u}}$ [39]. ${\tilde{T}}_{\mathrm{p}}^{\mathrm{t}:\mathrm{BEC}}$ is the solution of Eq. (4.2).

Figure 10

Averaged spin susceptibility $X_{\mathrm{t}}^{\mathrm{MF}}\left(T\right)$ in the mean-field approximation, where the Hartree-Fock mean-field self-energy ${\widehat{\mathrm{\Sigma }}}^{\mathrm{MF}}(r,T)$ in Eq. (4.1) is used for the ETMA one in Eq. (2.7).