Mermin–Wagner theorem (no continuous symmetry breaking in d ≤ 2)

For short-range systems with a continuous symmetry, spontaneous symmetry breaking and conventional long-range order are impossible in one and two dimensions at positive temperature.

Mermin–Wagner theorem (no continuous symmetry breaking in d ≤ 2)

Context

In lattice statistical mechanics, a spontaneous symmetry breaking phase is detected by a nonzero order parameter (e.g. magnetization) or by multiple infinite-volume Gibbs states .

For models with a continuous symmetry (see continuous symmetry on spins ), low-dimensional fluctuations can destroy long-range order even when interactions are ferromagnetic and short-range.

Theorem (Mermin–Wagner; lattice spin systems, informal standard form)

Consider a translation-invariant, finite-range (or sufficiently fast decaying) lattice spin system in dimension $d\le 2$ whose Hamiltonian is invariant under a nontrivial continuous compact Lie group (e.g. $O(n)$ rotations with $n\ge 2$ ). Then at any positive temperature:

There is no spontaneous breaking of that continuous symmetry.
In particular, any symmetry-breaking magnetization order parameter vanishes: $\langle \mathbf{S}_0\rangle = 0$ in every translation-invariant Gibbs measure .

A common corollary is the absence of conventional ferromagnetic long-range order:

\lim_{|x|\to\infty}\langle \mathbf{S}_0\cdot \mathbf{S}_x\rangle = 0 \quad (d\le 2,\ T>0),

where the bracket denotes an ensemble average and the left-hand side is a two-point correlation function limit.

Why it matters (examples and contrasts)

2D XY / 2D Heisenberg: no nonzero magnetization at $T>0$ , but the 2D XY model can still have a Kosterlitz–Thouless transition with quasi-long-range order (power-law correlations) rather than true long-range order.
Discrete symmetry is different: the theorem does not apply to discrete symmetries like the $\mathbb{Z}_2$ symmetry of the 2D Ising model , which does exhibit spontaneous magnetization below $T_c$ (see Onsager’s solution ).

Technical conditions (typical hypotheses)

Precise statements depend on the setting, but usually assume:

finite-range or summable interactions (a condition on the lattice Hamiltonian ),
continuous symmetry acting nontrivially on the single-site spin space,
finite temperature Gibbs equilibrium described via the DLR equations .

Prerequisites and connections (cross-links)

Equilibrium notion: thermodynamic equilibrium , finite-volume Gibbs measures , DLR condition .
Order and correlations: order parameter , correlation length .