6 EC
Semester 2, period 5
5354ANMI6Y
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.
Lecture notes
Lectures and programming exercises (in Python)
|
Activity |
Number of hours |
|
Computerpracticum |
22 |
|
Hoorcollege |
22 |
|
Tentamen |
4 |
|
Zelfstudie |
124 |
| Item and weight | Details |
|
Final grade | |
|
1 (100%) Tentamen |
The final exam includes a written part and a programming part.
In order to be admitted to the final exam, 2 specific exercises will need to be completed (to get a "pass") before their due dates. The details and deadlines will be communicated during the course.
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 |
Course introduction and overview, Python refresher, introduction to Monte Carlo, importance sampling, pseudo random numbers, Markov chains, Metropolis algorithm, autocorrelation effects, binning analysis, Jackknife analysis
|
Basic MC exercise, Metropolis algorithm, binning analysis
|
| 2 |
Ising model, single-spin flip Metropolis algorithm, critical behavior and universality, finite size effects, Finite 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 |
Monte Carlo code for the 2D Ising model & data analysis
Wolff algorithm and finite size scaling for the 2D Ising model
|
| 3 |
First order phase transitions, Wang Landau algorithm, Numerov algorithm for the quantum one-body problem, scattering and bound state problem in 1D and higher dimensions Variational solution, time-dependent Schrödinger equation, 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 |
Wolff algorithm (cont.)
Bound states in a finite harmonic potential well using the Numerov algorithm, anharmonic oscillator |
| 4 |
Exact diagonalization, Lanczos algorithm, Jordan-Wigner transformation, bit coding, exploiting symmetries Hartree-Fock method and derivation, configuration interaction, introduction to quantum Monte Carlo, transverse field quantum Ising model |
Exact diagonalization of the S=1/2 and S=1 Heisenberg spin chain Exact diagonalization (continued), |
| 5 |
World-line representations, the loop algorithm, the negative sign problem, stochastic series expansion, worm algorithm, variational Monte Carlo Introduction to tensor networks, diagrammatic notation, matrix product states, the area law of the entanglement entropy |
The 1D quantum Ising model, world-line configurations, the ALPS library Schmidt decomposition and entanglement entropy, decomposition of a state into an MPS, contraction a tensor network, drawing tensor network diagrams |
| 6 |
Canonical forms of matrix product states, compression of an MPS, matrix product operators, energy minimization algorithm, imaginary time evolution
|
Imaginary time evolution algorithm with matrix product states
|
| 7 |
Finite temperature tensor network algorithms, infinite matrix product states, projected entangled pair states & outlook |
Real time evolution |
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.