Temperature dependence of a vortex in a superfluid Fermi gas

The temperature dependence of an isolated quantum vortex, embedded in an otherwise homogeneous fermionic superfluid of infinite extent, is determined via the Bogoliubov–de Gennes (BdG) equations across the BCS-BEC crossover. Emphasis is given to the BCS side of this crossover, where it is physically relevant to extend this study up to the critical temperature for the loss of the superfluid phase, such that the size of the vortex increases without bound. To this end, two techniques are introduced. The first one solves the BdG equations with “free boundary conditions,” which allows one to determine with high accuracy how the vortex profile matches its asymptotic value at a large distance from the center, thus avoiding a common practice of constraining the vortex in a cylinder with infinite walls. The second one improves on the regularization procedure of the self-consistent gap equation when the interparticle interaction is of the contact type, and permits us to considerably reduce the time needed for its numerical integration by drawing elements from the derivation of the Gross-Pitaevskii equation for composite bosons starting from the BdG equations.

Figure 1

Profile of the order parameter $\Delta \left(\rho \right)$ (normalized to its asymptotic value ${\Delta }_0$ at the given temperature) for an isolated vortex versus the distance $\rho$ from the center. The weak-coupling case with ${\left(k_F{a}_F\right)}^{-1}=-1$ is considered for three different temperatures: (a) $T=0$; (b) $T=0.6T_c$; (c) $T=0.9T_c$. In the three cases, the calculation using “free boundary conditions” (full line) is compared against that using a cylindrical box (dashed line), with the value $R=25k_F^{-1}$ for the radius as typically taken in previous calculations (Ref. 14).

Figure 2

Gap parameter $\Delta \left(\rho \right)$ (normalized to its asymptotic value ${\Delta }_0$ at the given temperature) of an isolated vortex versus the distance $\rho$ from the center, for the coupling ${\left(k_F{a}_F\right)}^{-1}$: (a) $-2.0$; (b) $-1.0$; (c) $0.0$; (d) $+1.0$. For each coupling, three different temperatures are considered: $T=0$ (full lines); $T=0.5T_c$ (dashed lines); $T=0.9T_c$ (dashed-dotted lines).

Figure 3

Number density $n\left(\rho \right)$ (normalized to its asymptotic value $n_0$) for an isolated vortex versus the distance $\rho$ from the center, for the same couplings and temperatures of Fig. 2. The inset shows the density at the center of the vortex vs ${\left(k_F{a}_F\right)}^{-1}$, where our results at $T=(0,0.5,0.9)T_c$ from bottom to top (circles with interpolating dashed lines) are compared with those at $T=0$ from Ref. 14 (squares) and from Ref. 15 (triangles).

Figure 4

Number current $j\left(\rho \right)$ (normalized to its maximum value $j_{\max }$ at the given temperature) of an isolated vortex versus the distance $\rho$ from the center, for the same couplings and temperatures of Figs. 2 and 3. The maximum value of the current conventionally identifies the vortex radius $R_{\mathrm{v}}$.

Figure 5

(a) Gap parameter $\Delta \left(\rho \right)$, (b) number density $n\left(\rho \right)$, and (c) number current $j\left(\rho \right)$ of an isolated vortex versus the distance $\rho$ from the center, at zero temperature for the coupling ${\left(k_F{a}_F\right)}^{-1}=-1$. The results of the complete calculation (full lines) are contrasted with those of a calculation that excludes the contribution from the continuum part of the spectrum in Eqs. (B12), (C1), and (C2) (dashed lines).

Figure 6

(a) Gap parameter $\Delta \left(\rho \right)$ and (b) number current $j\left(\rho \right)$ of an isolated vortex versus the distance $\rho$ from the center, close to the critical temperature for the coupling ${\left(k_F{a}_F\right)}^{-1}=-1$. The results of the calculation of the BdG equations (full lines) are compared with those obtained by the Ginzburg-Landau (GL) theory (dashed lines). The maximum values ${\Delta }_0$ for $\Delta \left(\rho \right)$ and $j_{\max }$ for $j\left(\rho \right)$ correspond to the BdG calculation.

Figure 7

Same as Fig. 6 but for the coupling ${\left(k_F{a}_F\right)}^{-1}=-2$ closer to the BCS limit.

