Lieb-Liniger Hamiltonian

A cold bosonic gas confined in a waveguide or in a very elongated trap can be described in terms of a one-dimensional model if the energy of the motion in the long longitudinal direction is insufficient to excite the levels of transverse confinement. Further, if the range of interparticle interaction potential is much smaller than the characteristic length of the external confinement, one parameter is sufficient to describe the interaction potential, namely the one-dimensional $s$-wave scattering length. In this case the particle-particle interactions can be safely modeled by a $\delta$-pseudopotential. Such a system is described by the homogeneous Lieb-Liniger Hamiltonian

$$\hat H_{LL} = -\frac{\hbar^2}{2m}\sum\limits_{i=1}^N\frac{\partial^2}{\partial z^2_i} +g_{1D}\sum\limits_{i<j}\delta(z_i-z_j),$$ (5.1)

where the positive one-dimensional coupling constant is related to the one-dimensional $s$-wave scattering length $g_{1D} = -2\hbar^2/ma_{1D}$ (1.69) with $m$ being mass of an atom. In the presence of tight harmonic transverse confinement (we denote the oscillator length as $a_\perp$, the one-dimensional scattering length $a_{1D}$ was shown [Ols98] to experience a resonant behavior in terms of $a_{3D}$ due to virtual excitations of transverse oscillator levels

$$a_{1D} = -a_\perp\left(\frac{a_\perp}{a_{3D}}-1.0326\right)$$ (5.2)

By tuning the $a_{3D}$ by Feshbach resonance value of $a_{1D}$ can by widely varied. Without using the Feshbach resonance one typically has $a_{3D}\ll a_\perp$ condition fulfilled. In this situation the relation (5.2) simplifies $a_{1D} = -a_\perp^2/a_{3D}$ (compare with the mean-field result (1.125)).

All properties in this model depend only on one parameter, the dimensionless density $n_{1D}a_{1D}$. On the contrary to 3D case, where at low density the gas is weakly interacting, in 1D system small values of the gas parameter $n_{1D}a_{1D}$ mean strongly correlated system. This peculiarity of 1D system can be easily seen by comparing the characteristic kinetic energy $\hbar^2n^{2/D}/2m$, with $D$ being number of dimensions, to the mean-field interaction energy $g n$. The equation of state of this model first was obtained Lieb and Liniger [LL63] by using the Bethe ansatz formulation. The energy of the system is conveniently expressed as

$$\frac{E}{N} = e(n_{1D}\vert a_{1D}\vert)\frac{\hbar^2n_{1D}^2}{2m},$$ (5.3)

where the function $e(n_{1D}\vert a_{1D}\vert)$ is obtained by solving a system of LL integral equations. In the mean-field regime $n_{1D}a_{1D}\gg 1$ the energy per particle is linear in the density $E^{GP}/N = g_{1D} n_{1D}/2$, although in the strongly correlated regime the dependence is quadratic (1.102): $E^{TG}/N = \pi^2n_{1D}^2/6m$. Still an explicit general expression of the energy for an arbitrary value of $n_{1D}a_{1D}$ is not known. The dependence of the energy on the density resulting from numerical solution of the LL integral equations is plotted in Fig. 5.1. On the same figure we present the energy obtained by a different method of solving the Schrödinger equation, the Diffusion Monte Carlo method (see section 5.3). Results of both methods are in perfect coincidence.

Figure 5.1: Energy per particle: Bethe ansatz solution (solid line), DMC (circles), GP limit (dashed line), TG limit (dotted line). Energies are in units of $\hbar ^2/(ma_{1D}^2)$.

The chemical potential is defined as the derivative of the total energy with respect to the number of particles $\mu = \partial E/\partial N$. The healing length $\xi = \hbar/(\sqrt 2 mc)$ is related to the speed of sound $c$, which in turn can be extracted from the chemical potential $mc^2 = n_{1D}\frac{\partial}{\partial n_{1D}} \mu$.

These quantities can be obtained in an explicit way in the regime of strong interactions $n_{1D}a_{1D}\ll 1$. In this limit the energy of incident particle is not sufficient to tunnel through the particle-particle interaction potential. Two particle can never be at the same point in space, which together with spatial peculiarity of 1D system acts as an effective Fermi exclusion principle. Indeed, in this limit the system of bosons acquires many fermi-like properties. There exist a direct mapping of the wave function of strongly interacting bosons onto a wave function of noninteracting fermions due to Girardeau [Gir60]. We will refer to this limit as the Tonks-Girardeau limit. The speed of sound in this gas is related to the fermi momentum of the one-component fermi system at the same density $c = p_F/m = \pi n_{1D} \hbar/m$. The chemical potential equals to the fermi energy (1.101) $\mu = \pi^2n_{1D}^2/2m$ (see $n_{1D}a_{1D}\ll 1$ limit in the Fig. 5.1).

