The Bethe Ansatz

The ground state is obtained through choosing the following set of quantum numbers:

In the thermodynamic limit, we can immediately obtain an integral equation for the corresponding ground state rapidity density function. We shall look for a ground state with finite particle density $n$. It is then natural to look for a density ${\rho }_g(\lambda )$ which is nonzero only within a finite interval $\lambda \in [-{\lambda }_F,{\lambda }_F]$ such that

The integral equation l.bec therefore becomes

which is known as the Lieb equation. The Fermi momentum ${\lambda }_F$ can be viewed as an independent variable which (indirectly) specifies the particle density $n$, the relationship between these being

The density is a monotonically increasing function of ${\lambda }_F$ for $c>0$ (this will be proven later on), but the exact relationship cannot be written in closed form. In practice, one thus needs to solve l.le and l.ng self-consistently if one requires a specific particle density.

The Lieb equation is in fact identical to an equation known as Love's equation, occurring in classical electrostatics, and describing the capacitance of a circular capacitor. It does not admit a closed-form solution for $0<c<\mathrm{\infty }$. Expansions for small or large interactions can however be written down. See b-Gaudin for details.

Plots of the ground-state rapidity density function $\rho (\lambda )$ for a number of values of the interaction parameter can be found in Fig. FigLLgs. There, $k_F$ as a function of $c$ for unit filling $n=1$ is also plotted, together with the chemical potential $\mu$, as well as the total ($e$), kinetic ($t$) and potential ($v$) energy per unit lenght (in other words per particle at unit filling). These are simply calculated as

It is interesting to define the `maximally complicated' value of interaction for Lieb-Liniger as the point at which the kinetic energy density equals the interaction energy density (there is then no obvious starting point for any perturbative expansion). This occurs when $t=v\simeq 0.82668(1)$ at $\gamma \simeq 4.3485(1)$. Note that the interaction energy density has a maximum of $\simeq 0.8277(1)$ at a slightly higher value of the interaction $\gamma \simeq 4.6572(4)$, the kinetic energy density then being $\simeq 0.8824(2)$.

Table 1: Top left: the ground state rapidity density distribution $\rho (\lambda )$ for various values of the interaction parameter. Top right: Fermi rapidity ${\lambda }_F$ as a function of interaction parameter $c$ for unit filling $n=1$. Bottom left: chemical potential $\mu$ as a function of $\gamma$. Bottom right: total ($e$), kinetic ($t$) and potential ($v$) energy per particle as a function of $\gamma$.

The solution of the Lieb equation is particularly simple in the limit $c\to \mathrm{\infty }$, since the Cauchy kernel vanishes. One then gets

The Bethe equations l.bec then give the very simple hole density ${\rho }_h$ and total density ${\rho }_t$