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→Condensed matter physics
金属的超导性是一种类似于希格斯现象的凝聚态物质，其中库珀电子对的凝聚会自发地打破与光和电磁相关的U(1)规范对称。
金属的超导性是一种类似于希格斯现象的凝聚态物质，其中库珀电子对的凝聚会自发地打破与光和电磁相关的U(1)规范对称。
===Condensed matter physics===
===Condensed matter physics 凝聚态物理===
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.
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.
物质的大多数相态都可以通过自发对称性破缺的透镜来理解。例如，晶体是原子的周期性排列，并非在所有平移下(仅在晶格向量平移的一个小子集下)都是不变的。磁体有朝向特定方向的南极和北极，打破了旋转对称。除了这些例子，还有一大堆其他的物质对称性破缺相——包括液晶的向列相、电荷和自旋密度波、超流体等等。
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>
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====
有几个已知的例子是不能用自发对称破缺来描述的，包括:物质的拓扑有序相，如分数量子霍尔液体和自旋液体。这些状态并不破坏任何对称性，然而是物质的不同相。与自发对称破缺的情况不同，没有一个描述这种状态的一般框架。
====Continuous symmetry 连续对称性====
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.
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.
铁磁体是正则系统，它在居里温度以下和h = 0(其中h为外部磁场)的情况下自发打破自旋的连续对称性。在居里温度以下，系统的能量在磁化强度m(x)的反转下（使m(x) =−m(−x)）不变。当哈密顿量在反转变换下不变，而期望值不是恒定时，对称性在h→0时自发破坏。
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-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.
物质的自发对称性破缺相由一个序参量表征，它描述了打破所考虑的对称性的量。例如，在磁铁中，序参量是局部磁化强度。
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.
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.
连续对称的自发破缺不可避免地伴随着无间隙(意味着这些模式不需要花费任何能量来激发)南部-戈德斯通模式，它与序参量的缓慢、长波长波动有关。例如，晶体中的振动模式，即声子，与晶体原子的缓慢密度涨落有关。磁铁相关的戈德斯通模式是自旋振荡波，称为自旋波。对于序参量不是守恒量的对称性破缺态，Nambu-Goldstone模通常是无质量的，并以恒定速度传播。
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.)
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.)
由Mermin和Wagner提出的一个重要定理指出，在有限温度下，热激活的南布-戈德斯通模式的扰动破坏了长程有序，并阻止了一维和二维系统中对称性的自发破缺。类似地，即使是在零温度下，序参量的量子涨落阻止了一维系统中大多数类型的连续对称破缺。(一个重要的例外是铁磁体，其序参量磁化强度是一个精确的守恒量，不存在任何量子涨落。)
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>
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>
|arxiv = 0704.3435 |bibcode = 2007PhRvL..99c0602K |pmid=17678276|s2cid=37983980
|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]].
}}</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]].
其他长程相互作用系统，如圆柱曲面通过库仑势或汤川势相互作用，已被证明打破平移和旋转对称。结果表明，在对称哈密顿量存在的情况下，在无限体积的极限下，系统自发地采用手性构型，即打破镜面对称。
===Dynamical symmetry breaking 动力学对称性破缺===
===Dynamical symmetry breaking 动力学对称性破缺===