Introduction

Monte Carlo methods are very powerful tools for the investigation of quantum many body systems (for a review see, for example, [20]). The simplest of the quantum Monte-Carlo methods is the variational method (VMC). The idea of this method is to use an approximate wavefunction $\psi_T$ for the system (trial wavefunction) and then to sample the probability distribution $p(r)~=~\vert\psi(r)\vert^2$ and calculate averages of physical quantities over this distribution. The average of the local energy $E_L = \psi^{-1}_T H \psi_T$ gives an upper bound to the ground-state energy. In this method one must make a good guess for the trial wavefunction, and there is no regular way for doing it and further improving it. In VMC the closer is the trial wavefunction to the stationary eigenfunction the smaller is the energy variance $\langle H^2\rangle-\langle H\rangle^2$. In usual applications the trial wavefunction depends on the particle coordinates and on some external parameters $\psi_T = \psi_T(r_1,...,r_N,a,b,...)$. By minimizing the variational energy with respect to the external parameters one can optimize the wavefunction within the given class of wavefunctions considered. The Diffusion Monte Carlo method (DMC) can be successfully applied to the investigation of boson systems at low temperatures. It is based on solving the Schrödinger equation in imaginary time and allows us to calculate the exact (in statistical sense) value of the ground state energy. The DMC method will be extensively discussed in the next sections. The Path Integral Monte Carlo (PIMC) is based on carrying out discretized Feynman integral in the imaginary time which allows to calculate the density matrix of the system. The main advantage of this method is that it works at finite temperatures and one has access to the study of thermodynamic properties such as the critical behavior in the proximity of a phase transition ([21], [22]). In this study we use DMC method because we are interested in the ground state properties of the system.