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在'''<font color="#ff8000">自发对称性破缺</font>'''中，系统的运动方程是不变的，但系统发生了变化。这是因为系统的背景(时空)——真空态——是非恒定的。这种对称性破缺可以用一个序参量进行参数化。这类对称破缺的一个特殊情况是'''<font color="#ff8000">动力学对称性破缺</font>'''。
在'''<font color="#ff8000">自发对称性破缺</font>'''中，系统的运动方程是不变的，但系统发生了变化。这是因为系统的背景(时空)——真空态——是非恒定的。这种对称性破缺可以用一个序参量进行参数化。这类对称破缺的一个特殊情况是'''<font color="#ff8000">动力学对称性破缺</font>'''。
==Overview==
In [[explicit symmetry breaking]], if two outcomes are considered, 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{{clarify|date=December 2016}} are [[Invariant (physics)|invariant]] under a [[Symmetry (physics)|symmetry]] transformation.
The system, as a whole{{clarify|date=December 2016}}, 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==
===Sombrero 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.<ref>{{cite book |first=Gerald M. |last=Edelman |title=Bright Air, Brilliant Fire: On the Matter of the Mind |location=New York |publisher=BasicBooks |year=1992 |url=https://archive.org/details/brightairbrillia00gera |url-access=registration |page=[https://archive.org/details/brightairbrillia00gera/page/203 203] }}</ref>
[[Image:Mexican hat potential polar.svg|270px|thumb|left|Graph of Goldstone's "[[Sombrero|sombrero]]" potential function <math>V(\phi)</math>.]]
In the simplest idealized relativistic model, the spontaneously broken symmetry is summarized through an illustrative [[scalar field theory]]. The relevant [[Lagrangian (field theory)|Lagrangian]] of a scalar field <math>\phi</math>, which essentially dictates how a system behaves, can be split up into kinetic and potential terms,
{{NumBlk|::|<math>\mathcal{L} = \partial^\mu \phi \partial_\mu \phi - V(\phi).</math>|{{EquationRef|1}}}}
It is in this potential term <math>V(\phi)</math> that the symmetry breaking is triggered. An example of a potential, due to [[Jeffrey Goldstone]]<ref>{{Cite journal | last1 = Goldstone | first1 = J. | doi = 10.1007/BF02812722 | title = Field theories with " Superconductor " solutions | journal = Il Nuovo Cimento | volume = 19 | issue = 1 | pages = 154–164 | year = 1961 | bibcode = 1961NCim...19..154G | s2cid = 120409034 | url = http://cds.cern.ch/record/343400 }}</ref> is illustrated in the graph at the left.
{{NumBlk|::|<math>V(\phi) = -5|\phi|^2 + |\phi|^4 \,</math>.|{{EquationRef|2}}}}
This potential has an infinite number of possible [[minimum|minima]] (vacuum states) given by
{{NumBlk|::|<math>\phi = \sqrt{5} e^{i\theta} </math>.|{{EquationRef|3}}}}
for any real ''θ'' between 0 and 2''π''. The system also has an unstable vacuum state corresponding to {{nowrap|1=''Φ'' = 0}}. This state has a [[Unitary group|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 [[Goldstone boson|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 [[ferromagnet]]ic 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 [[Friedmann–Lemaître–Robertson–Walker metric|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 [[Goldstone boson|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 cell]]s 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==
[[File:Spontaneous symmetry breaking (explanatory diagram).png|thumb|right|250px|''Spontaneous symmetry breaking illustrated'': At high energy levels (''left''), the ball settles in the center, and the result is symmetric. At lower energy levels (''right''), the overall "rules" remain symmetric, but the symmetric "[[sombrero|Sombrero]]" enforces an asymmetric outcome, since eventually the ball must rest at some random spot on the bottom, "spontaneously", and not all others.]]
===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 magnetization|''spontaneous'']] symmetry breaking (SSB) because ''nothing'' (that we know of) breaks the symmetry in the equations.<ref name="Weinberg2011">{{cite book|author=Steven Weinberg|title=Dreams of a Final Theory: The Scientist's Search for the Ultimate Laws of Nature|url=https://books.google.com/books?id=Rsg3PE_9_ccC|date=20 April 2011|publisher=Knopf Doubleday Publishing Group|isbn=978-0-307-78786-6}}</ref>{{rp|194–195}}
====Chiral symmetry====
{{Main article|Chiral symmetry breaking}}
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 boson]]s 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====
{{Main article|Higgs mechanism|Yukawa interaction}}
The strong, weak, and electromagnetic forces can all be understood as arising from [[gauge symmetry|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 {{nowrap|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 [[Unified field theory#Modern progress|Weinberg–Salam theory]] predicts that, at lower energies, this symmetry is broken so that the photon and the massive W and Z bosons emerge.<ref>A Brief History of Time, Stephen Hawking, Bantam; 10th anniversary edition (1998). pp. 73–74.{{ISBN?}}</ref> 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, such as [[Fractional quantum Hall effect|fractional quantum Hall liquids]], and [[Quantum spin liquid|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.<ref name=chen>{{cite journal | last1 = Chen | first1 = Xie | author-link3 = Xiao-Gang Wen | last2 = Gu | first2 = Zheng-Cheng | last3 = Wen | first3 = Xiao-Gang | year = 2010 | title = Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order | journal = Phys. Rev. B | volume = 82 | issue = 15| page = 155138 | doi=10.1103/physrevb.82.155138|arxiv = 1004.3835 |bibcode = 2010PhRvB..82o5138C | s2cid = 14593420 }}</ref>
====Continuous symmetry====
The ferromagnet is the canonical system that spontaneously breaks the continuous symmetry of the spins below the [[Curie temperature]] and at {{nowrap|1=''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 {{nowrap|1=''m''('''x''') = −''m''(−'''x''')}}. The symmetry is spontaneously broken as {{nowrap|''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.
