Continuous symmetries in spin systems

Spin models with global continuous symmetry (e.g., O(n), U(1)) and the consequences for phases, correlations, and symmetry breaking.

Continuous symmetries in spin systems

A lattice spin system has a continuous symmetry when its Hamiltonian is invariant under a continuous group of transformations acting on all spins simultaneously. This changes what kinds of ordered phases can exist and how correlations behave, especially in low spatial dimensions.

Definition (global continuous symmetry)

Let $\Lambda$ be a lattice and spins $\sigma_i$ take values in a continuous manifold (e.g., $\sigma_i \in S^{n-1}\subset \mathbb{R}^n$ ). A typical nearest-neighbor Hamiltonian is

H(\sigma) = -\sum_{\langle i,j\rangle} J_{ij}\, \sigma_i\cdot \sigma_j - \sum_{i\in\Lambda} h\cdot \sigma_i .

For $h=0$ , this Hamiltonian is invariant under the action of the group $G=O(n)$ :

H(g\sigma) = H(\sigma)\quad\text{for all }g\in O(n),

where $(g\sigma)_i = g\,\sigma_i$ .
Special cases:

XY model: $n=2$ , symmetry group includes $SO(2)\cong U(1)$ (continuous rotations in the plane).
Heisenberg model: $n=3$ , symmetry group includes $SO(3)$ .

This is the natural setting for spontaneous symmetry breaking in models with non-discrete symmetries.

Consequences for order and correlations

Order parameter and symmetry

A natural order parameter is the magnetization

m(\sigma)=\frac{1}{|\Lambda|}\sum_{i\in\Lambda}\sigma_i.

In a symmetry-broken phase at zero field, typical infinite-volume states have nonzero $\langle m\rangle$ but the full Gibbs measure may be a mixture of rotated pure states.

Goldstone modes (heuristic)

If a continuous symmetry is broken, slowly varying rotations of the local orientation cost little energy, producing long-wavelength modes and strong fluctuations. These fluctuations strongly constrain long-range order in low dimensions.

Low-dimensional obstruction (key theorem)

For short-range, sufficiently regular interactions and continuous symmetries, the Mermin–Wagner theorem rules out spontaneous breaking of such symmetries in $d\le 2$ (at positive temperature). A common signature is that the two-point correlation function fails to approach a nonzero constant as $|i-j|\to\infty$ .

Quasi-long-range order and topological defects

In the 2D XY model, one can have power-law decay of correlations (no true long-range order) and a transition driven by unbinding of vortices; see Kosterlitz–Thouless transition and vortices .