Exact solution for 1D ``perturbative'' equation of state

We will start from very general equation of state of a homogeneous system which can be found in any type of first-order perturbation theory. In the zeroth approximation one has^1.121.13:

$$\mu_{\hom}^{(0)}=C_{1}(na)^{\gamma_1}\frac{\hbar ^{2}}{ma^{2}},$$ (1.136)

here $a$ is unit of length, $C_{1}$ is a numerical coefficient of the leading term in the chemical potential and $\gamma _{1}$ is the power of the dependence on the gas parameter $na.$ The next term of perturbation in general can be written as

$$\mu_{\hom}^{(1)}=C_{1}(na)^{\gamma_1}(1+C_{2}(na)^{\gamma _{2}}+...)%% \frac{\hbar ^{2}}{ma^{2}},$$ (1.137)

where $C_{2}(na)^{\gamma _{2}}\ll 1.$ We will use local density approximation (Sec. 1.6) in order to obtain properties of trapped system. The equation (1.126) can be inverted by using (1.137) to obtain the density profile $n(z)$:

$$n(z)a={\left( \frac{1}{{C_{1}}}\frac{\mu}{\hbar ^{2}/ma^{2}\... ...\right) \right)}^{\frac{{1}+ {{\gamma}_{2}}}{{{\gamma}_{1}}}},$$ (1.138)

here size of the cloud $R$ is related to the chemical potential $\mu =\frac{1}{2}m\omega ^{2}R^{2}$ (1.130).

The value of the chemical potential is fixed by the normalization condition (1.127). It is convenient to make use of the integral equality [GR80]

$$\int\limits_{-1}^{1}(1-x^{2})^{\alpha}\,dx=\frac{\sqrt{\pi}\Gamma (\alpha +1)}{\Gamma (\alpha +\frac{3}{2})},\qquad\alpha >-1$$ (1.139)

Thus we have restriction on the polytropic indices $\gamma_1>-1,\frac{\gamma_1+\gamma_2}{\gamma_1}>-1$. If those conditions are satisfied, then the leading contribution to the chemical potential is given by

$$\frac{\mu ^{(0)}}{\hbar ^{2}/ma^{2}}={\left( \frac{C_{1}^{\frac{1... ...}\gamma_1+1)}{\Delta_{1D}}^{2}\right)}^{\frac{2\,\gamma _{1}}{ 2+\gamma _{1}}},$$ (1.140)

where $\Delta _{1D}$ is the characteristic parameter of a one-dimensional trapped gas defined by (1.135).

In the next order of accuracy the chemical potential is given by

$$\frac{\mu ^{(1)}}{\hbar ^{2}/ma^{2}}=\frac{\mu ^{(0)}}{\hbar ^{2}... .../ma^{2}}\right) ^{\frac{3}{2}+\frac{{1}+\,{{\gamma}_{2}}}{\,{{\gamma}%% _{1}}}}$$ (1.141)

The mean square displacement $\left\langle z^{2}\right\rangle =\frac{1}{N} \int\limits_{-R}^{R}z^{2}n(z)\,\,dz$ is directly related to the potential energy of the oscillator confinement and is given by

$$\frac{\left\langle z^{2}\right\rangle}{R^{2}}=\frac{\,{{\gamma}_{... ...\hbar}^{2}/m~{a}^{2}}%% \right)}^{\frac{{{\gamma}_{2}}}{{{\gamma}_{1}}}}\right)$$ (1.142)

The frequencies of the collective oscillations can be predicted within LDA. The frequency of the breathing mode is inferred from the derivative of the mean square displacement $\Omega_{z}^{2}=-2\left\langle z^{2}\right\rangle \left/ \frac{\partial \left\langle z^{2}\right\rangle}{\partial \omega ^{2}}\right.$ [MS02] and equals to

$$\frac{\Omega _{z}^{2}}{\omega _{z}^{2}}=\left( 2+{{\gamma}_{1}}\r... ...ac{1}{{{%% \gamma}_{1}}})}\right)}^{\frac{2\,{{\gamma}_{2}}}{2+{{\gamma}_{1}}}}$$ (1.143)

The obtained formula is very general and gives an insight to many interesting cases where the perturbation theory can be developed. In the table (1.1) we summarize some of the examples.

Table 1.1: Summary for some of one-dimensional models where the expansion of the equation of state is known. The first column labels the considered model. The coefficients of the expansion are given in columns 2-5. The last column gives the predictions for the oscillation frequencies calculated calculated as (1.143). The parameter $\Delta _{1D}$ is defined by (1.132). Note that the presence of a term in the chemical potential independent of the density (for example, binding energy of a molecule) does not modify the frequencies of oscillations and is ignored.

Limit	$C_1$	$\gamma_1$	$C_2$	$\gamma_2$	$\Omega_z^2/\omega_z^2$
Lieb-Liniger: weak interaction	$2\pi^2$	$1$	$-\sqrt{2}/\pi$	-1/2	$$3+\frac{5(9{\pi )}^{1/3}\,}{32\,\sqrt{2}}/\,{\Delta_{1D} }^{2/3}$$
Lieb-Liniger: strong interaction	$\pi^2/2$	2	-8/3	1	$$4-\frac{128\sqrt{2}}{15\pi ^{2}}{\Delta_{1D} }$$
Attractive Fermi gas: strong interaction	$\pi ^{2}/32$	2	2/3	1	$$4+\frac{64\sqrt{2}}{15\pi ^{2}}{\Delta_{1D} }$$
Attractive Fermi gas: weak interaction	$\pi ^{2}/8$	2	$-8/\pi ^{2}$	-1	$$4+\frac{32}{3\pi ^{2}}/{\Delta_{1D}}$$
Repulsive Fermi gas: strong interaction	$\pi ^{2}/2$	2	$-8\ln (2)/3$	1	$$4-\frac{128\sqrt{2}\ln 2}{15\pi ^{2}}{\Delta_{1D}}$$
Repulsive Fermi gas: weak interaction	$\pi^2/8$	2	$8/\pi^2$	-1	$$4-\frac{32}{3\pi ^{2}}/{\Delta_{1D}}$$
Gas of Hard-Rods	$\pi^2/2$	2	8/3	1	$$4+\frac{128\sqrt{2}}{15\pi ^{2}}{\Delta_{1D}}$$