image: The phase diagram features eight distinct topological phases characterized by the triplet (ν0, νπ/2, ν+) , with phase boundaries θx±θy=nπ ( n∈Z). The symbols of blue circle, blue ring, red square, purple downward triangle, and magenta upward triangle denote the parameter sets (θx, θy)=(1.3π,0), (0.6π,0), (0.3π,1.2π), (1.1π,1.6π) and (-1.042π, 1.6π) used in this study.
Credit: ©Science China Press
Topological phases of matter have been the research frontier in condensed matter and quantum physics. A defining feature of these systems is that the topology of the bulk can determine the appearance of special electronic or wave modes at the boundaries, known as topological boundary states. According to the conventional bulk–boundary correspondence, bulk topological invariants dictate the existence and properties of such boundary states. However, in many complex systems, different types of topological boundary states must be characterized by different invariants—for example, zero-dimensional corner states, one-dimensional hinge states, or two-dimensional surface states. This order-specific description makes the theoretical understanding and experimental characterization particularly challenging when multiple types of boundary states coexist in the same system. To address this challenge, the research team developed a unified topological framework. In this approach, the complete hierarchy of topological boundary states in a two-dimensional Floquet crystal can be fully characterized by a compact “topological triplet” consisting of three complementary one-dimensional winding numbers. This set of invariants simultaneously predicts the existence and type of strong, weak, and higher-order topological boundary states, enabling a unified description of the full hierarchy of boundary phenomena. The framework significantly simplifies the characterization of complex topological systems and provides a new perspective for understanding the relationships between different types of topological boundary states.
To experimentally verify the theory, the researchers implemented a two-dimensional Floquet crystal by periodically controlling the propagation of photons in a discrete synthetic time dimension. This platform allows precise tuning of system parameters and enables direct observation of dynamical evolution. The experiments clearly revealed the characteristic dynamics of different types of topological boundary states, in agreement with theoretical predictions, thereby confirming the capability of the topological triplet to unify the description of multiple boundary orders.
This work provides a simple yet general framework for describing complex topological systems. It opens new avenues for the systematic exploration of higher-order topological phenomena and offers a versatile platform for investigating novel topological effects in photonics, quantum simulation, and other synthetic quantum systems.