Figure 8

Gap parameter $\Delta \left(\rho \right)$ (normalized to the Fermi energy $E_F$) of an isolated vortex versus the distance $\rho$ from the center, for three different temperatures and ${\left(k_F{a}_F\right)}^{-1}=-1$. The results of the fittings to extract the lengths $\xi$ and $\zeta$ in the two different intervals of $\rho$ (broken lines) are compared with those of the the full calculation (full lines). The value of the vortex radius $R_{\mathrm{v}}$ is marked in each case.

Figure 9

The values of the healing length $\xi$, as obtained from the fits (28) to the profiles of $\Delta \left(\rho \right)$ (dots) for four different couplings across unitarity, are shown versus the temperature $T$ (in units of the respective critical temperature $T_c$). These values are then fitted by the mean-field-like expression $\xi \left(T\right)\propto {(T_c-T)}^{-1/2}$ over the whole temperature range down to $T=0$ (dashed lines).

Figure 10

Ratio of the two healing lengths $\xi$ and $\zeta$ (stars) for four different couplings across unitarity versus the temperature $T$ (in units of the respective critical temperature $T_c$). The horizontal (dashed) lines mark the value unity for $\xi /\zeta$.

Figure 11

Healing length $\xi$ at zero temperature versus the coupling ${\left(k_F{a}_F\right)}^{-1}$, obtained from the present BdG calculation and from the approach of Ref. 34 referred to as PS. For comparison, the two expressions $k_F{\xi }^{\mathrm{BCS}}=E_F/\left(3{\Delta }_0\right)$ and $k_F{\xi }^{\mathrm{BEC}}=\sqrt{3\pi /\left(8k_F{a}_F\right)}$ that hold, respectively, in the asymptotic BCS and BEC limits are also reported. To account for their different definitions, the curves labeled BdG and PS are made to coincide in the extreme weak coupling by multiplying the BdG curve by a factor 1.257 (for consistency, the BEC curve is also multiplied by the same factor). In addition, the inset shows $\xi$ vs ${\left(k_F{a}_F\right)}^{-1}$ obtained from the BdG calculation at the finite temperatures $T=0.2T_c$ and $0.5T_c$ across the BCS-BEC crossover.

Figure 12

Partial radial profiles of the gap parameter ${\Delta }^{<}\left(\rho \right)$, density $n^{<}\left(\rho \right)$, and current $j^{<}\left(\rho \right)$ at zero temperature for three couplings ${\left(k_F{a}_F\right)}^{-1}=(-1.0,0.0,+1.0)$, obtained using different values of the upper limit $E_{\mathrm{ul}}$ for the energy. For convenience, the correspondence between the various values of $E_{\text{ul}}$ and the types of lines used in these plots for the three different couplings is reported separately in Table .

Figure 13

Dispersion relation $\varepsilon$ vs $k_{\perp }$ given by Eq. (16), with (a) $\tilde{\mu }>0$ and (b) $\tilde{\mu }<0$. These plots identify the six energy ranges I–VI, where proper selections of the wave vectors (A1) have alternatively to be done.

Figure 14

$\Delta \left(\rho \right)$ (in units of $E_F$) vs $\rho$ (in units of $k_F^{-1}$) obtained by various approximations at zero temperature for (a) ${\left(k_F{a}_F\right)}^{-1}=-1$; (b) ${\left(k_F{a}_F\right)}^{-1}=0$; (c) ${\left(k_F{a}_F\right)}^{-1}=+1$. Comparison is made between the “best” calculation with a large value of $E_c$ [that equals $9E_F$ in panels (a) and (b), and $18E_F$ in panel (c)], and less sophisticated calculations all with the smaller value $E_c=3E_F$ which include, respectively, the linear, the linear plus cubic, and the linear plus cubic plus Laplacian terms on the right-hand side of Eq. (B11).

Figure 15

(a) Number density $n\left(\rho \right)$ (normalized to its bulk value $n_0$) and (b) current density $j\left(\rho \right)$ (normalized to its maximum value at $R_{\mathrm{v}}$) vs $\rho$ (in units of $k_F^{-1}$) obtained at zero temperature and unitarity for an isolated vortex. Calculations corresponding to two values of the cutoff $E_c$ are compared with each other.

Figure 16

The values of the radius $R_{\mathrm{v}}$ of the vortex (dots) are shown versus the temperature $T$ (in units of the respective critical temperature $T_c$) for four different couplings across unitarity. These values are then fitted by the mean-field-like expression $R_{\mathrm{v}}\left(T\right)\propto {(T_c-T)}^{-1/2}$ over the whole temperature range down to $T=0$ (dashed lines).