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Spontaneous symmetry breaking is a spontaneous process by which a system in a symmetrical state ends up in an asymmetrical state.

Consider the bottom of an empty wine bottle, a symmetrical upward dome with a gutter for sediment. If a ball is placed at the peak of the dome, the situation is symmetrical. But the ball may spontaneously break this symmetry and roll into the gutter, a point of lowest energy. The bottle and the ball retain their symmetry, but the system does not.

Spontaneous symmetry breaking in physics
Particle physics

In particle physics the force carrier particles are defined by field equations with gauge symmetry. These equations predict that certain measurements will be the same at any point in the field. For instance, they may predict that the mass of two quarks is constant. Solving the equations to find the mass of each quark might give two solutions. In one solution, quark A is heavier than quark B. In the second solution, quark B is heavier than quark A by the same amount. The symmetry of the equations is not reflected by the individual solutions, but it is reflected by the range of solutions. An actual measurement reflects only one solution, representing a breakdown in the symmetry of the underlying theory. "Hidden" is perhaps a better term than "broken" because the symmetry is always there in the equations. This phenomenon is called spontaneous symmetry breaking because nothing (that we know) breaks the symmetry in the equations.
Higgs mechanism

The W and Z bosons are the elementary particles that mediate the weak interaction, while the photon is the particle that mediates the electromagnetic interaction. At energies much greater than 100 GeV all these particles behave in a similar manner. The Weinberg–Salam theory predicts that at lower energies this symmetry is broken and the photon and the W and Z bosons emerge.

Without spontaneous symmetry breaking, the Standard Model of elementary particle interactions predicts the existence of a number of particles. However, some particles (the W and Z bosons) would then be predicted to be massless, when, in reality, they are observed to have mass. To overcome this, spontaneous symmetry breaking in conjunction with the Higgs mechanism gives these particles mass. It also suggests the presence of a new, as yet undetected particle, the Higgs boson.

If the Higgs boson isn't found, it will mean that the simplest implementation of the Higgs mechanism and spontaneous symmetry breaking as they are currently formulated is invalid.

A detailed presentation of the Higgs mechanism is given in the article on the Yukawa interaction, illustrating how it further gives mass to fermions.
Chiral symmetry

Chiral symmetry breaking is an example of spontaneous symmetry breaking affecting the chiral symmetry of the strong interactions in particle physics. It is a property of quantum chromodynamics, the quantum field theory describing these interactions, and is responsible for the bulk of the mass (over 99%) of the nucleons, and thus of all common matter, as it converts very light bound quarks into heavy baryons. The approximate Nambu–Goldstone bosons in this spontaneous symmetry breaking process are the pions, whose mass is an order of magnitude lighter than the mass of the nucleons. It served as the prototype and significant ingredient of the Higgs mechanism underlying the electroweak symmetry breaking outlined above.
Condensed matter physics

Spontaneous symmetry breaking is found in all interesting long range order systems in condensed matter physics, such as magnetization.
Generalisation and technical usage

For spontaneous symmetry breaking to occur, there must be a system in which there are several equally likely outcomes. The system as a whole is therefore symmetric with respect to these outcomes (If we consider any two outcomes, the probability is the same. This contrasts sharply to Explicit symmetry breaking. ). However, if the system is sampled (i.e. if the system is actually used or interacted with in any way), a specific outcome must occur. Though the system as a whole is symmetric, it is never encountered with this symmetry, but only in one specific asymmetric state. Hence, the symmetry is said to be spontaneously broken in that theory. Nevertheless, the fact that each outcome is equally likely is a reflection of the underlying symmetry, which is thus often dubbed "hidden symmetry", and has crucial formal consequences. (See the article on the Goldstone boson).

When a theory is symmetric with respect to a symmetry group, but requires that one element of the group be distinct, then spontaneous symmetry breaking has occurred. The theory must not dictate which member is distinct, only that one is. From this point on, the theory can be treated as if this element actually is distinct, with the proviso that any results found in this way must be resymmetrized, by taking the average of each of the elements of the group being the distinct one.

The crucial concept in physics theories is the order parameter. If there is a field (often a background field) which acquires an expectation value (not necessarily a vacuum expectation value) which is not invariant under the symmetry in question, we say that the system is in the ordered phase, and the symmetry is spontaneously broken. This is because other subsystems interact with the order parameter which forms a "frame of reference" to be measured against, so to speak. In that case, the vacuum state does not obey the initial symmetry (which would put it in the Wigner mode), and, instead has the (hidden) symmetry implemented in the Nambu–Goldstone mode. Normally, in the absence of the Higgs mechanism, massless Goldstone bosons arise.

The symmetry group can be discrete, such as the space group of a crystal, or continuous (e.g., a Lie group), such as the rotational symmetry of space. However, if the system contains only a single spatial dimension, then only discrete symmetries may be broken in a vacuum state of the full quantum theory, although a classical solution may break a continuous symmetry.
A pedagogical example: the Mexican hat potential
Graph of a Mexican hat potential function.

In the simplest idealized relativistic model, the spontaneously broken field is described through a scalar field theory. In physics, one way of seeing spontaneous symmetry breaking is through the use of Lagrangians. Lagrangians, which essentially dictate how a system behaves, can be split up into kinetic and potential terms,

(1) $$\qquad \mathcal{L} = \partial^\mu \phi \partial_\mu \phi - V(\phi).$$

It is in this potential term (V(Φ)) that the symmetry breaking occurs. An example of a potential is illustrated in the graph at the right.

(2) $$\qquad V(\phi) = -10|\phi|^2 + |\phi|^4 \,$$

This potential has an infinite number of possible minima (vacuum states) given by

(3) $$\qquad \phi = \sqrt{5} e^{i\theta}$$

for any real θ between 0 and 2π. The system also has an unstable vacuum state corresponding to Φ = 0. This state has a U(1) symmetry. However, once the system falls into a specific stable vacuum state (amounting to a choice of θ), this symmetry will appear to be lost, or "spontaneously broken".

