Spontaneous symmetry breaking and symmetry groups

A symmetry of the Hamiltonian (given by a group action) can fail to be a symmetry of an infinite-volume equilibrium state, producing multiple pure phases and nonzero order parameters.

Spontaneous symmetry breaking and symmetry groups

Symmetry group of a model

Let a group $G$ act on configurations $\sigma$ (spins, particle configurations, fields). A Hamiltonian $H$ is $G$ -invariant if

H(g\cdot \sigma)=H(\sigma)\quad\text{for all }g\in G.

Examples:

Ising model: $G=\mathbb{Z}_2$ acts by global spin flip $\sigma_i\mapsto -\sigma_i$ .
$O(n)$ models: $G=O(n)$ acts by rotating vector spins.

This invariance induces a corresponding symmetry of finite-volume Gibbs measures when boundary conditions and external fields are also symmetric; see finite-volume Gibbs measures .

Definition: spontaneous symmetry breaking (SSB)

Spontaneous symmetry breaking occurs when:

the Hamiltonian is $G$ -invariant (and there is no explicit symmetry-breaking field), but
there exists an infinite-volume equilibrium state (Gibbs measure) that is not invariant under $G$ .

Formally, let $\mu$ be an infinite-volume Gibbs measure (see infinite-volume Gibbs measures and DLR equation ). Then SSB means there exists $g\in G$ such that

\mu\circ g^{-1}\neq \mu.

Order parameters and symmetry breaking

SSB is detected by a symmetry-breaking order parameter (see order parameter ), i.e., an observable $O(\sigma)$ whose expectation changes under $G$ :

In an Ising ferromagnet, a canonical choice is the magnetization density, leading to spontaneous magnetization in the ordered phase.

A symmetric Gibbs state can often be expressed as a mixture of symmetry-broken extremal states (pure phases). This is one way to connect SSB to the structure of phase coexistence; see phase transitions via Gibbs measures .

Example: Ising $\mathbb{Z}_2$ symmetry below critical temperature

For the ferromagnetic Ising model at zero field ( $h=0$ ), the Hamiltonian is invariant under global spin flip. In dimensions $d\ge 2$ and at low temperature, there are distinct infinite-volume Gibbs states selected by boundary conditions:

a “plus” phase with positive magnetization,
a “minus” phase with negative magnetization.

These two states are exchanged by the $\mathbb{Z}_2$ action, so each breaks the symmetry even though the model is symmetric.

Continuous symmetries and the role of dimension

For continuous symmetry groups (e.g., $O(n)$ ), long-range order can be obstructed in low dimensions by fluctuations:

In many short-range lattice systems with continuous symmetry, SSB is ruled out in $d\le 2$ under suitable hypotheses; see Mermin–Wagner theorem and continuous symmetries on spins .

Connection to Landau theory

Landau theory encodes SSB via the symmetry of the Landau free energy:

If symmetry forces $f(m)$ to be even in $m$ at zero field, symmetry breaking occurs when the global minimizers become nonzero and come in group-related families; see Landau free-energy functional .