Physics of Critical Points

Tomasz Wydro, The Statistical Physics Group, IJL, University of Lorraine, France.

Lectures 30 hours, lab 15 hours, consultation 15 hours

About the lectures

The behavior of critical systems was gradually understood in the second half of the XXth century. The history of this research and its methods, which were based mainly on the particular symmetries of the critical systems, are the exciting subject of this series of lectures.

Program of lectures

  • What are critical phenomena and why do we study them?
    • The first observations of phenomena associated with density fluctuations.
    • Marian Smoluchowski: why is the sky blue?
    • Critical opalescence.
    • Short trip into the land of Brownian motions.
  • Phase transitions, critical points and critical lines.
    • Examples of phase transitions: gas-liquid, ferromagnetic-paramagnetic, liquid crystalline phases, percolation, etc.
    • Critical systems as “undecided” systems: continuous phase transitions.
    • Description of a system at a critical point: order parameter, susceptibility, critical exponents, scaling.
  • Microscopic models.
    • A simplified description of the microscopic properties of the system: short range interactions, small number of internal degrees of freedom, simplified form of interaction.
    • Examples of lattice models: lattice gas, Ising and Potts models, n-vector model, percolation, polymers, etc.
  • Methods
    • Mean-field approach, high temperature series, transfer matrix, finite size scaling, exact solutions, Monte Carlo simulations, Metropolis, Wolff and non-local COP Monte Carlo algorithms.
  • Role of symmetry in the study of phase transitions.
    • Dilatation symmetry of critical systems. Renormalization group: an outline of the method. Relevant, irrelevant and marginal parameters. Universality.
    • Example: real space renormalization group for Ising model on the triangular lattice.
    • Hamiltonian limit. Quantum chain for the two-dimensional Ising model.
  • Exact solution of the two-dimensional Ising model.
    • Introduction to Grassmann variables.
    • Ising partition function in Grassmann variables.
    • Mirror ordering of operators.
    • Elimination of classical Ising spins and Gaussian integration over the Grassmann variables.
  • Polymers
    • Random walks.
    • Self-avoiding walks.
    • Mapping onto magnetic systems: the n=0 theorem for the n-vector model.
  • Beyond equilibrium phase transitions.
    • Dynamical systems, dynamical phase transitions.
    • Master equations.
    • Generating functions.
    • Example: irreversible aggregation, Smoluchowski constant kernel equation, product kernel, gelation.


The general goal of the lab work was to illustrate the behavior of critical system by Monte Carlo simulations of 1D and 2D Ising model close to criticality.

The C++ code written in the frame of the Lab is accessible in Google Code repository.