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Introduction

Quantum Monte Carlo methods (QMC) are very powerful tools for the investigation of quantum many body systems (for a review see, for example, [Cep95],[Gua98]). The usage of QMC techniques provides deep insight into understanding of the physical problem. It allows one to accomplish the ab initio calculation and, starting from a microscopic model (commonly a model Hamiltonian), earn knowledge of the macroscopic behavior of the system. Often it turns out that this approach is the only accessible tool for studying sophisticated problems, as in order to have a model, which can be solved analytically in exact way, one usually has to make severe assumptions, which can be relaxed in QMC. In many cases it is possible to construct analytically a perturbation theory, then its applicability is restricted by smallness of the perturbation parameter and also in cases like that QMC methods can be used to avoid the restrictions. The QMC techniques solve the many-body Schrödinger equation for the ground state and for excited states at zero temperature. Similar to other MC approaches, these techniques are based on stochastic numerical algorithms, which are powerful when one is treating systems with many degrees of freedom.

We are interested in exploring the quantum properties of systems. The quantum effects manifest the most at the lowest temperatures, when the system stays in the ground state. Thus we choose the Diffusion Monte Carlo method to address the problem. This method is exact2.1 for calculation a ground state energy of a bosonic system

In order to study a fermionic system we use Fixed-Note Monte Carlo technique (FN-MC), which is a modification of the DMC method. In general this approach gives an upper bound to the ground state energy, but with a good choice of the trial wave function the difference can be significantly minimized.

In this chapter we will start from the Variational Monte Carlo method which is applicable both for bosons and fermions. Then we will discuss bosonic Diffusion Monte Carlo method and fermionic Fixed-Node Monte Carlo method. We will address in details construction of the trial wave functions and, next, will discuss the implementation of the measurements of the quantities of interest.


next up previous contents
Next: Variational Monte Carlo Up: Quantum Monte Carlo technique Previous: Quantum Monte Carlo technique   Contents
G.E. Astrakharchik 15-th of December 2004