Spontaneously broken symmetry

Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or the Lagrangian obey symmetries, but the lowest-energy vacuum solutions do not exhibit that same symmetry. When the system goes to one of those vacuum solutions, the symmetry is broken for perturbations around that vacuum even though the entire Lagrangian retains that symmetry.

In explicit symmetry breaking, if we consider two outcomes, the probability of a pair of outcomes can be different. By definition, spontaneous symmetry breaking requires the existence of a symmetric probability distribution—any pair of outcomes has the same probability. In other words, the underlying laws are invariant under a symmetry transformation.

The system, as a whole, changes under such transformations.

Phases of matter, such as crystals, magnets, and conventional superconductors, as well as simple phase transitions can be described by spontaneous symmetry breaking. Notable exceptions include topological phases of matter like the fractional quantum Hall effect.

Consider a symmetric upward dome with a trough circling the bottom. If a ball is put at the very peak of the dome, the system is symmetric with respect to a rotation around the center axis. But the ball may spontaneously break this symmetry by rolling down the dome into the trough, a point of lowest energy. Afterward, the ball has come to a rest at some fixed point on the perimeter. The dome and the ball retain their individual symmetry, but the system does not.

In the simplest idealized relativistic model, the spontaneously broken symmetry is summarized through an illustrative scalar field theory. The relevant Lagrangian, which essentially dictates how a system behaves, can be split up into kinetic and potential terms,

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