6 EC
Semester 2, period 5
5354ANMI6Y
| Owner | Master Physics and Astronomy (joint degree) |
| Coordinator | dr. P.R. Corboz |
| Part of | Master Physics and Astronomy, track Theoretical Physics, year 1 |
Topics covered in this course include: Monte Carlo methods, cluster algorithms, Wang-Landau algorithm, classical spin systems (the Ising model and generalizations), critical phenomena, finite size scaling, variational methods, quantum many-body problems and effective lattice models, exact diagonalization, Quantum Monte Carlo and the negative sign problem, Hartree-Fock, the density matrix renormalization group (DMRG), and tensor network methods. For more details see https://staff.fnwi.uva.nl/p.r.corboz/teaching.htm
Lecture notes
The goal of this course is to become familiar with modern computational methods for the simulation of many-body systems in condensed matter physics, including systems from classical statistical physics and quantum many-body problems. Besides the theoretical understanding of these algorithms an important part of the course is to gain practical experience in computational physics by implementing algorithms (programming) and performing simulations.
At the end of the course, the student is able to
In the lectures the various algorithms as well as applications to many-body problems are introduced and discussed. The exercises consist mostly of programming tasks (in Python) where the student can implement an algorithm and use it to study many-body problems.
Activity | Number of hours |
Computerpracticum | 28 |
Hoorcollege | 28 |
Tentamen | 4 |
Zelfstudie | 108 |
Requirements concerning attendance (OER-B).
| Item and weight | Details |
|
Final grade | |
|
1 (100%) Tentamen |
The exam includes a written part and a programming part
Contact the course coordinator to make an appointment for inspection.
The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl
| Week | Topics lecture |
Exercises |
| 1 | Python refresher, numerical integration, introduction to Monte Carlo, importance sampling, pseudo random numbers, non-uniformly distributed random numbers | Numerical integration, testing pseudo random number generators, Monte Carlo integration and importance sampling |
| 2 | Markov chains, Metropolis algorithm, autocorrelation effects, binning analysis, Jackknife analysis, Ising model, single-spin flip Metropolis algorithm, critical behavior and universality, finite size effects | Metropolis algorithm, binning analysis, Monte Carlo code for the 2D Ising model & data analysis |
| 3 | Fnite size scaling analysis, binder cumulant, critical slowing down, Kandel-Domany framework, cluster algorithms (Swendson-Wang and Wolff), improved estimators, generalization of cluster algorithms, Potts models, O(N) models | Simulation of the 2D Ising model (cont.), Wolff algorithm and finite size scaling for the 2D Ising model |
| 4 | First order phase transitions and the Wang Landau method, Numerov algorithm for the quantum one-body problem, scattering and bound state problem in 1D and higher dimensions, variational solution, time-dependent Schrödinger equation | Wolff algorithm and finite size scaling for the 2D Ising mode (cont.) Bound states in a finite harmonic potential well using the Numerov algorithm |
| 5 |
Introduction to the quantum many-body problem, the general electronic structure problem, effective lattice models, the Hubbard, t-J, and derivation of the Heisenberg model, frustrated spin systems and quantum spin liquids, exact diagonalization, Lanczos algorithm, Jordan-Wigner transformation, bit coding, exploiting symmetries |
Exact diagonalization of the S=1/2 and S=1 Heisenberg spin chain |
| 6 | Hartree-Fock method and derivation, configuration interaction, introduction to quantum Monte Carlo, transverse field quantum Ising model | Hartree-Fock solution of the hydrogen and helium atom, the 1D quantum Ising model |
| 7 | The loop algorithm, the negative sign problem, stochastic series expansion, worm algorithm, quantum Monte Carlo simulations of He-4, superfluids and supersolids, introduction to tensor networks, diagrammatic notation, matrix product states, the area law of the entanglement entropy | Schmidt decomposition and entanglement entropy, decomposition of a state into an MPS, contraction a tensor network, drawing tensor network diagrams |
| 8 | Canonical forms of matrix product states, compression of an MPS, matrix product operators, energy minimization algorithm, imaginary time evolution, the multi-scale entanglement renormalization ansatz, extension to 2D: projected entangled pair states, contraction of a 2D tensor network, application: the Shastry-Sutherland model | Imaginary time evolution algorithm with matrix product states |
The schedule for this course is published on DataNose.
Recommendend prior knowledge: Basic programming skills and knowledge in statistical physics and basic quantum many-body physics (including second quantization) are required. The course 'Statistical Physics and Condensed Matter Theory I' from the first semester is recommended.