Advanced Numerical Methods in Many Body Physics

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 Advanced Matter and Energy Physics, Master Physics and Astronomy, track Theoretical Physics, Master Physics and Astronomy (joint degree), track General Physics and Astronomy,

Course manual 2023/2024

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, Hartree-Fock, configuration-interaction, Quantum Monte Carlo and the negative sign problem, Variational Monte Carlo, 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

120

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  
    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,

    Ising model, single-spin flip Metropolis algorithm, critical behavior and universality, finite size effects, Finite size scaling analysis, binder cumulant, critical slowing down, 

    Basic MC exercise, Metropolis algorithm, binning analysis

     

    Monte Carlo code for the 2D Ising model & data analysis

    		  
    2 Kandel-Domany framework, cluster algorithms (Swendson-Wang and Wolff), improved estimators, generalization of cluster algorithms, Potts models, O(N) models

     

    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

    Exact diagonalization of the S=1/2 and S=1 Heisenberg spin chain

     
    5

    Hartree-Fock method and derivation, configuration interaction, introduction to quantum Monte Carlo, transverse field quantum Ising model,

    world-line representations, the loop algorithm, the negative sign problem, stochastic series expansion, worm algorithm, variational Monte Carlo

    Exact diagonalization (continued),
    Hartree-Fock solution of the hydrogen and helium atom

    The 1D quantum Ising model, world-line configurations, the ALPS library

    		  
    6 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  
    7

    Canonical forms of matrix product states, compression of an MPS, matrix product operators, energy minimization algorithm, imaginary time evolution

    Finite temperature tensor network algorithms, infinite matrix product states, multi-scale entanglement renormalization ansatz, projected entangled pair states & outlook

    (cont.)

     

    Imaginary time evolution algorithm with matrix product states,
    contraction of 2D tensor networks with the corner-transfer matrix renormalization group method

     

    		  
    8      

     

    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 course 'Statistical Physics and Condensed Matter Theory I' from the first semester is recommended.

    Contact information

    Coordinator

    • dr. P.R. Corboz

    Staff

    • Stijn Kleijweg