Spontaneous 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]], have been shown to break translational and rotational symmetries.<ref>
{{cite journal
|last1=Kohlstedt |first1=K.L.
|last2=Vernizzi |first2=G.
|last3=Solis |first3=F.J.
|last4=Olvera de la Cruz |first4=M.
|year=2007
|title=Spontaneous Chirality via Long-range Electrostatic Forces
|journal=[[Physical Review Letters]]
|volume=99 |issue=3
|page=030602
|doi=10.1103/PhysRevLett.99.030602
|arxiv = 0704.3435 |bibcode = 2007PhRvL..99c0602K |pmid=17678276|s2cid=37983980
}}</ref> 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 in which the ground state of the system has reduced symmetry properties compared to its theoretical description (i.e., [[Lagrangian (field theory)|Lagrangian]]).
Dynamical breaking of a global symmetry is a spontaneous symmetry breaking, which 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 {{ref|note1}} 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, for example, [[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 that 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, in which a composite particle plays the role of the Higgs boson, are often referred to as "Composite Higgs models".)<ref>
{{cite journal
|author1=William A. Bardeen
|author2-link=Christopher T. Hill
|author2=Christopher T. Hill
|author3-link=Manfred Lindner
|author3=Manfred Lindner
|year=1990
|title=Minimal dynamical symmetry breaking of the standard model
|journal=[[Physical Review D]]
|volume=41 |issue=5 |pages=1647–1660
|bibcode=1990PhRvD..41.1647B
|doi=10.1103/PhysRevD.41.1647
|pmid=10012522
|author1-link=William A. Bardeen
}}</ref> Dynamical breaking of gauge symmetries is often due to creation of a [[fermionic condensate]] — e.g., the [[quark condensate]], which is connected to the [[Chiral symmetry breaking|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 [[Symmetry (physics)|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 [[Nambu–Goldstone boson|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|''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 boson]]s 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 mechanics|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 (physicist)|Makoto Kobayashi]] and [[Toshihide Maskawa]], of [[Kyoto University]], shared the other half of the prize for discovering the origin of the [[Explicit symmetry breaking|explicit breaking]] of CP symmetry in the weak interactions.<ref>{{cite web|author=The Nobel Foundation|title=The Nobel Prize in Physics 2008|url=http://nobelprize.org/nobel_prizes/physics/laureates/2008/index.html|work=nobelprize.org|access-date=January 15, 2008}}</ref> 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==
{{div col|colwidth=24em}}
* [[Autocatalytic reactions and order creation]]
* [[Catastrophe theory]]
* [[Chiral symmetry breaking]]
* [[CP-violation]]
* [[Fermi ball]]
* [[Gauge gravitation theory]]
* [[Goldstone boson]]
* [[Grand unified theory]]
* [[Higgs mechanism]]
* [[Higgs boson]]
* [[Higgs field (classical)]]
* [[Irreversibility]]
* [[Magnetic catalysis]] of chiral symmetry breaking
* [[Mermin–Wagner theorem]]
* [[Norton's dome]]
* [[Second-order phase transition]]
* [[Spontaneous absolute asymmetric synthesis]] in chemistry
* [[Symmetry breaking]]
* [[Tachyon condensation]]
* [[1964 PRL symmetry breaking papers]]
{{div col end}}
==Notes==
* {{note|note1}} Note that (as in fundamental Higgs driven spontaneous gauge symmetry breaking) the term "symmetry breaking" is a misnomer when applied to gauge symmetries.
==References==
{{Reflist}}
==External links==
{{wikiquote}}
* For a pedagogic introduction to electroweak symmetry breaking with step by step derivations, not found in texts, of many key relations, see http://www.quantumfieldtheory.info/Electroweak_Sym_breaking.pdf
* [http://hyperphysics.phy-astr.gsu.edu/hbase/forces/unify.html#c2 Spontaneous symmetry breaking]
* [http://prl.aps.org/50years/milestones#1964 Physical Review Letters – 50th Anniversary Milestone Papers]
* [http://cerncourier.com/cws/article/cern/32522 In CERN Courier, Steven Weinberg reflects on spontaneous symmetry breaking]
* [http://www.scholarpedia.org/article/Englert-Brout-Higgs-Guralnik-Hagen-Kibble_mechanism Englert–Brout–Higgs–Guralnik–Hagen–Kibble Mechanism on Scholarpedia]
* [http://www.scholarpedia.org/article/Englert-Brout-Higgs-Guralnik-Hagen-Kibble_mechanism_%28history%29 History of Englert–Brout–Higgs–Guralnik–Hagen–Kibble Mechanism on Scholarpedia]
* [https://arxiv.org/abs/0907.3466 The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles]
* [http://www.worldscinet.com/ijmpa/24/2414/S0217751X09045431.html 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]
* [http://www.datafilehost.com/download-7d512618.html 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}}
* [http://lanl.arxiv.org/abs/hep-th/9802142 Spontaneous Symmetry Breaking in Gauge Theories: a Historical Survey]
{{Standard model of physics}}
{{Quantum mechanics topics}}
{{DEFAULTSORT:Spontaneous Symmetry Breaking}}
[[Category:Theoretical physics]]
[[Category:Quantum field theory]]
[[Category:Standard Model]]
[[Category:Quantum chromodynamics]]
[[Category:Symmetry]]
[[Category:Quantum phases]]
==实例==
==实例==