Further, due to this mapping one knows the pair distribution function, which exhibits Friedel-like oscillations

$$g_2(z) = 1 - \frac{\sin^2 \pi n_{1D} z}{(\pi n_{1D} z)^2}$$ (5.4)

The static structure factor of the TG gas is given by

$$S(k)=\left\{ \begin{array}{cc} \vert k\vert/(2\pi n_{1D}),&\... ...pi n_{1D}\\ 1,&\vert k\vert>2\pi n_{1D}\\ \end{array}\right.$$ (5.5)

The one-body density matrix $g_1(z)$ was calculated in terms of series expansion at small and large distances [Len64,VT79,JMMS80]. Its slow long-range decay

$$g_1(z) = \frac{\sqrt{\pi e}2^{-1/3}A^{-6}}{\sqrt{zn}}, \quad n_{1D}\vert a_{1D}\vert\ll 1$$ (5.6)

leads to an infrared divergence in the momentum distribution $n(k) \propto 1/\sqrt{\vert k\vert}$.

Beyond the TG regime full expressions of the correlation functions are not known. The long-range asymptotics (i.e. distances much larger than the healing length $\xi$) can be obtained from the hydrodynamic theory of the low-energy phonon-like excitations [RC67,Sch77,Hal81,KBI93]. One finds following power-law decay (1.199)

$$g_1(z) = \frac{C_{asympt}}{\vert z n_{1D}\vert^\alpha},$$ (5.7)

where $\alpha = mc/(2\pi\hbar n_{1D})$ and coefficient $C_{asympt}$ is given by formula (5.15). In the TG regime $c = \pi\hbar n_{1D}/m$ and thus $\alpha=1/2$ as anticipated above. In the opposite GP regime ( $n_{1D}\vert a_{1D}\vert\gg 1$), the result is $\alpha = 1/(\pi\sqrt{2n_{1D}\vert a_{1Dd}\vert})$ which decreases as $n_{1D}\vert a_{1D}\vert$ increases. That means that in the Lieb-Liniger theory $\alpha\le 1/2$. Instead in the super-Tonks regime one deals with special situation $\alpha>1/2$ In the mean-field limit the relation of the coefficient of proportionality in Eq. 5.7 to $\alpha$ was established by Popov [Pop80] Of course power-law decay of the non-diagonal element of the one-body density matrix (5.6,5.7) does not support long-range order and excludes the existence of Bose-Einstein condensation in one dimension even at zero temperature [Sch63]. The behavior of the momentum distribution for $\vert k\vert\ll 1/\xi$ follows immediately from (5.7)

$$n(k) = C_{asympt} \left\vert\frac{2n_{1D}}{k}\right\vert^{1-\alph... ...\left(\frac{1}{2}-\frac{\alpha}{2}\right)}{\Gamma\left(\frac{\alpha}{2}\right)}$$ (5.8)

Furthermore, the hydrodynamic theory allows one to calculate the static structure factor in the long-wavelength regime $\vert k\vert\ll 1/\xi$. One finds the well-known Feynmann result [Fey54]

$$S(k) = \frac{\hbar\vert k\vert}{2mc}$$ (5.9)

Recently, the short range behavior of the one-, two-, and three-body correlation functions has also been described. The value at $z=0$ of the pair correlation function at arbitrary density can be obtained from the equation of state through the Hellmann-Feynman theorem [GS03b]:

$$g_2(0)=-\frac{(n_{1D}\vert a_{1D}\vert)^2}{2} e',$$ (5.10)

where the derivative of $e$ is taken with the respect to the gas parameter $n_{1D}\vert a_{1D}\vert$.

This quantity vanishes in the TG regime and approaches unity in the GP regime. The ``excluded volume'' correction (1.106) allows one to specify its behavior close to the TG region:

$$g_2(0) = \frac{\pi^2n_{1D}^2\vert a_{1D}\vert^2}{3},\qquad n_{1D}\vert a_{1D}\vert\ll 1$$ (5.11)

The $z=0$ value of the three-body correlation function was obtained in a perturbative manner in the regions of strong and weak interactions [GS03b]. Similarly to $g_2(0)$, it quickly decays in the TG limit

$$g_3(0) = \frac{(\pi n a_{1D})^6}{60}, \quad n_{1D}a_{1D}\ll 1,$$ (5.12)

and goes to a constant value in the GP regime:

$$g_3 = 1-\frac{6\sqrt{2}}{\pi\sqrt{n a_{1D}}},\quad n_{1D}a_{1D}\gg 1.$$ (5.13)

Furthermore, recently the first few terms of the short-range series expansion of the one-body correlation function have been calculated in [OD03]

$$g_1(z) = 1 -\frac{1}{2}(e+e' n_{1D}\vert a_{1D}\vert)\vert n_{1D}z\vert^2 + \frac{e'}{6} \vert n_{1D}z\vert^3$$ (5.14)

This expansion is applicable for distances $\vert n_{1D}z\vert \ll 1$ and arbitrary densities.