时间晶体：神奇新物种 | 诺奖得主Wilczek专栏

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Without space and time translation symmetry, experiments carried out in different places and at different times would not be reproducible. In their everyday work, scientists take those symmetries for granted. Indeed, science as we know it would be impossible without them. But it is important to emphasize that we can test space and time translation symmetry empirically. Specifically, we can observe behavior in distant astronomical objects. Such objects are situated, obviously, in different places, and thanks to the finite speed of light we can observe in the present how they behaved in the past. Astronomers have determined, in great detail and with high accuracy, that the same laws do in fact apply.

对称性破缺

SYMMETRY BREAKING

晶体因对称而美，但对于物理学家来说，晶体最显著的特征却是它们缺失了对称。

For all their aesthetic symmetry,  it is actually the way crystals lack symmetry that is, for physicists, their defining characteristic.

考虑一个特别简单的晶体。它是一维的，它的原子核规则地排列在一条直线上，相邻间距是 d【 因此，每个原子核的坐标是 nd ，其中 n 是整数】。如果我们将这个晶体往右平移一丁点儿，那么它和移动前是不一样的。只有平移了特定的距离 d ，我们才会得到相同的晶体。所以，我们的理想化晶体只具有部分平移对称性，这与前面介绍的正方形只具有部分旋转对称性是一样的道理。

Consider a drastically idealized crystal. It will be one-dimensional, and its atomic nuclei will be located at regular intervals along a line, separated by the distance  d.  (Their coordinates therefore will be nd,  where n is a whole number.) If we translate this crystal to the right by a tiny distance, it will not look like the same object. Only after we translate through the specific distance d will we see the same crystal. Thus, our idealized crystal has a reduced degree of spatial translation symmetry, similarly to how a square has a reduced degree of rotation symmetry.

物理学家认为，在晶体中，物理基本定律的平移对称性“破缺”了，只剩下部分平移对称性。这些遗留的对称性却描述了晶体的本质特征。事实上，一旦我们知道晶体的对称是平移距离 d 的整数倍，我们就知道晶体中原子的相对位置。

Physicists say that in a crystal the translation symmetry of the fundamental laws is "broken," leading to a lesser translation symmetry. That remaining symmetry conveys the essence of our crystal. Indeed, if we know that a crystal's symmetry involves translations through multiples of the distance  d,  then we know where to place its atoms relative to one another.

二维和三维的晶体会更复杂，它们的种类非常多，可以同时具有部分旋转和平移对称性。十四世纪的艺术家在装饰西班牙格拉纳达的阿尔罕布拉宫时，利用想象和经验发现了很多可能的二维晶体。而十九世纪的数学家则对三维晶体进行了分类。

Crystalline patterns in two and three dimensions can be more complicated, and they come in many varieties. They can display partial rotational and partial translational symmetry. The 14th-century artists who decorated the Alhambra palace in Granada, Spain, discovered many possible forms of two-dimensional crystals by intuition and experimentation, and mathematicians in the 19th century classified the possible forms of three-dimensional crystals.

2011年的夏天，我开了一门课，主要讲授物理中的对称。我在准备晶体分类那章时觉得相关的数学非常优雅。在备课过程中，我总是尝试从一个新的角度来审视我的课程，尽可能增加一些新的内容。我突然意识到，三维空间晶体的分类可以推广到四维时空晶体。

In the summer of 2011 I was preparing to teach this elegant chapter of mathematics as part of a course on the uses of symmetry in physics. I always try to take a fresh look at material I will be teaching and, if possible, add something new. It occurred to me then that one could extend the classification of possible crystalline patterns in three-dimensional space to crystalline patterns in four-dimensional spacetime.

我把相关的数学研究告诉了阿尔弗雷德·萨皮尔（ Alfred Shapere ），他曾是我的学生，现在是我亲密的合作者。他目前在肯塔基大学工作。他希望我先回答两个基本的物理问题：

时空晶体能描述什么实际的物理体系？

这些晶体会引导我们发现不同的物质状态吗？

When I mentioned this mathematical line of investigation to Alfred Shapere, my former student turned valued colleague, who is now at the University of Kentucky, he urged me to consider two very basic physical questions. They launched me on a surprising scientific adventure:

What real-world systems could crystals in spacetime describe?

Might these patterns lead us to identify distinctive states of matter?

这两个问题带我走上一个充满惊喜的科学历程。

They launched me on a surprising scientific adventure.

第一个问题的答案相当直接。既然普通晶体是物体在空间的有序排列，那么时空晶体应该是事件在时空中的有序排列。

The answer to the first question is fairly straight-forward. Whereas ordinary crystals are orderly arrangements of objects in space, spacetime crystals are orderly arrangements of events in spacetime.

