Static response, collective frequencies, and ground-state thermodynamical properties of spin-saturated two-component cold atoms and neutron matter

The thermodynamical ground-state properties and static response in both cold atoms at or close to unitarity and neutron matter are determined using a recently proposed density functional theory (DFT) based on the $s$-wave scattering length $a_s$, effective range $r_e$, and unitary gas limit. In cold atoms, when the effective range may be neglected, we show that the pressure, chemical potential, compressibility modulus, and sound velocity obtained with the DFT are compatible with experimental observations or exact theoretical estimates. The static response in homogeneous infinite systems is also obtained and a possible influence of the effective range on the response is analyzed. The neutron matter differs from unitary gas due to the noninfinite scattering length and to a significant influence of effective range, which affects all thermodynamical quantities as well as the static response. In particular, we show for neutron matter that the latter response recently obtained in auxiliary-field diffusion Monte Carlo (AFDMC) can be qualitatively reproduced when the $p$-wave contribution is added to the functional. Our study indicates that the close similarity between the exact AFDMC static response and the free-gas response might stem from the compensation of the $a_s$ effect by the effective range and $p$-wave contributions. We finally consider the dynamical response of both atoms or neutron droplets in anisotropic traps. Assuming the hydrodynamical regime and a polytropic equation of state, a reasonable description of the radial and axial collective frequencies in cold atoms is obtained. Following a similar strategy, we estimate the equivalent collective frequencies of neutron drops in anisotropic traps.

Figure 1

The pressure (a), the isothermal compressibility (b), the chemical potential (c), and the sound velocity (d) as function of $-{\left(a_s{k}_F\right)}^{-1}$ obtained from the functional Eq. (1) assuming $r_e=0$ (blue long dashed line). The quantity $P_0$ is defined as $P_0={(\mu /{\mu }_{\mathrm{FG}})}^{5/3}{P}_{\mathrm{FG}}$. For comparison, different experimental data or theoretical estimates are shown: (a) black filled circles are data from [22], QMC calculations from [23] (red open circles), results obtained from a Noziére-Schmitt-Rink approximation (blue open triangles) [24], or with Green-function method (De Dominicis-Martin formalism) (green open squares) [25]; (b) gray filled diamonds from [26]; (c) calculation from [27] (open orange squares), and the green dotted line is obtained from the best fit to the QMC calculations [21] (data taken from Fig. 3 of Ref. [24]); (d) brown filled squares and purple triangles are data, respectively, from Ref. [28] and Ref. [29].

Figure 2

Pressure as a function of $-{\left(a_s{k}_F\right)}^{-1}$ (a) or as a function of $-\left(a_s{k}_F\right)$ (b) obtained from the functional Eq. (1) assuming $r_e=0$ fm (blue long dashed line). For comparison, the black short dashed and the black short dot-dashed lines correspond to the Taylor expansion to first order in $\left(a_s{k}_F\right)$ or to first order in ${\left(a_s{k}_F\right)}^{-1}$, respectively. The red solid line corresponds to the neutron matter case assuming $r_e=2.716$ fm. In both cases, $a_s=-18.9$ fm.

Figure 3

The pressure $[X=P]$ (blue solid line), chemical potential $[X=\mu ]$ (red dashed line), and inverse compressibility $[X=1/\kappa ]$ (green dot-dashed line) obtained at unitarity using the functional Eq. (8) as a function of $\left(r_e{k}_F\right)$. The arrow indicates the unitary limit for $r_e=0$.

Figure 4

The pressure (a), the inverse of the compressibility (b), the chemical potential (c), and the sound velocity (d) obtained in neutron matter as function of $-\left(a_s{k}_F\right)$ using the functional Eq. (1) (red solid line). The case with $r_e=0$ (cold atom reference) is displayed with blue dashed line. The green dot-dashed line corresponds to the result obtain by adding to the functional the leading order contribution of the $p$ wave (see text). In this figure, we use $a_s=18.9$ fm and $r_e=2.716$ fm so that $-\left(a_s{k}_F\right)=10$, 20, and 30 correspond, respectively, to $\rho =0.005$, 0.040, and $0.135{\mathrm{fm}}^{-3}$.

Figure 5

Static response of neutron matter at density $\rho =0.1{\mathrm{fm}}^{-3}$ obtained with the Skyrme EDF using the Sly4 (red thick solid line), ${\mathrm{SkM}}^{*}$ (blue dashed line), SkP (gray thin solid line), and SIII (green dot-dashed line) sets of Skyrme parameters (see Ref. [38] for their values). Note that the EDF results are obtained by neglecting the spin–orbit term in the functional. Adding the spin-orbit contribution does not change significantly the result. The black circles (with error bars) are those of Ref. [11], while the black dotted line is the free Fermi-gas static response.

Figure 6

(a) Evolution of the effective mass $m^{*}/m$ in neutron matter as a function of $-\left(a_s{k}_F\right)$ (with $a_s=-18.9$ fm) using Skyrme EDF for the set of parameters used in Fig. 5 with the same lines convention. (b) Evolution of the effective mass in neutron matter deduced from selected many-body calculations: blue triangles [39], red circles [40], green squares [41], black diamonds [42].

