content_image/435530.jpg content_image/435543.jpg content_image/435544.jpg content_image/435546.jpg content_image/435547.jpg content_image/435545.jpg content_image/435548.jpg content_image/435531.jpg content_image/435532.jpg content_image/435549.jpg content_image/435550.jpg content_image/435551.jpg
<img src='content_image/435530.jpg'>

Eq. (11), we get the free energy density for the ASF state 

<img src='content_image/435543.jpg'>

From the thermodynamical relation n = − ( ∂f/∂µ ), we find that 

<img src='content_image/435544.jpg'>

where n is the density of (bare) atoms and 

Using Euler’s relation f = − P , we get the equation of state at T = 0 for the ASF phase 

<img src='content_image/435546.jpg'>

with µ as a function of n and δ given by Eq. (24). Fi- nally, we may calculate the sound velocity v$_{s}$ through the relation v 2 s = ( n/m ) ∂µ/∂n , and obtain 

<img src='content_image/435547.jpg'>

with µ as a function of n and δ given by Eq. (24). 

The atom density n gives rise to a characteristic length scale n − 1 /d and a characteristic energy scale n 2 $^{/d}$/ (2 m ). By measuring the lengths and energies in units of these characteristic scales, setting d = 3, and using Eq. (10) to replace δ by ν , Eqs. (24), (25), and (26) can be written as 

<img src='content_image/435545.jpg'>

<img src='content_image/435548.jpg'>

<img src='content_image/435531.jpg'>

<img src='content_image/435532.jpg'>

where x = bmδa$_{0}$n − 1 / 3 = n − 1 / $^{3}$ν with ν = − 1 /a$_{s}$ , b = 2 /π and y = 2 mµn − 2 / $^{3}$, and 

where ˜ P = 2 mn − 5 / $^{3}$P and ˜ v$_{s}$ = 2 mv$_{s}$n − 1 / $^{3}$. Equations (27) — (29) exhibit the universal scaling forms for µ , P and v$_{s}$ . These universal scaling forms are due to the presence of a nontrivial zero-density fixed point. The results are shown in Figs. 2, 3, and 4. 

In the unitary limit δ = 0 Eqs. (24) and (25) reduce to 

<img src='content_image/435549.jpg'>

<img src='content_image/435550.jpg'>

<img src='content_image/435551.jpg'>

