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Global symmetry found to be not completely necessary for the protection of topological boundary states

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Schematic representation classifying disorders in the symmetry-protected topological phase (SPT phase). The overlap region surrounded by red, gray and green lines represents a symmetry protected topological phase with topological invariant and corresponding topological boundary conditions. The gray line is a set of perturbations that affect the boundary conditions, but do not violate the topological invariant of the overall system. The red and green line regions show the limit states protected by their respective associated subsymmetry. The illustrated example shows two sets of perturbations that satisfy the subsymmetry but destroy the topological invariant of the overall system. In this case, the subsymmetry protects the boundary conditions. Credit: Domenico Bongiovanni and co-authors

An international team led by researchers from Nankai University in China and the University of Zagreb in Croatia, together with a team at the Institut national de la recherche scientifique (INRS) in Canada, led by Robert Morandotti, made an important step forward in the study of topological phases. Their findings were recently published in Nature Physics.

In the last decade, topological photonics has attracted increasing attention due to its unique prospects to achieve high-performance light manipulation in terms of robustness and stability.

Discoveries in topological photonics have paved the way for the development of a new generation of photonic devices, such as topological lasers and cavities, with topologically protected states that are immune to disturbances and defects. The concept of topology in physics is inherited from mathematics, where topology is used to study the geometrical properties of an object in relation to quantities that are conserved under continuous deformation.

Two objects are topologically identical when the surface of one can be continuously deformed into the surface of the other and vice versa, e.g. a coffee cup and a torus are equivalent from the point of view of topology. In physics, the concept of topology is used to describe the characteristics of the energy band, which leads to the prediction of new topological states of matter and different topological materials.

Different topological phases (trivial and non-trivial) are distinguished by the appropriate introduction of quantized topological invariants, which enable the establishment of a connection between extensive properties and the appearance of features at the boundary of these materials, known as the bulk-boundary correspondence. In this sense, the most distinctive feature of non-trivial topology is the existence of robust topological boundary conditions protected by specific spatial and/or intrinsic symmetries.

In general, in symmetry-protected topological phase (SPT phase) systems, a close connection between topological boundary conditions, topological invariants, and one or more overall symmetries is believed to be necessary to maintain topological protection against perturbations.

As a consequence, both topological invariants and topological limit states are irreversibly affected by any distortion that breaks the underlying symmetry. In this paper, an international research team has challenged this traditional shared belief, thereby broadening the understanding of SPT border states. They found that even if a system no longer has quantized topological invariants and some kind of global symmetry, topological limit states can still exist in appropriate subspaces, protected by so-called subsymmetries.

“Our discovery challenges the common thinking of a symmetry-protected topological phase in topology and restores the correspondence of topological invariants and limit states,” said Domenico Bongiovanni, one of the principal investigators, a postdoctoral researcher at INRS-EMT. “Our idea has the potential to explain the topological origin of many unconventional states and may find application in a variety of platforms and physical systems.”

The researchers, introducing and exploring the concept of subsymmetry, discovered that global symmetry in the traditional sense is not entirely necessary to protect topological boundary states. In this sense, the topological boundary conditions are preserved as long as the symmetries of the specific subspaces are satisfied, even when the overall topological invariants no longer exist.

The research team cleverly designed and fabricated photonic lattice structures using a cw-laser writing technique to satisfy the conditions of different subspace symmetries. Experiments demonstrated proof of concept with the two most typical topological lattices: one-dimensional SSH and two-dimensional Kagome lattices.

In addition, the team innovatively introduced the concept of long-range symmetry into the Kagome lattice model, which resolves current controversies about the existence and topological protection of higher-order topological states in the Kagome lattice.

This study not only challenges the traditional understanding of symmetry-protected topological states, but also provides new ideas for research and application of topological states in various physical backgrounds. It is expected that this impact of this work will further promote the development of topological photonics and its cutting-edge interdisciplinary fields, as well as the research and development of a new generation of topological photonic devices based on subsymmetry-protected boundary states.

More information:
Ziteng Wang et al, Topological states protected by subsymmetry, Nature Physics (2023). DOI: 10.1038/s41567-023-02011-9

Journal Information:
Nature Physics

Provided by the National Institute for Scientific Research – INRS

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