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 for the system (trial
wavefunction) and then to sample the probability distribution
and calculate averages of physical quantities over
this distribution. The average of the local energy
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
. In usual applications the trial
wavefunction depends on the particle coordinates and on some
external parameters
.
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.
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