For other uses see Symmetry breaking (disambiguation).
In physics, symmetry breaking is a phenomenon where a disordered but symmetric state collapses into an ordered, but less symmetric state.[1] This collapse is often one of many possible bifurcations that a particle can take as it approaches a lower energy state. Due to the many possibilities, an observer may assume the result of the collapse to be arbitrary. This phenomenon is fundamental to quantum field theory (QFT), and further, contemporary understandings of physics.[2] Specifically, it plays a central role in the Glashow–Weinberg–Salam model which forms part of the Standard model modelling the electroweak sector.In an infinite system (Minkowski spacetime) symmetry breaking occurs, however in a finite system (that is, any real super-condensed system), the system is less predictable, but in many cases quantum tunneling occurs.[3] Symmetry breaking and tunneling relate through the collapse of a particle into non-symmetric state as it seeks a lower energy.[4]
Symmetry breaking can be distinguished into two types, explicit and spontaneous. They are characterized by whether the equations of motion fail to be invariant, or the ground state fails to be invariant.
This section describes spontaneous symmetry breaking. This is the idea that for a physical system, the lowest energy configuration (the vacuum state) is not the most symmetric configuration of the system. Roughly speaking there are three types of symmetry that can be broken: discrete, continuous and gauge, ordered in increasing technicality.
An example of a system with discrete symmetry is given by the figure with the red graph: consider a particle moving on this graph, subject to gravity. A similar graph could be given by the function
f(x)=(x2-a2)2
x=0
x=\pma
x=\pma
An example with continuous symmetry is given by a 3d analogue of the previous example, from rotating the graph around an axis through the top of the hill, or equivalently given by the graph
f(x,y)=(x2+y2-a2)2
Gauge symmetry breaking is the most subtle, but has important physical consequences. Roughly speaking, for the purposes of this section a gauge symmetry is an assignment of systems with continuous symmetry to every point in spacetime. Gauge symmetry forbids mass generation for gauge fields, yet massive gauge fields (W and Z bosons) have been observed. Spontaneous symmetry breaking was developed to resolve this inconsistency. The idea is that in an early stage of the universe it was in a high energy state, analogous to the particle being at the top of the hill, and so had full gauge symmetry and all the gauge fields were massless. As it cooled, it settled into a choice of vacuum, thus spontaneously breaking the symmetry, thus removing the gauge symmetry and allowing mass generation of those gauge fields. A full explanation is highly technical: see electroweak interaction.
See main article: Spontaneous symmetry breaking. In spontaneous symmetry breaking (SSB), the equations of motion of the system are invariant, but any vacuum state (lowest energy state) is not.
For an example with two-fold symmetry, if there is some atom which has two vacuum states, occupying either one of these states breaks the two-fold symmetry. This act of selecting one of the states as the system reaches a lower energy is SSB. When this happens, the atom is no longer
Z2
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 (QFT), the Lagrangian
L
G
G
G
H
Within the context of gauge symmetry however, SSB is the phenomenon by which gauge fields 'acquire mass' despite gauge-invariance enforcing that such fields be massless. This is because the SSB of gauge symmetry breaks gauge-invariance, and such a break allows for the existence of massive gauge fields. This is an important exemption from Goldstone's Theorem, where a Nambu-Goldstone Boson can gain mass, becoming a Higgs Boson in the process.[5]
Further, 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.
Spontaneous symmetry breaking is also associated with phase transitions. For example in the Ising model, as the temperature of the system falls below the critical temperature the
Z2
See main article: Explicit symmetry breaking.
In explicit symmetry breaking (ESB), 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 often studied when the Hamiltonian can be written
H=H0+Hint
Here
H0
G
The
Hint
G
This is essentially the paradigm for perturbation theory in quantum mechanics. An example of its use is in finding the fine structure of atomic spectra.
Symmetry breaking can cover any of the following scenarios:
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[6] and soon later Liouville,[7] 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.