English translation of the original Chinese article. Publication dates and the extent of recovered text are preserved. Figures retain their original labels. Read the Chinese original.
Department of Physics, University of Toronto (affiliation credited in the original manuscript)
Keywords: topology, topological phases, phase transitions
What is topology, and what is a topological phase? Bystanders were utterly bewildered as to why the 2016 Nobel Prize in Physics went to these three gentlemen. When graphene won, at least we could understand something: they had peeled graphene off with adhesive tape! Emboldened by my single encounter with one of the laureates, F. D. M. Haldane, I will venture a brief account of the past and present of topological phases. That year's laureates were David J. Thouless, F. Duncan M. Haldane and J. Michael Kosterlitz. Their prize recognized theoretical discoveries of topological phase transitions and topological phases of matter.[4]
Once upon a time, there was a village called Condensed-Matter Physics. A Russian gentleman named L. Landau lived there. One question interested him: what changes when water turns into ice? In the village, this was called a phase transition. Some children might ask, “Doesn't it simply change from something that flows to something fixed? What's there to study?” Yet when a liquid becomes a crystal, its molecules acquire periodic order, breaking the original continuous translational symmetry. Landau's theory described many phase transitions through order parameters and symmetry breaking, becoming an important language through which the village understood matter. Later, people discovered that the world of phase transitions was richer than this framework alone could describe.[5]
Although “topology” sounds abstract, it can begin with a simple question about shape. A rubber ball cannot be transformed into a ring with a hole in the middle by smooth stretching or compression alone, unless it is cut or glued. Properties that remain unchanged under these deformations are topological properties; in this example, the number of holes is a topological invariant. In physics, topology may be applied to the structure of fields or quantum states in matter. The outward shape of a material, or the total number of vortices in an ordinary lake, cannot simply serve as a classification rule for all topological phases.[5]
Vortices in the lake beside the village help us imagine topological defects, but this is only a metaphor. In systems such as two-dimensional superfluids, Kosterlitz and Thouless studied vortices and antivortices in a field. At low temperatures, these form mutually bound vortex pairs. When the temperature rises through the phase transition, the pairs unbind, allowing vortices to move independently. This change is called the Kosterlitz–Thouless, or KT, transition. In recognition of Berezinskii's related contribution, it is also often called the BKT transition. It provides a mechanism for phase transitions beyond the usual picture of symmetry breaking.[1][4]
Haldane followed a different research path, revealing the distinct properties of integer-spin and half-integer-spin chains in one-dimensional antiferromagnets. The gapped state of integer-spin chains became an important example of topological quantum matter.[2]In 1988, he also proposed a two-dimensional model showing that quantized Hall conductance could arise without an applied magnetic field, providing a theoretical foundation for the quantum anomalous Hall effect and later research into topological materials.[3]The achievements of these three researchers along different paths jointly advanced the field.
Topological phases give condensed-matter physics a new way to describe matter and have driven research into topological materials. “Topological order” has a separate, specific meaning in physics; here, the broader term “topological phase” is used for the topics discussed above. People hope that this research will eventually find applications in materials and electronics. When it may reach mobile phones or computer chips still depends on progress in particular materials, devices and engineering. Step by step, complex and abstract ideas in physics are moving closer to everyday life.[4]
References
[1] Kosterlitz, J. M.; Thouless, D. J. Ordering, metastability and phase transitions in two-dimensional systems. Journal of Physics C: Solid State Physics 6, 1181–1203 (1973).
[2] Haldane, F. D. M. Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis Néel State. Physical Review Letters 50, 1153 (1983). View source
[3] Haldane, F. D. M. Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the “Parity Anomaly”. Physical Review Letters 61, 2015 (1988). View source
[4] Royal Swedish Academy of Sciences. Nobel Prize in Physics 2016: Popular Science Background. View source
[5] Nobel Prize in Physics 2016: Award Ceremony Speech. View source
The text was recovered from a local Word manuscript of the same title credited to 陈一阁. In October 2026, its scientific statements were corrected and additional sources supplied using the official Nobel Prize explanation and original papers. The final text published on the original site has not yet been obtained for comparison, so this version may differ from the one published at the time. The historical date is retained from the original site's index.
Editorial note: On October 10, 2026, statements about symmetry, vortices and the researchers' contributions were corrected using official Nobel Prize materials and original papers, with the historical timeframe made explicit. The author credit was restored from the local manuscript of the same title.
