Spontaneous symmetry breaking occurs when the action and equations have a symmetry but a chosen ground state is invariant only under a proper subgroup. Acting with the broken symmetry generates a degenerate vacuum manifold.
The vacuum manifold is the space of field configurations that minimize the energy. If a group acts transitively and a chosen vacuum has stabilizer , its connected component is the homogeneous space .
In a relativistic quantum field theory, each spontaneously broken generator of a continuous internal symmetry produces a massless scalar excitation. For , the standard setting therefore has Goldstone bosons.
A nonlinear sigma model has fields valued in a curved target space such as a vacuum manifold . Its leading action is quadratic in spacetime derivatives but nonlinear in any unconstrained coordinates used for the target.
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Spontaneous symmetry breaking is a phenomenon that occurs in various fields of physics, particularly in condensed matter physics, particle physics, and cosmology. It describes a situation in which a system that is symmetric under some transformation settles into an asymmetrical state. Despite the underlying laws or equations being symmetric, the actual observed state of the system does not exhibit this symmetry.
