Spontaneous symmetry breaking (SSB) is an important phenomenon in quantum physics, appearing in many areas.

There are two ways in which a symmetry which is present in a model on the level of the Lagrangean will fail to manifest itself in the quantum states. One is quantum anomaly, and the second is SSB. However, whereas quantum anomaly destroys also the conservation laws associated with the symmetry via Noether's theorem (at least on the local level), spontaneously broken symmetries still correspond to a conserved local current.

SSB can apply both to global and local (gauge) symmetry, and both to continuous and discrete symmetry.

In case of a global symmetry group G, SSB occurs when the groundstate of model |0> fails to be G-invariant. In general, it would be invariant under a certain subgroup H of G. As G is a symmetry group, for any element g of G, g|0> is also a groundstate of the model. Therefore, we necessarily obtain multiple groundstates. Due to the cluster decomposition principle, those different groundstate will necessarily lie in different superselection sectors. Therefore, any element g of G that doesn't also belong to H transforms the states of one superselection sector into the states of another. As a result, when we limit our view to a single superselction sector (what in reality we have to do), the symmetry transformation vanishes.

One might ask in what kind circumstances would |0> fail to be G-invariant. In fact, it is easy to produce examples of this kind. The simplest thing is a scalar field phi_{n} with a potential V(phi_{n}) s.t. V is G invariant however its minima aren't. The classical example of this sort is the so-called Mexican hat potential

V(phi_{n}) = (SUM_{n}phi_{n}^{2} - a^{2})^{2}

where a is some real constant. If phi_{n} is an
N-dimensional vector (i.e. the index n takes the values 1..N), V is invariant under the full N-dimensional rotation group SO(N) (and even O(N), but we may ignore this for our purpose), however, its minima (the points satisfying

SUM_{n}phi_{n}^{2} = a^{2}

are only symmetric under an SO(N-1) subgroup.

Of course, in reality what matters is the quantum potential, rather than potential appearing in the Lagrangean. Therefore, quantizations may results in destroying or creating SSB. The later case is sometimes called "radiative symmetry breaking". Note that quantization may lead to SSB via other mechanisms also, such as the appearance of an unsymmetric quadratic fermion vacuum condensate (in case there are fermions in the model). This is usually called "dynamical symmetry breaking". At any rate, it is comparatively easy to prove that in our example one would get SSB for a sufficiently high value of a.

An important phenomenon associated with the SSB of global continuous symmetry is the appearance of so-called Goldstone bosons. Those are the massless particles corresponding to excitations of the field along the directions tangent to the vacuum manifold. The later is the set of groundstates resulting from SSB, having the geometry of the quotient space G/H. For the Mexican hat example, G/H is the (N-1)-sphere.

SSB of local (gauge) symmetry doesn't lead to multiple vacua as gauge symmetry doesn't act on the physical states anyway. However, it may still happen unsymmetric vacuum expectation values appear in gauge theory (to produce an example, just couple a "Mexican hat" type field to a gauge field). In such cases we can fix the gauge symmetry by demanding the field that develops the expectation values to choose everywhere the same point on the "vacuum manifold" G/H. In such a model, no Goldstone bosons appear (as we just "gauged away" the excitations which produced them before), however, the gauge boson (which otherwise has to be massless) gains mass! This may be demonstrated by substituting the vacuum expectation values into the Lagrangean: the result is a mass-term for the gauge field.

It's important to emphasize that although I used the language of quantum field theory most way through this write-up, the concepts described apply in exactly the same way to models of the kind found in solid state physics. A few real-life examples of SSB would be:

- The chiral symmetry of the quark field in quantum chromodynamics. It is the reason chiral symmetry is not manifest in the particle multiplets, although the chiral currents are still conserved.

- The rotational symmetry of space in the Heisenberg (XXX) model of ferromagnets. It is the reason spontaneous magentization appears within each domain, and therefore also the reason for the hysteresis loop.

- The electromagnetic U(1) gauge symmetry in superconductors. There the Cooper-pairs result in the appearance of the vacuum expectation values and all the special properties of superconductors (such as zero resistance and magnetic flux quantization) can be traced back to the SSB.

- The breaking of the electroweak SU(2) X U(1) group down to the U(1) group of electromagnetism (in our notations, G = SU(2) X U(1), H = U(1)) in the Glashow-Weinberg-Salam model.