The Heisenberg Model & Classical Spin Systems

6.1 Classical Spin Models

The Ising model restricts spins to two states. Real magnetic materials often have continuous spin degrees of freedom. Classical spin models capture this:

These are the \(O(n)\) models, characterised by the symmetry group of the spin space.

6.2 The Classical Heisenberg Model

Hamiltonian

$$H = -J \sum_{\langle i,j \rangle} \mathbf{S}_i \cdot \mathbf{S}_j - \mathbf{h} \cdot \sum_i \mathbf{S}_i, \quad |\mathbf{S}_i| = 1. \tag{6.1}$$

Each spin is a unit vector on \(S^2\). For \(J > 0\) (ferromagnetic) in 3D, there is a phase transition at finite \(T_c\). In 1D and 2D, the Mermin-Wagner theorem forbids continuous symmetry breaking at any \(T > 0\).

Note

Mermin-Wagner Theorem In a 2D system with a continuous symmetry (\(n \geq 2\)) and short-range interactions, there is no spontaneous symmetry breaking at any finite temperature. Long-range order is destroyed by spin-wave fluctuations. The 2D Ising model (discrete symmetry) is exempt.

Physical Examples

Note

MnO and FeF₂ MnO has a rock-salt structure with Mn²⁺ ions carrying spin \(S = 5/2\). Below the Néel temperature \(T_N = 122\) K, spins order antiferromagnetically (\(J < 0\)) in (111) planes — well described by the classical Heisenberg model. FeF₂ is a nearly ideal Ising antiferromagnet (\(T_N = 78\) K) due to strong single-ion anisotropy.

Note

Spin-Ice Materials: Dy₂Ti₂O₇ In spin-ice, magnetic moments sit on a pyrochlore lattice and are constrained to point along local ⟨111⟩ axes. The ice rule (2-in, 2-out per tetrahedron) maps to the proton disorder in water ice. MC simulations with dipolar interactions reproduce the broad specific heat anomaly, diffuse neutron scattering, and emergent magnetic monopole excitations.

6.3 MC Updates for Continuous Spins

Random Rotation on \(S^2\)

A Metropolis update must propose \(\mathbf{S}’\) from the surface of the unit sphere. Do not sample angles uniformly — this oversamples the poles.

Correct uniform sampling on \(S^2\): $$\cos\theta \sim \mathcal{U}[-1,1], \quad \phi \sim \mathcal{U}[0, 2\pi]. \tag{6.2}$$

In practice, propose a small rotation:

$$\mathbf{S}’_i = \mathbf{S}_i + \delta \mathbf{v}, \quad \text{then normalise}, \tag{6.3}$$

where \(\delta \mathbf{v}\) is drawn from a small cube. Tune \(\delta\) for 40–50% acceptance.

The energy change: $$\Delta H = -J (\mathbf{S}’i - \mathbf{S}i) \cdot \mathbf{h}{\rm eff}, \quad \mathbf{h}{\rm eff} = \sum_{j \in \partial i} \mathbf{S}_j + \mathbf{h}/J. \tag{6.4}$$

Over-Relaxation

The over-relaxation (or microcanonical) step reflects the spin about its local field direction:

$$\mathbf{S}’i = 2(\hat{\mathbf{h}}{\rm eff} \cdot \mathbf{S}i)\hat{\mathbf{h}}{\rm eff} - \mathbf{S}_i. \tag{6.5}$$

This is always accepted (\(\Delta H = 0\)) and moves the spin to the “opposite side” of its local field. Combined with Metropolis steps in ratio \(\sim 4:1\) (over-relaxation: Metropolis), it dramatically reduces autocorrelation time while maintaining ergodicity.

6.4 The XY Model and the Kosterlitz-Thouless Transition

The 2D XY model has spins \(\mathbf{S}_i = (\cos\theta_i, \sin\theta_i)\):

$$H = -J \sum_{\langle i,j \rangle} \cos(\theta_i - \theta_j). \tag{6.6}$$

Despite Mermin-Wagner (no long-range order), the 2D XY model has a remarkable topological phase transition at \(T_{\rm KT} = \pi J / 2\) (Kosterlitz-Thouless 1973, Nobel Prize 2016).

