Special Topics in Complex Systems

Interactive demos for a graduate course in the Department of Physics, Gyeongsang National University. Eight browser simulations follow the course from percolation on trees, lattices and networks to the Ising model and the renormalization group.

Created by Claude Opus 5.5, based on the lecture notes by Prof. Sang Hoon Lee.

Block-spin renormalization of the 2D Ising model. Each row runs a Monte Carlo simulation on a 128 × 128 lattice, and each step to the right replaces every 2 × 2 block by the sign of its majority spin. Below Tc the coarse-grained configurations flow toward a uniform, ordered state; above Tc they flow toward uncorrelated noise. Only at Tc, where the correlation length diverges, do they look statistically the same at every scale.

From demo 8, The renormalization group for the Ising model.

Percolation

  1. 1

    Percolation on the Bethe lattice

    An exactly solvable case: on a tree with coordination number z, the percolation threshold, the order parameter and the mean cluster size follow from simple self-consistency equations.

    Open demo Source on GitHub
    pc = 1z − 1
  2. 2

    Percolation on the 2D square lattice

    Clusters on the square lattice near the threshold, the emergence of a spanning cluster, and critical exponents beyond mean-field theory.

    Open demo Source on GitHub
    P∞ ∝ (p − pc)β,  β = 536
  3. 3

    Percolation on networks

    Percolation on random networks with arbitrary degree distributions, the Molloy–Reed criterion, and the robustness of heterogeneous networks.

    Open demo Source on GitHub
    φc = ⟨k⟩⟨k2⟩ − ⟨k⟩

The Ising model

  1. 4

    The Ising model: introduction

    The model, its Hamiltonian and partition function, and why a system of interacting binary spins is the standard testing ground for phase transitions.

    Open demo Source on GitHub
    H = −J ∑⟨ij⟩ sisj − h ∑i si
  2. 5

    The 1D Ising model

    The transfer-matrix solution in one dimension, and why there is no ordered phase at any nonzero temperature.

    Open demo Source on GitHub
    ξ = −1ln tanh(βJ)
  3. 6

    Mean-field and Landau theory of the Ising model

    The self-consistent mean-field equation, the Landau free energy expanded in the order parameter, and the resulting mean-field exponents.

    Open demo Source on GitHub
    m = tanh[β(qJm + h)]
  4. 7

    The 2D Ising model

    Monte Carlo simulations on the square lattice, Onsager's exact critical temperature, and finite-size behavior near the transition.

    Open demo Source on GitHub
    kBTc/J = 2ln(1 + 2) ≈ 2.269
  5. 8

    The renormalization group for the Ising model

    Coarse-graining by block spins, the flow of couplings under repeated transformations, fixed points, and critical exponents from the linearized flow.

    Open demo Source on GitHub
    ν = ln bln λt

References

  • Kim Christensen and Nicholas R. Moloney, Complexity and Criticality (Imperial College Press, 2005).
  • Prof. Sang Hoon Lee, lecture notes for Special Topics in Complex Systems.