The Spin Model: Rules, Hamiltonian, and Thermal Noise

The Ising model is spare. It has one type of element (a spin), one interaction (nearest-neighbor coupling), one source of randomness (thermal noise), and one control parameter (temperature). This minimality is the point. Every result derived from the model --- spontaneous magnetization, the critical point, universality --- follows from this specification and nothing else.

The Lattice and the Spins

The model places a binary variable at each site of a regular lattice. The variable is called a “spin” by convention, taking values +1 (up) or -1 (down). In the standard two-dimensional case, the lattice is a square grid with L x L sites and periodic boundary conditions (the top row is adjacent to the bottom row; the left column is adjacent to the right column), forming a torus. Each site has exactly four nearest neighbors.

The one-dimensional case is a ring of N sites, each with two neighbors. The three-dimensional case is a cubic lattice with six neighbors per site. Higher dimensions are defined analogously.

The coupling constant J measures the interaction strength between neighboring spins. When J > 0 (the ferromagnetic case), aligned neighbors lower the energy. When J < 0 (the antiferromagnetic case), anti-aligned neighbors lower the energy. The ferromagnetic case is the one that produces the phase transition and is the focus of nearly all applications beyond condensed matter physics.

An external magnetic field h may be applied to all sites, favoring one spin direction. Most of the model’s interesting behavior occurs at h = 0, where the Hamiltonian is symmetric under global spin flip (replacing every +1 with -1 and vice versa).

The Hamiltonian

The total energy of a spin configuration is given by:

H = -J sum_{<i,j>} s_i s_j - h sum_i s_i

The first sum runs over all nearest-neighbor pairs (each pair counted once). When two neighboring spins are aligned (both +1 or both -1), their product is +1 and the energy contribution is -J (negative, lowering the total energy). When they are anti-aligned, the product is -1 and the contribution is +J (positive, raising the energy).

The ground state --- the configuration of minimum energy at h = 0 --- has all spins aligned: either all +1 or all -1. The ground state energy is -2dNJ for a d-dimensional lattice with N sites, where each site has 2d neighbors. There are exactly two ground states, related by global spin flip. This twofold degeneracy is the symmetry that spontaneous magnetization breaks.

The partition function Z = sum over all configurations of exp(-H/kT) weights each configuration by its Boltzmann factor. At low temperature (large beta = 1/kT), configurations near the ground state dominate. At high temperature (small beta), all 2^N configurations contribute roughly equally and the system is disordered.

Temperature and the Balance Between Order and Disorder

Temperature parameterizes the competition between two forces.

The coupling pulls spins into alignment. A spin surrounded by aligned neighbors would pay an energy cost of 8J (in 2D) to flip. At low temperature, this cost is large relative to kT, and flips are rare. The system remains in or near the ground state.

Thermal noise randomly disrupts alignment. At high temperature, the energy cost of a flip is small relative to kT, and flips are frequent. Any spin is nearly as likely to be up as down, regardless of its neighbors. The magnetization --- the average spin value M = (1/N) sum(s_i) --- is zero because the contributions of up and down spins cancel.

The reduced temperature T/T_c (or equivalently the ratio kT/J) is the single dimensionless parameter that controls the model’s behavior at h = 0. Below T_c, order dominates. Above T_c, disorder dominates. At T_c, the system is critical.

Monte Carlo Simulation: The Metropolis Algorithm

The Ising model at finite temperature is not deterministic --- it samples configurations from the Boltzmann distribution. The standard simulation method is the Metropolis algorithm, introduced by Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller in 1953:

  1. Select a spin at random.
  2. Compute the energy change delta_E from flipping that spin. For the 2D Ising model, delta_E = 2 J s_i (sum of four neighbors), which takes only five possible values: -8J, -4J, 0, 4J, 8J.
  3. If delta_E <= 0, accept the flip (it lowers or maintains the energy).
  4. If delta_E > 0, accept the flip with probability exp(-delta_E / kT). Generate a uniform random number r in [0,1]; accept if r < exp(-delta_E / kT).
  5. Repeat.

This algorithm satisfies detailed balance: the transition rate from configuration A to configuration B, weighted by the Boltzmann probability of A, equals the reverse rate weighted by the probability of B. In the long run, the algorithm samples configurations with frequency proportional to their Boltzmann weight, which is the correct equilibrium distribution.

The Glauber algorithm (heat bath dynamics) is an alternative: the flip probability is set to 1/(1 + exp(beta * delta_E)), which also satisfies detailed balance. For the Ising model the two algorithms produce the same equilibrium statistics but may differ in dynamical properties.

Thermalization. Starting from a random configuration, the system must run for many Monte Carlo steps before reaching equilibrium. The number of steps required (the equilibration time) depends on temperature and system size. Near T_c, thermalization is slow --- this is critical slowing down, where the autocorrelation time diverges as a power law in the system size. The Wolff cluster algorithm (1989) mitigates critical slowing down by flipping entire clusters of aligned spins rather than individual spins, but it is limited to models with the appropriate symmetry.

Finite-size effects. A finite system cannot exhibit a true phase transition --- the partition function is a finite sum of analytic terms and cannot produce the non-analytic behavior (divergent susceptibility, discontinuous magnetization) that defines a phase transition. The transition is rounded and shifted. Finite-size scaling theory (Fisher, 1972) provides a systematic framework for extracting infinite-system critical behavior from finite-system simulation data, by studying how observables scale with system size L near the apparent transition point.

Further Reading