Spontaneous symmetry breaking is a spontaneous
process of symmetry breaking, by which a physical
system in a symmetric state ends up in an
asymmetric state.
In particular, it can describe systems where
the equations of motion or the Lagrangian
obey symmetries, but the lowest-energy vacuum
solutions do not exhibit that same symmetry.
When the system goes to one of those vacuum
solutions, the symmetry is broken for perturbations
around that vacuum even though the entire
Lagrangian retains that symmetry.
== Overview ==
In explicit symmetry breaking, if we consider
two outcomes, the probability of a pair of
outcomes can be different.
By definition, spontaneous symmetry breaking
requires the existence of a symmetric probability
distribution—any pair of outcomes has the
same probability.
In other words, the underlying laws are invariant
under a symmetry transformation.
The system, as a whole, changes under such
transformations.
Phases of matter, such as crystals, magnets,
and conventional superconductors, as well
as simple phase transitions can be described
by spontaneous symmetry breaking.
Notable exceptions include topological phases
of matter like the fractional quantum Hall
effect.
== Examples ==
=== 
Mexican hat potential ===
Consider a symmetric upward dome with a trough
circling the bottom.
If a ball is put at the very peak of the dome,
the system is symmetric with respect to a
rotation around the center axis.
But the ball may spontaneously break this
symmetry by rolling down the dome into the
trough, a point of lowest energy.
Afterward, the ball has come to a rest at
some fixed point on the perimeter.
The dome and the ball retain their individual
symmetry, but the system does not.In the simplest
idealized relativistic model, the spontaneously
broken symmetry is summarized through an illustrative
scalar field theory.
The relevant Lagrangian of a scalar field
ϕ
{\displaystyle \phi }
, which essentially dictates how a system
behaves, can be split up into kinetic and
potential terms,
It is in this potential term
V
(
ϕ
)
{\displaystyle V(\phi )}
that the symmetry breaking is triggered.
An example of a potential, due to Jeffrey
Goldstone is illustrated in the graph at the
right.
This potential has an infinite number of possible
minima (vacuum states) given by
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 Lorentz 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.
== Spontaneous symmetry breaking in physics
==
=== 
Particle physics ===
In particle physics the force carrier particles
are normally specified by field equations
with gauge symmetry; their equations predict
that certain measurements will be the same
at any point in the field.
For instance, field equations might 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 a better term than "broken", because
the symmetry is always there in these equations.
This phenomenon is called spontaneous symmetry
breaking (SSB) because nothing (that we know
of) breaks the symmetry in the equations.
==== 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 100 times heavier
constituents of 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.
==== Higgs mechanism ====
The strong, weak, and electromagnetic forces
can all be understood as arising from gauge
symmetries.
The Higgs mechanism, the spontaneous symmetry
breaking of gauge symmetries, is an important
component in understanding the superconductivity
of metals and the origin of particle masses
in the standard model of particle physics.
One important consequence of the distinction
between true symmetries and gauge symmetries,
is that the spontaneous breaking of a gauge
symmetry does not give rise to characteristic
massless Nambu–Goldstone physical modes,
but only massive modes, like the plasma mode
in a superconductor, or the Higgs mode observed
in particle physics.
In the standard model of particle physics,
spontaneous symmetry breaking of the SU(2)
× U(1) gauge symmetry associated with the
electro-weak force generates masses for several
particles, and separates the electromagnetic
and weak forces.
The W and Z bosons are the elementary particles
that mediate the weak interaction, while the
photon 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
so that the photon and the massive W and Z
bosons emerge.
In addition, fermions develop mass consistently.
Without spontaneous symmetry breaking, the
Standard Model of elementary particle interactions
requires 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
is augmented by the Higgs mechanism to give
these particles mass.
It also suggests the presence of a new particle,
the Higgs boson, detected in 2012.
Superconductivity of metals is a condensed-matter
analog of the Higgs phenomena, in which a
condensate of Cooper pairs of electrons spontaneously
breaks the U(1) gauge symmetry associated
with light and electromagnetism.
=== Condensed matter physics ===
Most phases of matter can be understood through
the lens of spontaneous symmetry breaking.
For example, crystals are periodic arrays
of atoms that are not invariant under all
translations (only under a small subset of
translations by a lattice vector).
Magnets have north and south poles that are
oriented in a specific direction, breaking
rotational symmetry.
In addition to these examples, there are a
whole host of other symmetry-breaking phases
of matter including nematic phases of liquid
crystals, charge- and spin-density waves,
superfluids and many others.
