Monte Carlo study of quantum phase transitions at zero temperature

In this Ph.D. thesis quantum Monte Carlo methods are applied to investigate the properties of a number of ultracold quantum systems. In Chapter 1 we discuss the analytical approaches and approximations used in the subsequent Chapters; also we describe the general concepts of the two-particle scattering problem as a tool to construct Jastrow terms in trial wave functions. Chapter 2 explains in details the Quantum Monte Carlo methods employed in our calculations from the theoretical and practical points of view. In Chapter 3 we explain the Ewald summation technique, applied to a power-law 1/|r|k interaction potential, and a generic approach to obtain the Ewald terms. The obtained expressions of this analytic work are implemented into simulations of different physically relevant systems (Rydberg atoms and Yukawa particles). Chapter 5 is devoted to the modelling of a system, governed by the model potential between Rydberg atoms $1/r^6$. The phase diagram of the system is obtained for a relevant range of densities and temperatures, combining quantum simulations at low temperature and classical treatment at higher temperature. A special attention is paid to the classical description of this system, composed of Rydberg atoms, and its comparison to the quantum system. In Chapter 4 we present the simulation of a system with the Yukawa interaction potential. The following Chapter 6 presents the results of the Quantum Monte Carlo simulations of molecular para-hydrogen at zero and finite temperatures, performed in our Group. Conclusions are drawn in Chapter 7.