中国物理学家发现，临界点——即物理系统迅速转变状态的阈值——可以拥有类似于甜甜圈孔的平滑数学结构，这挑战了这两种现象必须分开研究的固有观念。
Chinese physicists have discovered that a tipping point – a threshold where a physical system rapidly shifts states – can possess a smooth mathematical structure resembling a doughnut’s hole, challenging a belief that the two phenomena must be studied separately.
几十年来，主流物理学家大多将描述物理系统在临界点如何变化的临界现象，以及研究几何性质的拓扑学，视为两个独立的研究领域。
For decades, mainstream physicists have mostly treated critical phenomena, which describe how physical systems change at critical points, and topology, which studies the properties of geometry, as separate areas of research.
拓扑学的一个经典图示是球体与甜甜圈的区别。由于两者的孔洞数量不同，除非对表面进行切割或粘合，否则仅通过拉伸或弯曲是无法将它们互相转换的。
A classic illustration of topology is the difference between a sphere and a doughnut. The two cannot be transformed into each other by stretching or bending alone because they have a different number of holes, unless the surface is cut or glued.
这项由新加坡和中国机构的研究人员开展的研究于10月7日发表在《自然》杂志上，新加坡南洋理工大学的程哲宇和张修海为共同第一作者。
The study by researchers at Singaporean and Chinese institutions was published in Nature on October 7, with Cheng Zheyu and Zhang Xiuhai, both affiliated with Nanyang Technological University in Singapore, as co-first authors.
程哲宇及其同事着手调查，甜甜圈状的拓扑性质是否可以存在于临界点——即物理系统改变其行为的阈值。
Cheng and his colleagues set out to investigate whether doughnut-like topological properties could exist at critical points – the thresholds at which physical systems change their behaviour.
程哲宇告诉《南华早报》：“我们的研究发现，临界点本身可以具备拓扑结构。”
“Our research found that the critical point itself can have topology,” Cheng told the South China Morning Post.
这些发现可能会促使研究人员重新审视研究了数个世纪的物理系统，以探究他们是否忽略了临界点处以及更广泛的经历临界转变的系统中的基本性质。
The findings could prompt researchers to revisit physical systems studied for centuries to investigate whether they have overlooked fundamental properties at critical points and, more broadly, in systems undergoing critical transitions.
程哲宇表示：“这项工作意义重大，因为它挑战了物理学两个领域长期以来的分立状态，并可能为临界点数学性质的研究开辟一个新的方向。”
“The work was significant because it challenged a long-standing separation between two areas of physics, potentially opening a new direction for research into the mathematical properties of tipping points,” Cheng said.
为了说明临界现象的工作原理，程哲宇提到了团簇伊辛链，这是一个代表一排微小磁铁的模型，每个磁铁都可以朝上或朝下，它们之间的相互作用会影响其集体行为。
To illustrate how critical phenomena work, Cheng pointed to the cluster-Ising chain, a model representing a row of tiny magnets, each of which can point up or down, with interactions between them influencing their collective behaviour.
随着这些相互作用发生变化，系统可以从一种状态转变到另一种状态，临界点标志着这种转变，并且可能具有研究人员此前忽视的拓扑性质。
As these interactions change, the system can shift from one state to another, with the critical point marking the transition and potentially possessing topological properties that researchers had previously overlooked.
中国内地和香港的研究人员在同日于《自然》杂志上发表的另一项独立研究，在一维和二维系统中报告了相关发现，他们使用工程声子晶体来研究临界拓扑，并直接成像了不同状态边界上独特的拓扑模式。
A separate study by researchers in mainland China and Hong Kong published in Nature on the same day reported related findings in one- and two-dimensional systems, using engineered phononic crystals to investigate critical topology and directly image distinctive topological modes along boundaries between different states.
这两项研究共同为一个此前少有涉足的物理学领域提供了实验证据。
Together, the studies provide experimental evidence for a previously underexplored area of physics.
程哲宇表示：“这是一个非常年轻的领域，我们仍在探索中，”并补充说，科学家们才刚刚开始研究其他因临界行为而被研究的物理系统是否也可能具有拓扑性质。
“This is a very young field, and we are still exploring it,” Cheng said, adding that scientists were beginning to investigate whether other physical systems studied for their critical behaviour might also possess topological properties.
这些发现还提出了一个更广阔的问题：拓扑学能否帮助科学家理解自然界和人类社会中更复杂系统里的临界点？
The findings also raise a broader question: could topology help scientists understand tipping points in more complicated systems found in nature and society?
在复杂系统研究中，科学家们会研究可能表明系统何时接近临界点的预警信号，例如生态系统的退化或气候行为的转变。
In complex-systems research, scientists study warning signals that may indicate when a system is approaching a tipping point, such as the degradation of an ecosystem or a shift in climate behaviour.
程哲宇表示，临界现象可能与这一更广泛的领域相关，但他无法确认复杂系统中的临界点是否也具备拓扑性质。
Cheng said critical phenomena could be connected to this broader field, but he could not confirm whether tipping points in complex systems also possessed topological properties.
到目前为止，其团队的实验一直集中在相对简单的物理模型上，而不是自然界或社会中发现的更复杂的系统。
His team’s experiments have so far focused on relatively simple physical models rather than the more complicated systems found in nature or society.
研究人员还使用了基于熵的分析——一种测量随机性和无序性的技术——来研究临界点，将其工作与统计物理学中使用的数学工具联系起来。然而，这些发现不能直接应用于生态、气候或社会系统。目前，其主要贡献在于基础科学，而非直接的实际应用。程哲宇说：“这个领域还太年轻，短期内看不到任何应用。”
The researchers have also used entropy-based analysis – a technique for measuring randomness and disorder – to study critical points, linking their work to mathematical tools used in statistical physics. However, the findings cannot be directly applied to ecological, climatic or social systems. For now, the main contribution is to fundamental science rather than an immediate practical application. “It is too young to see any applications in the short term,” Cheng said.
他说，其团队的下一步将是研究涉及多体相互作用的系统，从而使研究人员能够探索更复杂的物理行为。
His team’s next step would be to investigate systems involving many-body interactions, allowing researchers to explore more complicated physical behaviour, he said.