我们效仿上面对普通晶体的讨论，先考虑一维时空晶体来找找感觉。这个特殊情况下，时空晶体就成了纯粹的时间晶体。我们这时需要寻找的系统应该这样：它的状态每隔一段时间就会重复。令人尴尬的是，这样的系统早已为人熟知。比如，地球在空间中的姿态每隔一天就重复一 遍，地球与太阳的相对位置每隔一年也重复一次。

As we did for ordinary crystals, we can get our bearings by considering the one-dimensional case, in which spacetime crystals simplify to purely time crystals. We are looking, then, for systems whose overall state repeats itself at regular intervals. Such systems are almost embarrassingly familiar. For example, Earth repeats its orientation in space at daily intervals, and the Earth-sun system repeats its configuration at yearly intervals.

发明家和科学家在过去几十年里发展了很多时钟系统，这些时钟每重复一次的时间间隔的精度越来越高。单摆和弹簧钟已经被基于（传统）晶体振动的晶钟超越，后者又被基于原子振动的原子钟超越了。原子钟已经取得了令人惊叹的精度，但我们有很多理由去继续提高精度——我们后面将会看到，在这个问题上，时间晶体极有可能会帮上忙。

Inventors and scientists have, over many decades, developed systems that repeat their arrangements at increasingly accurate intervals for use as clocks. Pendulum and spring clocks were superseded by clocks based on vibrating (traditional) crystals, and those were eventually superseded by clocks based on vibrating atoms. Atomic clocks have achieved extraordinary accuracy, but there are important reasons to improve them further-and time crystals might help, as we will see later.

一些大家熟知的真实体系则是高维时空晶体。比如下图中的平面声波，其曲面的高度表示随空间和时间变化的密度。更复杂的时空晶体可能很难在自然界找到，但它们可能成为艺术家和工程师追求的目标——想象一下，一个会动的增强版阿尔罕布拉宫也是一个时空晶体。

Some familiar real-world systems also embody higher-dimensional spacetime crystal patterns. For example, the pattern shown here can represent a planar sound wave, where the height of the surface indicates compression as a function of position and time. More elaborate spacetime crystal patterns might be difficult to come by in nature, but they could be interesting targets for artists and engineers-imagine a dynamic Alhambra on steroids.

对于这类时空晶体，我们只是新瓶装旧酒，换了一个不同的标签。而回答萨皮尔的第二个问题则会将我们带入一个真正创新的物理领域。为此，我们现在必须介绍一个概念：对称性自发破缺。

These types of spacetime crystals, though, simply repackage known phenomena under a different label. We can move into genuinely new territory in physics by considering Shapere's second question. To do that, we must now bring in the idea of  spontaneous  symmetry breaking.

对称性自发破缺

SPONTANEOUS SYMMETRY BREAKING

当液体或气体冷却成晶体时，一件非常基本且神奇的事发生了：晶体——这个物理定律的解——具有的对称性少于物理定律本身的对称性。由于这个对称性的减少只是通过降温而获得的，在这个过程中并没有其他外界因素的干预，于是我们认为在晶体形成过程中，物质“自发”破坏了空间平移对称性。

When a lIquId  or gas cools into a crystal, something fundamentally remarkable occurs: the emergent solution of the laws of physics-the crystal-displays less symmetry than the laws themselves. As this reduction of symmetry is brought on just by a decrease in temperature, without any special outside intervention, we can say that in forming a crystal the material breaks spatial translation symmetry "spontaneously."

晶体形成的一个重要特征是物质系统的行为有一个急剧的变化，或者按专业说法，一个急剧的相变。在临界温度上（这个温度的高低取决于系统的化学成分和环境压强），系统是液体；临界温度下，系统则变成了晶体——晶体的各种性质都和液体非常不同。这个相变可以预测，并伴有能量的释放（一般是以热的形式）。环境条件的微小变化会让物质重组，成为非常不同的材料，比如水变冰。人们虽然很熟悉这个现象但依然会觉得神奇。

An important feature of crystallization is a sharp change in the system's behavior or, in technical language, a sharp phase transition. Above a certain critical temperature (which depends on the system's chemical composition and the ambient pressure), we have a lIquId; below it we have a crystal-objects with quite different properties. The transition occurs predictably and is accompanied by the emission of energy (in the form of heat). The fact that a small change in ambient conditions causes a substance to reorganize into a qualitatively distinct material is no less remarkable for being, in the case of water and ice, very familiar.

晶体的刚性是另一个不同于液体和气体的性质。从微观上看，晶体之所以有刚性是因为晶体中的原子在很大范围内的有序排列，任何试图破坏这种有序性的行为都会遭到晶体的抵抗。

The rigidity of crystals is another emergent property that distinguishes them from lIquIds and gases. From a microscopic perspective, rigidity arises because the organized pattern of atoms in a crystal persists over long distances and the crystal resists attempts to disrupt that pattern.

我们刚刚讨论了晶体形成的三个特征——减少的对称性、急剧的相变和刚性——它们是紧密相关的。这三个特征都源于一个基本原则，原子“希望”按照一个能量尽可能小的方式排列。在不同的外部条件（比如不同的压强和温度）下，原子会按不同的方式排列——这些就是不同的 “相”。当外界条件改变时，我们经常会看到急剧的相变。有序排列的形成要求原子们集体行动，整个材料中的原子都会被要求按照同样的方式排列。这种排列即使受到小的扰动，也会自动恢复。

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