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## Summary

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In physics, symmetry breaking is a phenomenon in which (infinitesimally) small fluctuations acting on a system crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observer unaware of the fluctuations (or "noise"), the choice will appear arbitrary. This process is called symmetry "breaking", because such transitions usually bring the system from a symmetric but disorderly state into one or more definite states. The phenomenon is part of most theories of everything. Symmetry breaking is thought to play a major role in pattern formation. A ball is initially located at the top of the central hill (C). This position is an unstable equilibrium: a very small perturbation will cause it to fall to one of the two stable wells left (L) or right (R). Even if the hill is symmetric and there is no reason for the ball to fall on either side, the observed final state is not symmetric.

In his 1972 Science paper titled "More is different" Nobel laureate P.W. Anderson used the idea of symmetry breaking to show that even if reductionism is true, its converse, constructionism, which is the idea that scientists can easily predict complex phenomena given theories describing their components, is not.

Symmetry breaking can be distinguished into two types, explicit symmetry breaking and spontaneous symmetry breaking, characterized by whether the equations of motion fail to be invariant or the ground state fails to be invariant.

## Explicit symmetry breaking

In explicit symmetry breaking, the equations of motion describing a system are variant under the broken symmetry. In Hamiltonian mechanics or Lagrangian Mechanics, this happens when there is at least one term in the Hamiltonian (or Lagrangian) that explicitly breaks the given symmetry.

In the Hamiltonian setting, this is most often studied when the Hamiltonian can be written $H=H_{0}+H_{\text{int}}$ .

Here $H_{0}$  is a 'base Hamiltonian', which has some manifest symmetry. More explicitly, it is symmetric under the action of a (Lie) group $G$ . Often this is an integrable Hamiltonian.

The $H_{\text{int}}$  is a perturbation or interaction Hamiltonian. This is not invariant under the action of $G$ . It is often proportional to a small, perturbative parameter.

## Spontaneous symmetry breaking

In spontaneous symmetry breaking, the equations of motion of the system are invariant, but the system is not. This is because the background (spacetime) of the system, its vacuum, is non-invariant. Such a symmetry breaking is parametrized by an order parameter. A special case of this type of symmetry breaking is dynamical symmetry breaking.

In the Lagrangian setting of quantum field theory, the Lagrangian $L$  is a functional of quantum fields which is invariant under the action of a symmetry group $G$ . However, the ground state configuration (the vacuum expectation value) of the fields may not be invariant under $G$ , but instead partially breaks the symmetry to a subgroup $H$  of $G$ . This is spontaneous symmetry breaking.

Outside of gauge symmetry, spontaneous symmetry breaking is associated with phase transitions. For example in the Ising model, as the temperature of the system falls below the critical temperature the $\mathbb {Z} _{2}$  symmetry of the vacuum is broken, giving a phase transition of the system.

Within the context of gauge symmetry, spontaneous symmetry breaking is the mechanism by which gauge fields can 'acquire a mass' despite gauge-invariance enforcing that such fields be massless. This is because spontaneous symmetry breaking of gauge symmetry breaks the gauge-invariance, allowing the gauge fields to be massive. Also, in this context the usage of 'symmetry breaking', while standard, is a misnomer, as gauge 'symmetry' is not really a symmetry but a redundancy in the description of the system. Mathematically, this redundancy is a choice of trivialization, somewhat analogous to redundancy arising from a choice of basis.

## Examples

Symmetry breaking can cover any of the following scenarios:

• The breaking of an exact symmetry of the underlying laws of physics by the apparently random formation of some structure;
• A situation in physics in which a minimal energy state has less symmetry than the system itself;
• Situations where the actual state of the system does not reflect the underlying symmetries of the dynamics because the manifestly symmetric state is unstable (stability is gained at the cost of local asymmetry);
• Situations where the equations of a theory may have certain symmetries, though their solutions may not (the symmetries are "hidden").

One of the first cases of broken symmetry discussed in the physics literature is related to the form taken by a uniformly rotating body of incompressible fluid in gravitational and hydrostatic equilibrium. Jacobi and soon later Liouville, in 1834, discussed the fact that a tri-axial ellipsoid was an equilibrium solution for this problem when the kinetic energy compared to the gravitational energy of the rotating body exceeded a certain critical value. The axial symmetry presented by the McLaurin spheroids is broken at this bifurcation point. Furthermore, above this bifurcation point, and for constant angular momentum, the solutions that minimize the kinetic energy are the non-axially symmetric Jacobi ellipsoids instead of the Maclaurin spheroids.