In fact, any other choice of θ would have exactly the same energy, implying the existence of a massless Nambu–Goldstone boson, the mode running around the circle at the minimum of this potential, and indicating there is some memory of the original symmetry in the Lagrangian.
Other examples

For ferromagnetic materials, the underlying laws are invariant under spatial rotations. Here, the order parameter is the magnetization, which measures the magnetic dipole density. Above the Curie temperature, the order parameter is zero, which is spatially invariant, and there is no symmetry breaking. Below the Curie temperature, however, the magnetization acquires a constant nonvanishing value, which points in a certain direction (in the idealized situation where we have full equilibrium; otherwise, translational symmetry gets broken as well). The residual rotational symmetries which leave the orientation of this vector invariant remain unbroken, unlike the other rotations which do not and are thus spontaneously broken.
The laws describing a solid are invariant under the full Euclidean group, but the solid itself spontaneously breaks this group down to a space group. The displacement and the orientation are the order parameters.
General relativity has a Lorenz symmetry, but in FRW cosmological models, the mean 4-velocity field defined by averaging over the velocities of the galaxies (the galaxies act like gas particles at cosmological scales) acts as an order parameter breaking this symmetry. Similar comments can be made about the cosmic microwave background.
For the electroweak model, as explained earlier, a component of the Higgs field provides the order parameter breaking the electroweak gauge symmetry to the electromagnetic gauge symmetry. Like the ferromagnetic example, there is a phase transition at the electroweak temperature. The same comment about us not tending to notice broken symmetries suggests why it took so long for us to discover electroweak unification.
In superconductors, there is a condensed-matter collective field ψ, which acts as the order parameter breaking the electromagnetic gauge symmetry.
Take a thin cylindrical plastic rod and push both ends together. Before buckling, the system is symmetric under rotation, and so visibly cylindrically symmetric. But after buckling, it looks different, and asymmetric. Nevertheless, features of the cylindrical symmetry are still there: ignoring friction, it would take no force to freely spin the rod around, displacing the ground state in time, and amounting to an oscillation of vanishing frequency, unlike the radial oscillations in the direction of the buckle. This spinning mode is effectively the requisite Nambu–Goldstone boson.
Consider a uniform layer of fluid over an infinite horizontal plane. This system has all the symmetries of the Euclidean plane. But now heat the bottom surface uniformly so that it becomes much hotter than the upper surface. When the temperature gradient becomes large enough, convection cells will form, breaking the Euclidean symmetry.
Consider a bead on a circular hoop that is rotated about a vertical diameter. As the rotational velocity is increased gradually from rest, the bead will initially stay at its initial equilibrium point at the bottom of the hoop (intuitively stable, lowest gravitational potential). At a certain critical rotational velocity, this point will become unstable and the bead will jump to one of two other newly created equilibria, equidistant from the center. Initially, the system is symmetric with respect to the diameter, yet after passing the critical velocity, the bead ends up in one of the two new equilibrium points, thus breaking the symmetry.

Nobel Prize

On October 7, 2008, the Royal Swedish Academy of Sciences awarded the 2008 Nobel Prize in Physics to three scientists for their work in subatomic physics symmetry breaking. Yoichiro Nambu, 87, of the University of Chicago, won half of the prize for the discovery of the mechanism of spontaneous broken symmetry in the context of the strong interactions, specifically chiral symmetry breaking. Physicists Makoto Kobayashi and Toshihide Maskawa shared the other half of the prize for discovering the origin of the explicit breaking of CP symmetry in the weak interactions. This origin is ultimately reliant on the Higgs mechanism, but, so far understood as a "just so" feature of Higgs couplings, not a spontaneously broken symmetry phenomenon.

Autocatalytic reactions and order creation
Catastrophe theory
Chiral symmetry breaking
CP-violation
Dynamical symmetry breaking
Explicit symmetry breaking
Gauge gravitation theory
Goldstone boson
Grand unified theory
Magnetic catalysis of chiral symmetry breaking
Mermin-Wagner theorem
Second-order phase transition
Symmetry breaking
Tachyonic field
Tachyon condensation
Vacuum fluctuation
J. J. Sakurai Prize for Theoretical Particle Physics
Higgs mechanism
Higgs boson
1964 PRL symmetry breaking papers

Notes

^ Gerald M. Edelman, Bright Air, Brilliant Fire: On the Matter of the Mind (New York: BasicBooks, 1992) 203
^ Stephen Weinberg, Dreams of a Final Theory, Hutchinson, 1993, p.153-160
^ A Brief History of Time, Stephen Hawking, Bantam; 10th anniversary edition (September 1, 1998)
^ The Nobel Foundation. "The Nobel Prize in Physics 2008". nobelprize.org. Retrieved January 15, 2008.

Spontaneous symmetry breaking
Physical Review Letters - 50th Anniversary Milestone Papers
In CERN Courier, Steven Weinberg reflects on spontaneous symmetry breaking
Englert-Brout-Higgs-Guralnik-Hagen-Kibble Mechanism on Scholarpedia
History of Englert-Brout-Higgs-Guralnik-Hagen-Kibble Mechanism on Scholarpedia
The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles
International Journal of Modern Physics A: The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles
Guralnik, G S; Hagen, C R and Kibble, T W B (1967). Broken Symmetries and the Goldstone Theorem. Advances in Physics, vol. 2 Interscience Publishers, New York. pp. 567-708 ISBN 0-470-17057-3
Spontaneous Symmetry Breaking in Gauge Theories: a Historical Survey