Quantum Fields on the Lattice


Dénes Sexty & Alexander Rothkopf, Summer term 2015


  • Lecture: Monday, 11:30-13:00, Phil 19 / Seminarraum
  • Exercises: Monday,14:15-16:00, CIP-Pool Phil 12 (starting Mon. April 20th)
    You will need your 'Uni-ID' to log in to the computer system.
  • Klausur: Preliminary date July 31st, 13:00h in the CIP-Pool Phil 12
  • Programming take-home exam: PDF hand in solutions until July 30th or at the Klausur on July 31st.


  • Introduction
    • Field content of the standard model, non-perturbative phenomena of current interest, the concept of lattice discretization, development of computational ressources
      Lecture note as PDF
  • The Ising Model
    • 1d solution: direct and transfer-matrix, 2d-Ising low- and high-temperature expansion, dimerization, phase diagram
      Lecture note as PDF

      Tutorial - Introduction to C: Note as PDF
  • Monte Carlo Methods
    • Markov chains, Detailed balance, Metropolis algorithm, Hybrid Monte Carlo, Langevin equation, Fokker-Planck equation
      Lecture note as PDF (incl. complete proof of detailed balance, see also Bhanot 1986.)

      Tutorial - Towards 1d-Ising with Metropolis: Note as PDF, Sample Program: source
  • Solving the linear and non-linear Schroedinger equation
    • Recap on partial differential equations, physics motivation: quantum mechaniscs, Bose-Einstein condensation, discretization schemes
      Lecture note as PDF

      Tutorial - Solving the 2d-Poisson equation: Note as PDF, Sample Program: source
  • Path integral on the lattice
    • PDE solvers continued; path integral for quantum mechanics, euclidean formulation, partition function (transfer matrix, reflection positivity)
      Lecture note as PDF (typo corrected in last line of ADI solver)

      Tutorial - Eigenvalues and -vectors for the 1d-harmonic oscillator, forward Euler time evolution,
      Sample Program Eigenvectors, -values: source
  • Scalar Fields I
    • reflection positivity, transfer operator, free scalar field, continuum limit
      Lecture note as PDF

      Tutorial - Thomas algorithm for periodic boudnary conditions, Crank-Nicolson time evolution, Sample Program : source
  • Scalar Fields II
    • spontaneous symmetry breaking, spin system, critical point, continuum limit, lattice perturbation theory, renormalization hopping parameter expansion, triviality
      Lecture note as PDF

      Tutorial - HMC algorithm for O(1) theory in 1 Euclidean dimension, Sample Program : source
  • Scalar Fields III - Data Analysis
  • Gauge fields on the lattice I
    • Concept of gauge theory, parallel transport, Haar measure, Wilson action, Gauge fixing
      Lecture note as PDF
      Tutorial - Simulating SU(2) gauge theory : Exercise Sheet
  • Gauge fields on the lattice II
    • Landau gauge fixing, Kogut-Sussking Hamiltonian, Improved actions, Wilson loop and strong coupling expansion, Polyakov loop and center symmetry, scale setting
      Lecture note as PDF

      Tutorial - Heatbath for SU(2) - Simulating SU(3) gauge theory : Exercise Sheet
  • Fermions on the lattice
    • Naive fermions, doubling, Chiral symmetry, Wilson fermions, staggered fermions,Fermion matrix inversion
      Lecture note as PDF

      Tutorial - Conjugate Gradient algorithm for sparse matrices: code , Exercise Sheet
  • Non-equilibrium QFT
    • scalar fields in cosmology and in cold atom systems, gauge fields in heavy-ion collisions, path integral on the Schwinger-Keldysh time contour, classical statistical approximation and its simulation, classical gauge fields in Hamiltonian and Lagrangian formulation
      Lecture note as PDF

      Tutorial - High performance computing with OpenMPI: Exercise Sheet
      Sample program: code, Guide to collective communication
  • Nonzero chemical potential


    Lattice theory

  • Lecture notes on lattice theory by Prof. M. Laine (recommended reading)
  • I. Montvay, G. Münster: Quantum Fields on a Lattice
    Cambridge, UK: Univ. Pr. (1994) 491 p. (Cambridge monographs on mathematical physics)
  • Heinz J. Rothe: Lattice Gauge Theories, An introduction
    World Scientific Lecture Notes in Physics - Vol 74
  • Jan Smit: Introduction to quantum fields on a lattice: A robust mate
    Cambridge Lect.Notes Phys. 15 (2002)
  • C. Gattringer, C.B. Lang: Quantum Chromodynamics on the Lattice
    Lecture Notes in Physics 788, Springer

  • C Programming

  • For those with no C programming experience please work through:
    C programming tutorial
  • Brian W. Kernighan, Dennis M. Ritchie: The C Programming Language (2nd edition)
    ISBN-13: 007-6092003106 ISBN-10: 0131103628
    Solution to exercises Examples
  • K.N. King: C Programming: A Modern Approach
    ISBN-13: 978-0393969450 ISBN-10: 0393969452
  • K. Reek: Pointers on C
    ISBN-13: 978-0673999863 ISBN-10: 0673999866
  • Z. A. Shaw: Learn C The Hard Way
    See especially Chapter 4 for the use of the debugger "Valgrind"

  • Algorithms and Implementations

  • W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, Third Edition (2007)
    ISBN-10: 0521880688 ISBN-13: 978-0521880688


Knowledge of quantum mechanics and statistical physics is required. Basic knowledge of quantum field theories is useful.