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Quantum Monte Carlo Method

We use Diffusion Monte Carlo (DMC) technique in order to obtain the ground state properties of the system. A good choice of the trial wave-function is crucial for the efficiency of the calculation. In order to prove that our trial wave-function is indeed very close to the true ground state wave function we perform calculation of the variational energy $E_{VMC}$ which provides an upper-bound to the ground state energy (see Table 5.1). We find that the variational energy is at maximum $2\%$ higher than the energy of the DMC calculation, which coincides with the exact solution based on the use of the Bethe ansatz (see, also, Fig. 5.1).

In a homogeneous system we use the Bijl-Jastrow construction (2.37) of the trial wave function. The construction of the two-body term $f_2(z)$ is described in Sec. 2.5.4.5. The $\vert z\vert < R$ part corresponds to the exact solution of the two-body problem and provides a correct description of short-range correlations. Long-range correlations arising from phonon excitations are instead accounted for by the functional dependence of $f(z)$ for $z>Z$ [RC67]. The value of the matching point $R$ is a variational parameter which we optimize using the variational Monte Carlo (VMC). The TG wave function (2.5.4.1) is obtained as a special case of our trial wave function (2.55) for $R = B =
L/2$ and $kL = \pi$.


Table 5.1: Energy per particle for different values of the dimensionless density $n_{1D}a_{1D}$: exact result $E_{LL}$ obtained by solving Lieb-Liniger equations, variational result $E_{VMC}$ obtained by optimization the trial wave function 2.55. Variational energy gives the upper bound to the exact energy. DMC calculation recovers the exact result $E_{LL}$.
$n_{1D}a_{1D}$ $E_{LL}/N$ $E_{VMC}/N$
$10^{-3} $ $1.6408~10^{-6} $ $1.64(1)~10^{-6}$
$0.03 $ $1.3949~10^{-3} $ $1.3956(3)~10^{-3}$
$0.3 $ $9.0595~10^{-2} $ $9.089(3)~10^{-2}$
$1$ $0.5252 $ $0.535(3) $
$30 $ $26.842 $ $27.121(3) $
$10^3 $ $981.15 $ $981.72(6) $


The level of accuracy of the trial wave function is particularly important for the calculation of the of $g_1(z)$. Instead, the pair distribution function $g_2(z)$ is calculated using the method of ``pure'' estimators, unbiased by the choice of the trial wave function [CB95]. Due to non local property of the one-body density matrix, the function $g_1(z)$ can instead by obtained only through the extrapolation technique. For an operator $\hat A$, which does not commute with the Hamiltonian, the output of the DMC method is a ``mixed'' estimator $\langle\Psi_0\vert\hat A\vert\psi_T\rangle$. Combined together with the variational estimator $\langle\psi_T\vert\hat A\vert\psi_T\rangle$ obtained from the VMC calculation it can be used for extrapolation to the ``pure'' estimator by the rule $\langle\Psi_0\vert\hat A\vert\Psi_0\rangle = 2\,\langle\Psi_0\vert\hat
A\vert\psi_T\rangle-\langle\psi_T\vert\hat A\vert\psi_T\rangle$. Of course, this procedure is very accurate only if $\psi_T \simeq
\Psi_0$. We find that DMC and VMC give results for $g_1(z)$ which are very close and we believe that the extrapolation technique is in this case exact.

We consider $N$ particles in a box of size $L$ with periodic boundary conditions. In the construction of the trial wave function we have ensured that the two-body term $f_2$ is uncorrelated at the boundaries $f_2(\pm L/2) = 1$. In order to estimate properties of an infinite system we we increase number of particles and study convergence in the quantities of interest. The dependence on the number of particles (finite size effects) are more pronounced at the large density where the correlations extend up to large distances. Out of the quantities we measured, the one-body density matrix is the most sensitive to finite size corrections. As an example in Fig. 5.2 we show $g_1(z)$ at density $n_{1D}a_{1D} = 30$ for 50, 100, 200 and 500 particles and make comparison it with the asymptotic $z\to\infty$ behavior. We find largest finite size effects near the maximal distance $L/2$ for which the one-body density matrix can be calculated. Also we see a dependence of the slope on the number of particles. Already for 500 particles we find the correct slope of the one-body density matrix. For the smaller densities, where finite size effects are smaller, it is sufficient to have $N = 500$.

Figure 5.2: Example of the finite size effects in the calculation of the one-body density matrix at $n_{1D}a_{1D} = 30$.


next up previous contents
Next: Homogeneous system Up: Ground state properties of Previous: Lieb-Liniger Hamiltonian   Contents
G.E. Astrakharchik 15-th of December 2004