The model and method

We consider a 1D system of $N$ spinless bosons described by the following contact-interaction Hamiltonian

$$H=-\frac{\hbar^2}{2m}\sum_{i=1}^{N}\frac{\partial^2}{\partial z_i^2}+g_{1D}\sum_{i<j}\delta(z_{ij}) \;,$$ (6.1)

where $m$ is the mass of the particles, $z_{ij}=z_i-z_j$ denotes the interparticle distance between particle $i$ and $j$ and $g_{1D}$ is the coupling constant which we take large and negative. The study of the scattering problem of two particles in tight waveguides yields the a relation of the effective 1D coupling constant $g_{1D}$ in terms of the 3D $s$-wave scattering length $a_{3D}$ [Ols98]. The relation is given by the formula (8.4), where $a_\perp=\sqrt{\hbar/m\omega_\perp}$ is the characteristic length of the transverse harmonic confinement producing the waveguide. The confinement induced resonance is located at the critical value $a_{3D}^c$ and corresponds to the abrupt change of $g_{1D}$ from large positive values ( $a_{3D}\lesssim a_{3D}^c$) to large negative values ( $a_{3D} \gtrsim a_{3D}^c$). The renormalization (8.4) of the effective 1D coupling constant has been recently confirmed in a many-body calculation of Bose gases in highly elongated harmonic traps using quantum Monte Carlo techniques [ABGG04b,ABGG04a].

For positive $g_{1D}$, the Hamiltonian (6.1) corresponds to the Lieb-Liniger (LL) model (5.1). The ground state and excited states of the LL Hamiltonian have been studied in detail [LL63,Lie63] and, in particular, the TG regime corresponds to the limit $g_{1D}=+\infty$. The ground state of the Hamiltonian (6.1) with $g_{1D}<0$ has been investigated by McGuire [McG64] and one finds a soliton-like state with energy $E/N=-mg_{1D}^2(N^2-1)/24\hbar^2$. The lowest-lying gas-like state of the Hamiltonian (6.1) with $g_{1D}<0$ corresponds to a highly-excited state that is stable if the gas parameter $na_{1D}\ll 1$, where $n$ is the density and $a_{1D}$ is the 1D effective scattering length defined in Eq. (8.4). This state can be realized in tight waveguides by crossing adiabatically the confinement induced resonance. The stability of the gas-like state can be understood from a simple estimate of the energy per particle. For a contact potential the interaction energy is given by (1.21) $E_{int}/N=g_{1D} n g_2(0)/2$, where the $g_2(0)=\langle\hat\Psi^\dagger(z)\hat\Psi^\dagger(z)\hat\Psi(z)\hat\Psi(z)\rangle/n^2$, is the value at zero of the two-body correlation function (1.12) and $\hat\Psi^\dagger$, $\hat\Psi$ are the creation and annihilation particle operators (1.1). In the limit $g_{1D}\to-\infty$ one can use for the correlation function the result in the TG regime (5.11)[GS03b], which does not depend on the sign of $g_{1D}$. In the same limit the kinetic energy can be estimated by (1.102): $E_{kin}/N\simeq\pi^2\hbar^2n^2/(6m)$, corresponding to the energy per particle of a TG gas. For the total energy $E=E_{kin}+E_{int}$ one finds the result (1.105) $E/N\simeq\pi^2\hbar^2n^2/(6m)-\pi^2\hbar^2n^3a_{1D}/(3m)$, holding for $na_{1D}\ll 1$. For $na_{1D}<0.25$ this equation of state yields a positive compressibility $mc^2=n\partial\mu/\partial n$, where $\mu=dE/dN$ is the chemical potential and $c$ is the speed of sound, corresponding to a stable gas-like phase. We will show that a more precise estimate gives that the gas-like state is stable against cluster formation for $na_{1D}\lesssim 0.35$.

The analysis of the gas-like equation of state is carried out using the VMC technique. The trial wave function employed in the calculation is of the form (2.63). For $g_{1D}<0$ ($a_{1D}>0$) the wave function $f(z)$ changes sign at a nodal point which, for $R_{m}\gg a_{1D}$, is located at $\vert z\vert=a_{1D}$. In the attractive case the wave function has a node which means that the variational calculation can be easily done, while without additional modifications the DMC can not be done. The variational energy is calculated through the expectation value of the Hamiltonian (6.1) on the trial wave function (2.14). In the calculations we have used $N=100$ particles with periodic boundary conditions and because of the negligible dependence of the variational energy on the parameter $R_{m}$ we have used in all simulations the value $R_{m}=L/2$, where $L$ is the size of the simulation box. Calculations carried out with larger values of $N$ have shown negligible finite size effects.