Figure 7

Static response of unitary gas (UG) obtained with the functional Eq. (8), assuming $r_e=0$ (black solid line) or $r_e\to +\infty$ (green dot-dashed line). The latter case corresponds to the maximum effect induced by $r_e$ and corresponds to an effective Bertsch parameter equal to 0.66. The free Fermi-gas response (black dotted line) is also shown as a reference. For comparison, the red filled circles corresponds to the static response at unitarity obtained in Ref. [43], using the SLDA where a slightly different value of the Bertsch parameter ${\xi }_0=0.42$ was assumed. For completeness, we also show (blue dashed line) the UG result obtained with Eq. (8) for ${\xi }_0=0.42$.

Figure 8

Static response function obtained with the functional Eq. (1) as function of $q/k_F$ for (a) $\rho =0.04{\mathrm{fm}}^{-3}$ and (b) $\rho =0.1{\mathrm{fm}}^{-3}$. The blue dashed line and red solid line correspond, respectively, to the UG case (cold atom reference) and $r_e=2.716\mathrm{fm}$ (neutron matter case). Note that at the density considered, the UG case cannot be distinguished from the neutron matter assuming $r_e=0$. At both densities, the AFDMC results of Refs. [10, 11] are shown with blue open circles. The green dot-dashed line finally display the result obtained by adding the $p$-wave contribution to the functional Eq. (1). Consistently with the use of the AV8 interaction in the AFDMC calculation, we use a value $a_p^3=0.0916$ fm for the $p$-wave scattering volume.

Figure 9

Evolution of the normalized static response as a function of density for $q=k_F$ (same as Fig. 15 of Ref. [11]). The AFDMC results of Ref. [11] is shown by blue open circles. The result of the functional Eq. (1) with $r_e=2.176$ fm and $r_e=0$ fm are shown by red solid and blue dashed line, respectively. The result obtained by adding the $p$-wave contribution to the functional is shown by green dot-dashed line. The black dotted line corresponds to the free-Fermi-gas limit.

Figure 10

Evolution of the nonadiabatic index $\mathrm{\Gamma }$ at unitarity (red solid line) deduced from the functional Eq. (8) as a function of $\left(r_e{k}_F\right)$. The arrow indicates the unitary limit for $r_e=0$. For comparison, the black short dashed line corresponds to the Taylor expansion of $\xi$ to second order in $\left(r_e{k}_F\right)$.

Figure 11

Adiabatic index as a function of $-{\left(a_s{k}_F\right)}^{-1}$ (a) or as a function of $-\left(a_s{k}_F\right)$ (b) obtained from the functional Eq. (1) assuming $r_e=0$ fm (blue long dashed line). For comparison, the black short dashed and the black short dot-dashed lines correspond to the Taylor expansion to first order in $\left(a_s{k}_F\right)$ or to first order in ${\left(a_s{k}_F\right)}^{-1}$, respectively. The red solid line corresponds to the neutron matter case assuming $r_e=2.716$ fm. In both cases, $a_s=-18.9$ fm.

Figure 12

${\omega }_{rad}^p/{\omega }_0$ (a) and ${\omega }_{ax}^p/\left(\lambda {\omega }_0\right)$ (b) as function of $-{\left(a_s{k}_F\right)}^{-1}$ obtained with the functional Eq. (1) with $r_e=0$ (blue dashed line). The symbols are experimental data: triangles from Ref. [55], circles from Ref. [56], and squares from Ref. [57]. (All data sets are taken from Fig. 3 of Ref. [58]).

Figure 13

${\omega }_{\mathrm{rad}}^p/{\omega }_0$ (a) and ${\omega }_{\mathrm{ax}}^p/\left(\lambda {\omega }_0\right)$ (b) as function of $\left(r_e{k}_F\right)$ obtained with the functional Eq. (1) at unitarity ${\left(a_s{k}_F\right)}^{-1}=0$.

Figure 14

${\omega }_{\mathrm{rad}}^p/\left(\lambda {\omega }_0\right)$ (respectively, ${\omega }_{\mathrm{ax}}^p/{\omega }_0$) as a function of $-\left(a_s{k}_F\right)$ (a) (respectively, (b)) obtained in neutron matter using $a_s=-18.9\mathrm{fm}$ and $r_e=2.716\mathrm{fm}$ (red solid line) in the functional Eq. (1), while the blue long-dashed line corresponds to the result obtained with $r_e=0$. For comparison, we also show the result of the Skyrme Sly5 parameter sets (black filled circles) and the result obtained by adding to the functional the leading order $p$-wave contribution (green dot-dashed line).

Figure 15

Dynamical structure function at unitarity obtained from the functional Eq. (1) (red solid line) compared to the experimental results of Ref. [59] (blue triangles). ${\varepsilon }_F={\hslash }^2{k}_F^2/2m$ is the Fermi energy.

Figure 16

Static structure function obtained with the functional (with $r_e=0$) (red solid line) as a function of $q/k_F$ compared to the diffusion Monte Carlo result of Ref. [62] (blue circles). For comparison, we also show the result obtained in Ref. [60] with the SLDA (green dashed line).