Symmetry breaking

Symmetry breaking occurs when a system’s state or governing conditions have less symmetry than the laws or conditions that permit it. It can be spontaneous, explicit, or induced by an external influence.

Spontaneous symmetry breaking: A system settles into a state that lacks a symmetry preserved by its governing laws. It is the central case: the laws remain symmetric while the realized state does not.
Symmetry (physics): An invariance of a physical system or its laws under a specified transformation. Breaking is defined by comparing a state or condition with a symmetry of its laws.
Ising model: A mathematical model of interacting spins that can point in one of two directions on a lattice. Below its critical temperature, the model can develop a magnetization that breaks spin-flip symmetry.
Symmetry restoration: A change in which a symmetry absent from a system’s state or effective description becomes manifest again. It describes the reverse pattern, such as magnetization disappearing above a critical temperature.
Order parameter: A quantity that distinguishes phases or states, often taking a nonzero value when a symmetry is broken. Its value can make the transition from a symmetric state to a broken-symmetry state measurable.
Group theory: The mathematical study of groups, structures that describe transformations and their composition. Symmetries are organized into groups, making the lost transformations precise.
Curie temperature: The temperature above which a ferromagnetic material loses its spontaneous magnetization. Crossing it illustrates the loss of an ordered, symmetry-broken magnetic phase.
Symmetry protection: The persistence of a property or degeneracy because symmetry forbids perturbations that would remove it. It shows that symmetry can stabilize features rather than be broken to produce them.
Vacuum expectation value: The expectation value of a quantum field or operator in the vacuum state. A nonzero value can select a vacuum that does not share the theory’s symmetry.
Phase transition: A change between distinct phases of matter as external conditions vary. Many symmetry-breaking changes occur at phase transitions, though the ideas are not identical.

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