The past decade has witnessed dramatic progress related to various aspects of emergent topological polar textures in oxide nanostructures displaying vortices, skyrmions, merons, hopfions, dipolar waves, or labyrinthine domains, among others. For a long time, these nontrivial structures (the electric counterparts of the exotic spin textures) were not expected due to the high energy cost associated with the dipolar anisotropy: the smooth and continuous evolution of the local polarization to produce topologically protected structures would result in a large elastic energy penalty. However, it was discovered that the delicate balance and intricate interplay between the electric, elastic, and gradient energies can be altered in low-dimensional forms of ferroelectric oxide nanostructures. These can be tuned to manipulate order parameters in ways once considered impossible. This review addresses the historical context that provided the fertile background for the dawning of the polar topological era. This has been possible thanks to a fruitful, positive feedback between theory and experiment: advances in materials synthesis and preparation (with a control at the atomic scale) and characterization have come together with great progress in theoretical modeling of ferroelectrics at larger length and timescales. An in-depth scientific description to formalize and generalize the prediction, observation, and probing of exotic, novel, and emergent states of matter is provided. Extensive discussions of the fundamental physics of such polar textures, a primer explaining the basic topological concepts, an explanation of the modern theoretical and computational methodologies that enable the design and study of such structures, what it takes to achieve deterministic, on-demand control of order-parameter topologies through atomically precise synthesis, the range of characterization methods that are key to probing these structures, and their thermodynamic field-driven (temperature-driven, stress-driven, etc.) susceptibilities are included. The new emergent states of matter join together with exotic functional properties (such as chirality, negative capacitance, and coexistence of phases) that, along with their small size and ultrafast dynamical response, make them potential candidates in multifunctional devices. Finally, some open questions and challenges for the future are presented, underlining the interesting future that is anticipated in this field.

中文翻译：

---

极性氧化物纳米结构的拓扑相

在过去的十年中，氧化物纳米结构中出现的拓扑极性纹理的各个方面取得了巨大的进展，显示出涡旋、斯格明子、半子、hopfion、偶极波或迷宫域等。长期以来，由于与偶极各向异性相关的高能量成本，这些非平凡的结构（奇异自旋纹理的电对应物）并不被预期：局部极化的平滑和连续演化以产生拓扑保护的结构将导致大的弹性能量损失。然而，人们发现，在低维形式的铁电氧化物纳米结构中，电能、弹性能和梯度能之间的微妙平衡和复杂的相互作用可以改变。这些可以被调整来以曾经被认为不可能的方式操纵订单参数。这篇综述探讨了为极地拓扑时代的到来提供肥沃背景的历史背景。这要归功于理论和实验之间富有成效的积极反馈：材料合成和制备（在原子尺度上进行控制）和表征方面的进步与更大长度和时间尺度的铁电体理论建模的巨大进展相结合。提供了深入的科学描述，以形式化和概括对奇异的、新颖的和紧急的物质状态的预测、观察和探测。对这种极性纹理的基础物理的广泛讨论，解释基本拓扑概念的入门读物，对能够设计和研究这种结构的现代理论和计算方法的解释，如何实现确定性的、按需的控制包括通过原子级精确合成的有序参数拓扑、探测这些结构的关键表征方法的范围，以及它们的热力学场驱动（温度驱动、应力驱动等）敏感性。新的物质涌现状态与奇特的功能特性（例如手性、负电容和相共存）结合​​在一起，再加上它们的小尺寸和超快的动态响应，使它们成为多功能设备的潜在候选者。最后，提出了一些未来的开放性问题和挑战，强调了该领域预期的有趣的未来。