Static structure factor of a trapped Tonks-Girardeau gas

The chemical potential of the Tonks-Girardeu gas is known due to fermion-bosonic mapping [Gir60]. It equals to the fermi energy of a one-dimensional spinless fermi gas and is given by the formula (1.101). The dependency on the density is simple and lies within class of functions (1.136) for which the LDA problem was solved in Sec. 1.6.2. The TG gas is described by the subsequent set of parameters: $C_1 = \pi^2/2,\gamma_1 = 2,C_2=0$.

The value of the chemical potential of the TG gas in a trap is immediately found from Eq. 1.140 and equals to $\mu = N\hbar\omega_z$. Its integration with the respect of the number of particles gives the total energy in the LDA^1.15:

$$\frac{E}{N}=N\frac{\hbar\omega_z}{2}$$ (1.151)

The density profile is a semicircle

$$n(z) = n_0\left(1-\frac{z^2}{R_z^2}\right)^{1/2},$$ (1.152)

with system size given by (1.130) $R_z = \sqrt{2N}a_z$ and the density in the center equal to $n_0a_z = \sqrt{2N}/\pi$. The mean square radius of the trapped system is given then by

$$\sqrt{\langle z^2\rangle}=\sqrt{\frac{N}{2}}a_z$$ (1.153)

The static structure factor of a uniform system depends on value of momentum $\k$ and on the density (i.e. on the value of the fermi momentum $k_f$) as given by formula (5.5). We approximate the static structure factor in a trap by averaging it over the density profile (1.152):

$$S^{LDA}(k) = \frac{1}{2R_z}\int\limits_{-R}^R S(k,n(z))\,dz =\fra... ...}{\sqrt{8N}}\arcsin\sqrt{1-\frac{(ka_z)^2}{8N}} +1-\sqrt{1-\frac{(ka_z)^2}{8N}}$$ (1.154)

It is easy to check that it vanishes for small momenta $S^{LDA}(k) \to 0, k\to 0$, while it saturates to unity at large values of momenta $S^{LDA}(k) = 1,\vert k\vert>\sqrt{8N}/a_z$.