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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.[1][2][3] 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.
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.[1][2][3] 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.
自发对称破缺是一个自发的对称破缺过程，它使处于对称状态的物理系统最终处于非对称状态。特别地，它可以描述运动方程或拉格朗日方程服从某种对称性，但最低能量真空解不具有该对称性的系统。当系统进入其中一个真空解时，真空解周围的扰动会破坏系统对称性，尽管整个拉格朗日方程仍然保持了对称性。
==Overview==
==Overview==
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]].
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==
==Examples 例子==
===Sombrero potential===
===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>
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>.]]
考虑一个对称向上的圆顶，底部环绕着一个槽。如果把一个球放在圆顶的最顶端，这个系统是围绕中心轴旋转对称的。但球体可能会自发地打破这种对称性，因为它会沿着穹顶滚动到能量最低的槽中。然后，球在圆周上某个固定的点上停下来。圆顶和球保持了各自的对称，但系统却没有保持对称性。
[[Image:Mexican hat potential polar.svg|270px|thumb|left|Graph of Goldstone's "[[sombrero]]" potential function <math>V(\phi)</math>.|链接=Special:FilePath/Mexican_hat_potential_polar.svg]]
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,
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,
在最简单的理想相对论模型中，可以用一个解释性的标量场理论总结自发破对称性。一个标量场 <math>\phi</math>的拉格朗日量从本质上决定了系统的行为，它可以分解成动能项和势能项：
{{NumBlk|::|<math>\mathcal{L} = \partial^\mu \phi \partial_\mu \phi - V(\phi).</math>|{{EquationRef|1}}}}
{{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.
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.
正是在势能项 <math>V(\phi)</math> 中触发了对称性破缺。例如作图所示的 [[Jeffrey Goldstone]] 给出的势能函数：
{{NumBlk|::|<math>V(\phi) = -5|\phi|^2 + |\phi|^4 \,</math>.|{{EquationRef|2}}}}
{{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
This potential has an infinite number of possible [[minimum|minima]] (vacuum states) given by
该是函数具有无穷数量的最小值点（真空态）当
{{NumBlk|::|<math>\phi = \sqrt{5} e^{i\theta} </math>.|{{EquationRef|3}}}}
{{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".
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".
对于0到2π之间的任何实数θ。系统也有一个不稳定的真空状态，对应于Φ = 0。这个状态具有U(1)对称。然而，一旦系统落入某个稳定真空状态(相当于选择θ)，这种对称性就会消失，或者说“自发破缺”。
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.
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.
事实上，任何其他θ的选择都将具有完全相同的能量，这意味着无质量的南部-戈德斯通玻色子的存在，这种模式在势能的最小值绕圆运动，并表明存在拉格朗日方程中原始对称性的一些记忆。
===Other examples===
===Other examples===
==Spontaneous symmetry breaking in physics==
==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.]]
[[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.|链接=Special:FilePath/Spontaneous_symmetry_breaking_(explanatory_diagram).png]]
===Particle physics===
===Particle physics===