Course manual 2021/2022

Course content

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

Study materials

Other

Lecture notes

Objectives

Explain, implement, and apply computational methods to study many-body systems, ranging from Monte Carlo simulations in classical statistical physics to tensor network algorithms for quantum many-body systems
Explain the challenges and the physics of (selected) many-body systems
Interpret and analyze numerical simulation results

Teaching methods

Lecture
Computer lab session/practical training

Lectures and programming exercises (in Python)

Learning activities

Activity	Number of hours
Computerpracticum	24
Hoorcollege	24
Tentamen	4
Zelfstudie	116

Attendance

Requirements concerning attendance (OER-B).

In addition to, or instead of, classes in the form of lectures, the elements of the master’s examination programme often include a practical component as defined in article A-1.2 of part A. The course catalogue contains information on the types of classes in each part of the programme. Attendance during practical components is mandatory.

Assessment

Item and weight	Details
Final grade
1 (100%) Tentamen

The final exam includes a written part and a programming part.

Assignments

In order to be admitted to the final exam, 3 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.

Fraud and plagiarism

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

Course structure

Week	Topics lecture	Exercises	Due date
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	21.4 (23:59)
3	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	Simulation of the 2D Ising model (cont.) Wolff algorithm and finite size scaling for the 2D Ising model
4	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, exact diagonalization, Lanczos algorithm, Jordan-Wigner transformation, bit coding, exploiting symmetries	Bound states in a finite harmonic potential well using the Numerov algorithm Exact diagonalization of the S=1/2 and S=1 Heisenberg spin chain	12.5 (23:59)
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 (continued)
6	Hartree-Fock method and derivation, configuration interaction, introduction to quantum Monte Carlo, transverse field quantum Ising model, the loop algorithm, the negative sign problem, stochastic series expansion, worm algorithm,	Hartree-Fock solution of the hydrogen and helium atom, the 1D quantum Ising model
7	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	23.5. (23:59)
8	Canonical forms of matrix product states, compression of an MPS, matrix product operators, energy minimization algorithm, imaginary time evolution, projected entangled pair states & outlook	Imaginary time evolution algorithm with matrix product states

Timetable

The schedule for this course is published on DataNose.

Additional information

Recommendend prior knowledge: Basic programming skills and knowledge in statistical physics and basic quantum many-body physics (including second quantization) are required. The courses 'Condensed Matter Theory I+2' from the first semester are recommended.

Owner	Master Physics and Astronomy (joint degree)
Coordinator	dr. P.R. Corboz
Part of	Master Physics and Astronomy, track Advanced Matter and Energy Physics, Master Physics and Astronomy, track Theoretical Physics, Master Physics and Astronomy (joint degree), track General Physics and Astronomy,

Advanced Numerical Methods in Many Body Physics