Teaching plan for the course unit

 General information

Course unit name: Monte Carlo Methods

Course unit code: 574646

Coordinator: Bruno Julia Diaz

Department: Department of Quantum Physics and Astrophysics

Credits: 3

Single program: S

 Estimated learning time Total number of hours 75

 Face-to-face and/or online activities 26
 -  Lecture Face-to-face and online 20 -  Lecture with practical component Face-to-face and online 6
 Supervised project 15
 Independent learning 34

 Competences to be gained during study

 Capacity to develop Monte Carlo algorithms to solve quantum mechanical problems

 Learning objectives
 Referring to knowledge To develop a broad and unified perspective on stochastic methods which can be used for evaluation of multi-dimensional integrals and finding optimal parameters in a many-parametric space. To become familiar with classical Monte Carlo method which can be used to sample properties of many-body systems in thermal equilibrium  To become familiar with quantum Monte Carlo method which can be used to sample properties of quantum many-body systems in the ground state and in thermal equilibrium   Referring to abilities, skills To be able to develop, debug and use Monte Carlo code for classical or quantum systems To gain an experience of preparing a Final project which includes development of the code, analysis of the results, preparation of a poster or an online presentation

 Teaching blocks

1. Introduction

*  1.1 Random processes. Discrete and continuous random variables.
1.2 Probability distribution function. Its moments.

1.3 Normal distribution. Box-Muller random generator.

1.4 Central Limit Theorem.

2. Evaluation of integrals

*  2.1 Crude Monte Carlo method (hit-or-miss)
2.2 Monte Carlo method with importance sampling

2.3 Estimation of statistical variance

3. Random walks and Metropolis algorithm

*  3.1 Diffusion equation. Brownian motion.
3.2 Discrete and continuous random walks.

3.3. Random walk solution to Laplace equation

3.4 Importance sampling. Detailed balance condition.

3.5 Metropolis algorithm. Generation of random numbers according to simple laws of the probability distribution.

4. Classical Monte Carlo method for many-body systems.

*  4.1 Classical Monte Carlo method. Maxwell-Boltzmann distribution.
4.2 Observables in classical systems. Energy. Density profile. Pair distribution function. Static structure factor.

4.3 Thermal phase transitions

4.4 Simulated annealing method

5. Variational Monte Carlo methods for bosons and fermions.

*  5.1 Observables in quantum systems. Hamiltonian. Energy. One-body density matrix. Momentum distribution.
5.2 Two observables for the kinetic energy.

5.3 Variational principle. Upper bound to the ground-state energy.

5.4 Variational Monte Carlo method for bosons.

5.5 Jastrow and Nosanow trial wave functions. Gas-solid quantum phase transition.

5.6 Variational Monte Carlo method for fermions. Slater determinants. Fixed node approximation.

6. Diffusion Monte Carlo method.

*  6.1 Imaginary-time projection method. Exact estimator for the ground-state energy.
6.2 Diffusion Monte Carlo method for bosons. Drift Force. Local energy. Branching.

7. Path Integral Monte Carlo method

*  7.1 Path Integral formalism
7.2 Path Integral Monte Carlo method

7.3 Observables.

 Teaching methods and general organization

 Lectures where theoretical contents of the subject are presented.  Practical exercise classes in which students have to participate and develop their own codes..  Individual final projects will be provided.

 Official assessment of learning outcomes

 Exercises will be provided during the course .  At the end of the course each student will be given a personal final project, to be carried out and defended.