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 Theoretical Physics,

Course manual 2019/2020

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

28

Hoorcollege

28

Tentamen

20mins

Zelfstudie

108

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%)

    individual oral exam

    The grading of the final exam is "pass" or "fail"

    Inspection of assessed work

    Contact the course coordinator to make an appointment for inspection.

    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, Markov chains, Metropolis algorithm, autocorrelation effects, binning analysis Numerical integration, testing pseudo random number generators, Monte Carlo integration and importance sampling, Metropolis algorithm, binning analysis  
    2 Jackknife analysis, Ising model, single-spin flip Metropolis algorithm, critical behavior and universality Monte Carlo code for the 2D Ising model & data analysis 16/4, 23:59
    3 Finite size effects & 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, first order phase transitions and the Wang Landau method, 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 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

    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 7/5, 23:59
    6 The loop algorithm, the negative sign problem, stochastic series expansion, worm algorithm, quantum Monte Carlo simulations of He-4, superfluids and supersolids Hartree-Fock solution of the hydrogen and helium atom, the 1D quantum Ising model with QMC  
    7 Introduction to tensor networks, diagrammatic notation, matrix product states, the area law of the entanglement entropy, 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 Schmidt decomposition and entanglement entropy, decomposition of a state into an MPS, contraction a tensor network, drawing tensor network diagrams 18/5, 23:59
    8 2D tensor networks, projected entangled pair states, contraction of a 2D tensor network, application: the Shastry-Sutherland model 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 course 'Statistical Physics and Condensed Matter Theory I' from the first semester is recommended.

    Contact information

    Coordinator

    • dr. P.R. Corboz