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===Other examples===
===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.
* 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.
* 对于铁磁性材料，其基本定律在空间旋转下是不变的。在这里，序参量是衡量磁偶极子密度的磁化强度。在居里温度以上，序参量为零，具有空间不变性，不存在对称性破缺。然而，在居里温度以下，磁化强度变成一个恒定的非零值，指向一个特定的方向(在有充分平衡的理想情况下；否则，平移对称性也会破缺)。使该向量方向不变的旋转对称性仍然保留，而其他旋转对称性自发破缺。
* 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.
* 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.
* 描述固体的定律在完整的欧几里得群下是不变的，但固体本身会自发地将这个群分解为一个空间群。其中位移和方向是序参量。
* 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.
* 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.
* 广义相对论具有洛伦兹对称性，但在FRW宇宙模型中，定义为星系速度的平均值(星系在宇宙尺度上的行为就像气体粒子) 的平均 4-速度场，作为序参量会打破这种对称性。对于宇宙微波背景辐射也有类似的评论。
* 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.
* 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.
* 对于电弱模型，如前面所解释的，希格斯场的一个分量提供了将电弱规范对称性破缺到电磁规范对称性的序参量。和铁磁的例子一样，在电弱温度下也有相变。同样的关于我们不倾向于注意破缺对称性的评论，也说明了为什么我们花了这么长时间才发现电弱统一。
* In superconductors, there is a condensed-matter collective field ψ, which acts as the order parameter breaking the electromagnetic gauge symmetry.
* In superconductors, there is a condensed-matter collective field ψ, which acts as the order parameter breaking the electromagnetic gauge symmetry.
* 在超导体中有一个凝聚态集体场ψ，它是打破电磁规范对称性的序参量。
* 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]].
* 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]].
* 拿一个细长的圆柱形塑料杆，把两端推到一起。在屈曲之前，系统在旋转下是对称的，因此可见圆柱对称性。但在弯曲之后，它看起来就不同了，而且是不对称的。然而，圆柱对称性的特征仍然存在：忽略摩擦，杆可以不受外力自由地自旋，在时间上取代基态，等于一个频率趋于零的振荡，而不是沿屈曲方向的径向振荡。这种自旋模式实际上是必需的南部-戈德斯通玻色子。
* 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 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.
* 考虑无限水平面上的一层均匀的流体。这个系统具有欧几里得平面的所有对称性。但是现在均匀地加热底部表面，使它变得比上表面热得多。当温度梯度足够大时，就会形成对流单元，打破了欧几里得对称。
* 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.
* 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.
* 考虑一个围绕某个竖直的直径旋转的圆形箍上的珠子。当旋转速度从静止逐渐增加时，珠子最初会停留在环底部的初始平衡点(直观上稳定，重力势最低)。在一定的临界旋转速度下，这一点将变得不稳定，珠子将跳到另外两个新创建的离中心等距离的平衡点中的一个。起初，系统相对直径是对称的，但在通过临界速度后，珠子最终停留在两个新的平衡点中的一个，从而打破了对称性。
==Spontaneous symmetry breaking in physics==
==Spontaneous symmetry breaking in physics==