There are several known examples of matter
that cannot be described by spontaneous symmetry
breaking, including: topologically ordered
phases of matter like fractional quantum Hall
liquids, and spin-liquids.
These states do not break any symmetry, but
are distinct phases of matter.
Unlike the case of spontaneous symmetry breaking,
there is not a general framework for describing
such states.
==== Continuous symmetry ====
The ferromagnet is the canonical system which
spontaneously breaks the continuous symmetry
of the spins below the Curie temperature and
at h = 0, where h is the external magnetic
field.
Below the Curie temperature the energy of
the system is invariant under inversion of
the magnetization m(x) such that m(x) = −m(−x).
The symmetry is spontaneously broken as h
→ 0 when the Hamiltonian becomes invariant
under the inversion transformation, but the
expectation value is not invariant.
Spontaneously-symmetry-broken phases of matter
are characterized by an order parameter that
describes the quantity which breaks the symmetry
under consideration.
For example, in a magnet, the order parameter
is the local magnetization.
Spontaneously breaking of a continuous symmetry
is inevitably accompanied by gapless (meaning
that these modes do not cost any energy to
excite) Nambu–Goldstone modes associated
with slow long-wavelength fluctuations of
the order parameter.
For example, vibrational modes in a crystal,
known as phonons, are associated with slow
density fluctuations of the crystal's atoms.
The associated Goldstone mode for magnets
are oscillating waves of spin known as spin-waves.
For symmetry-breaking states, whose order
parameter is not a conserved quantity, Nambu–Goldstone
modes are typically massless and propagate
at a constant velocity.
An important theorem, due to Mermin and Wagner,
states that, at finite temperature, thermally
activated fluctuations of Nambu–Goldstone
modes destroy the long-range order, and prevent
spontaneous symmetry breaking in one- and
two-dimensional systems.
Similarly, quantum fluctuations of the order
parameter prevent most types of continuous
symmetry breaking in one-dimensional systems
even at zero temperature (an important exception
is ferromagnets, whose order parameter, magnetization,
is an exactly conserved quantity and does
not have any quantum fluctuations).
Other long-range interacting systems such
as cylindrical curved surfaces interacting
via the Coulomb potential or Yukawa potential
has been shown to break translational and
rotational symmetries.
It was shown, in the presence of a symmetric
Hamiltonian, and in the limit of infinite
volume, the system spontaneously adopts a
chiral configuration, i.e. breaks mirror plane
symmetry.
=== Dynamical symmetry breaking ===
Dynamical symmetry breaking (DSB) is a special
form of spontaneous symmetry breaking where
the ground state of the system has reduced
symmetry properties compared to its theoretical
description (Lagrangian).
Dynamical breaking of a global symmetry is
a spontaneous symmetry breaking, that happens
not at the (classical) tree level (i.e. at
the level of the bare action), but due to
quantum corrections (i.e. at the level of
the effective action).
Dynamical breaking of a gauge symmetry [1]
is subtler.
In the conventional spontaneous gauge symmetry
breaking, there exists an unstable Higgs particle
in the theory, which drives the vacuum to
a symmetry-broken phase (see e.g. Electroweak
interaction).
In dynamical gauge symmetry breaking, however,
no unstable Higgs particle operates in the
theory, but the bound states of the system
itself provide the unstable fields that render
the phase transition.
For example, Bardeen, Hill, and Lindner published
a paper which attempts to replace the conventional
Higgs mechanism in the standard model, by
a DSB that is driven by a bound state of top-antitop
quarks (such models, where a composite particle
plays the role of the Higgs boson, are often
referred to as "Composite Higgs models").
Dynamical breaking of gauge symmetries is
often due to creation of a fermionic condensate;
for example the quark condensate, which is
connected to the dynamical breaking of chiral
symmetry in quantum chromodynamics.
Conventional superconductivity is the paradigmatic
example from the condensed matter side, where
phonon-mediated attractions lead electrons
to become bound in pairs and then condense,
thereby breaking the electromagnetic gauge
symmetry.
== 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.
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 specifies
a "frame of reference" to be measured against.
In that case, the vacuum state does not obey
the initial symmetry (which would keep it
invariant, in the linearly realized Wigner
mode in which it would be a singlet), and,
instead changes under the (hidden) symmetry,
now implemented in the (nonlinear) 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.
== 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, 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, of Kyoto University, 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.
== See also ==
== 
Notes ==
^ Note that (as in fundamental Higgs driven
spontaneous gauge symmetry breaking) the term
"symmetry breaking" is a misnomer when applied
to gauge symmetries
