Topological foundations of ferroelectricity

Abstract

The 21st century has witnessed a revolutionary shift in the understanding of properties of matter driven by the application of topological principles. While the traditional approach to material science has been focusing on local interactions, topology introduces a global, non-local description in which the geometry of a material profoundly influences its properties. Ferroelectric materials, with their spontaneous electric polarization, have long been essential for understanding fundamental physical phenomena, which have led to numerous practical applications. Recent discoveries have revealed that nanostructured ferroelectrics host a wealth of fundamental topological states, which effectively enrich the landscape of ferroelectric research. This Review explores the topological foundation of ferroelectricity, rooted in the electrostatic essence of these materials. Drawing upon the analogy between the hydrodynamics of incompressible fluids and the electrostatics of polarization fields, we establish a comprehensive framework for classifying the complex topological states observed in ferroelectrics. We demonstrate that the rich diversity of polarization structures can be exhaustively described using the advanced topological approach. By extending fundamental topological concepts such as helicity, fibration, foliation, and ergodicity, we offer a systematic analysis of the topological textures in ferroelectrics. This work provides a coherent framework for understanding and manipulating topological structures in nanostructured ferroelectrics, paving the way for innovations in materials science and technology.