# Spontaneous breaking of symmetry

Spontaneous symmetry breaking is a concept of theoretical physics that plays an important role in the standard model of elementary particle physics ( Higgs mechanism ). One speaks of spontaneous symmetry breaking when the ground state (the state of lowest energy) of a physical system has fewer symmetries than the underlying equations of motion .

Symmetries are important physical properties of a system as they can, for example, determine the fulfillment of conservation laws or the existence of elementary particles .

The concept also plays a role in solid state physics , where it originated. Cools e.g. For example, if a ferromagnet drops below the Curie temperature , a “spontaneous magnetization” will develop - even with an internal magnetic field of any weakness - oriented in its direction, which breaks the previously existing rotational symmetry . The direction of the spontaneous magnetization can be specified by the weak magnetic field , while the amount is independent of it.

## example

"Sombrero" potential ; the possible stable ground states lie on a circle around the axis of symmetry. (See also Higgspotential )

The picture illustrates a rotationally symmetrical potential. As a mechanical analogue, this potential can be thought of as a surface on which a ball rolls, for example the inwardly curved bottom of a bottle or a sombrero . In this case, the potential depends on two position coordinates and . In particle physics, a complex field (or two real fields and ) is considered instead of the position coordinates . A certain value corresponds to a field configuration from which predictions can be made for quantities to be observed in the experiment. For complex fields ${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle \ phi}$${\ displaystyle \ phi _ {1}}$${\ displaystyle \ phi _ {2}}$

${\ displaystyle \ phi (r, \ theta) = re ^ {\ mathrm {i} \ theta}}$

one can get such a potential through the equation

${\ displaystyle V (\ phi) = - u_ {0} | \ phi | ^ {2} + | \ phi | ^ {4} \,}$

describe, where in the figure was selected. ${\ displaystyle u_ {0} = 10}$

This potential has stable local minimum values ​​on a whole circle around the axis of symmetry, at the "bottom" of the potential. In the mechanical analogue, these are the places where a ball can rest. In particle physics, the minimum values ​​correspond to configurations that the field can assume in a vacuum; one therefore speaks of the "vacuum circle". The local minima and maxima of the field are where the first derivative of the potential (the gradient ) vanishes:

${\ displaystyle V '(\ phi) = \ left (-2 \, u_ {0} +4 | \ phi | ^ {2} \ right) | \ phi | \, {\ stackrel {!} {=}} \, 0 \ ,,}$

This is the case on the circle around the origin with the radius : ${\ displaystyle \ textstyle {\ sqrt {u_ {0} / 2}}}$

${\ displaystyle \ phi = {\ sqrt {\ frac {u_ {0}} {2}}} \, e ^ {\ mathrm {i} \ theta} \ Rightarrow | \ phi | = {\ sqrt {\ frac { u_ {0}} {2}}}}$.

In addition, there is an unstable steady state exactly in the center of the potential surface, at the top of the bulge. It is now assumed that the ball or the field is initially in this state. At the slightest disturbance, the ball will roll away from this point and finally - braked by friction - stop at a point on the minimum circle. Similarly, the field will spontaneously change from the unstable state (the so-called “false vacuum”) to a stable basic state on the vacuum circuit. However, the configuration has lost its rotational symmetry ("symmetry breaking"), because the system has decided on a certain position on the circle, which is therefore distinguished from all other positions on the circle.

## Consequences

The spontaneous breaking of global continuous symmetries results in the gold stone theorem . This means that for every broken generator of the symmetry group there is a massless scalar Goldstone boson.

The gold stone theorem does not apply to local gauge symmetric theories . Instead there is the phenomenon of the Higgs mechanism , in which the gauge fields interact with a scalar Higgs field. The Goldstone bosons of the spontaneous symmetry break in the Higgs field become an additional longitudinal degree of polarization freedom of the calibration fields, which thereby acquire a mass. Without spontaneous symmetry breaking, the calibration fields are massless and would have only two transverse polarizations ( left-handed modes with spin antiparallel to the direction of propagation and right-handed modes with spin parallel to the direction of propagation).

## Nobel Prizes

Yōichirō Nambu was awarded the 2008 Nobel Prize in Physics for his idea of ​​spontaneous symmetry breaking . For the Higgs mechanism, François Englert and Peter Higgs received the 2013 Nobel Prize in Physics.

## Significance for other branches of physics

The concept of spontaneous symmetry breaking also plays an essential role in other areas of physics, especially in statistical physics (e.g. critical behavior during phase transitions ), solid-state physics (e.g. in the theory of superconductivity ) and in the Particle physics (e.g. in the Higgs mechanism mentioned above).