Vortices

A vortex is a topological defect where the spin winds by \(2\pi\) around a loop: $$\oint \nabla \theta \cdot d\mathbf{l} = 2\pi n, \quad n \in \mathbb{Z}. \tag{6.7}$$

Below \(T_{\rm KT}\): vortices bound in pairs (vanishing net topological charge). Algebraic long-range order.

Above \(T_{\rm KT}\): free vortices proliferate, destroying quasi-long-range order. The transition is driven by vortex unbinding.

Note

Superfluid Helium Films The 2D superfluid transition of \(^4\)He adsorbed on a substrate (e.g., graphite) belongs to the XY universality class. The superfluid stiffness \(\rho_s\) jumps discontinuously at \(T_{\rm KT}\) — the “universal jump” predicted by Nelson and Kosterlitz and confirmed by Rudnick (1978). MC simulations of the 2D XY model reproduce this jump.

Measuring the KT Transition

The helicity modulus (spin-wave stiffness): $$\Upsilon = J\left[\langle \cos(\theta_i - \theta_j) \rangle - \frac{J}{T} \langle \sin^2(\theta_i - \theta_j) \rangle\right] \tag{6.8}$$

(averaged over horizontal bonds). The universal jump condition: \(\Upsilon(T_{\rm KT}) = 2T_{\rm KT}/\pi\). This gives a sharp MC signature of \(T_{\rm KT}\).

6.5 Frustrated Spin Systems

When competing interactions cannot all be simultaneously satisfied, the system is frustrated.

Triangular Antiferromagnet

For \(J < 0\) on a triangular lattice, not all pairs of neighbours can be antiparallel. The ground state is the 120° structure; entropy remains finite at \(T = 0\). MC reveals a continuous spectrum of degenerate ground states.

Heisenberg Antiferromagnet on the Kagome Lattice

Extreme frustration — the classical ground-state degeneracy is exponentially large. The system remains disordered (a classical spin liquid) down to \(T = 0\). MC simulations show no peak in \(C_V\), a signature of the absence of order.

Note

Spin Ice: Emergent Monopoles In Dy₂Ti₂O₇, the “2-in, 2-out” ice rule is violated at finite \(T\) by topological excitations that act as magnetic monopoles (Castelnovo, Moessner & Sondhi 2008). MC simulations with dipolar interactions reproduce the monopole density, their diffusion constant, and the anomalous low-\(T\) specific heat.

6.6 Spin-Spin Correlation Functions

The spin-spin correlation function:

$$G(\mathbf{r}) = \langle \mathbf{S}0 \cdot \mathbf{S}{\mathbf{r}} \rangle \tag{6.9}$$

characterises spatial ordering. In the ordered phase: \(G(r) \to M^2\) as \(r \to \infty\). At \(T_c\): \(G(r) \sim r^{-(d-2+\eta)}\) with anomalous exponent \(\eta\).

The correlation length \(\xi\) is extracted from the exponential decay in the disordered phase:

$$G(r) \sim e^{-r/\xi} \quad (T > T_c). \tag{6.10}$$

6.7 Computational Lab — Class 11

Objectives:

1. Simulate the 3D classical Heisenberg model on an \(L = 16\) cubic lattice. Using Metropolis + over-relaxation, compute \(\langle |\mathbf{M}| \rangle\) and \(\chi\) as a function of \(T\). Compare \(T_c\) to the literature value \(T_c \approx 1.443 J/k_B\).

2. Simulate the 2D XY model on an \(L = 64\) lattice. Compute the helicity modulus \(\Upsilon(T)\) and locate \(T_{\rm KT}\). Compare to the theoretical prediction \(T_{\rm KT} = \pi J / 2 \approx 1.571 J/k_B\).

3. Visualise vortices in the 2D XY model above and below \(T_{\rm KT}\) by plotting the local winding number on each plaquette. Count free vortices as a function of \(T\).

Summary

• The classical Heisenberg model (continuous spins on \(S^2\)) requires correct uniform sampling of \(S^2\).
• Over-relaxation dramatically reduces \(\tau_{\rm int}\) at low cost.
• The 2D XY model: KT transition at \(T_{\rm KT} = \pi J/2\), driven by vortex unbinding.
• Frustration (triangular AF, kagomé, spin ice) leads to macroscopic ground-state degeneracy and exotic physics.
• Spin-spin correlation functions and helicity modulus are key MC observables.