Global symmetry was found to be incompletely necessary for the protection of topological boundary states

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Schematic illustration classifying perturbations into the symmetry-protected topological phase (SPT phase). The overlaid region surrounded by red, gray, and green lines represents a symmetry-protected topological phase with topological invariants and corresponding topological boundary states. The gray line is a set of perturbations which influence the boundary states but do not break the topological invariant of the overall system. The red and green line areas show boundary states protected by their associated sub-symmetry. The illustrated example shows two sets of perturbations which satisfy the sub-symmetry but destroy the topological invariant of the overall system. In this case, subsymmetry protects the boundary states. Credits: 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 the team from the Institut national de la recherche scientifique (INRS) in Canada, led by Roberto Morandotti, has taken an important step forward in the study of phase topology. Their results were recently published in Physics of nature.

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

Breakthroughs in topological photonics have paved the way for the development of a new generation of photonic devices, such as topological lasers and cavities, characterized by 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 geometric properties of an object related to quantities that are conserved under continuous deformation.

Two objects are topologically identical when the surface of one can be continuously deformed into that of the other and vice versa, for example, a coffee cup and a torus are topologically equivalent. In physics, the concept of topology is employed to describe the characteristics of the energy band, leading to the prediction of new topological states of matter and various topological materials.

The different topological phases (trivial and non-trivial) are distinguished by suitably introducing quantized topological invariants, which allow establishing a link between the mass properties and the emergence of the characteristic at the boundary of these materials, known as mass-boundary correspondence. In this regard, the most distinctive feature of a non-trivial topology is the existence of robust topological boundary states protected by specific spatial and/or intrinsic symmetries.

In general, in symmetry-protected topological phase (SPT phase) systems, it is believed that the close relationship between topological boundary states, topological invariants and one or more overall symmetries is indispensable for maintaining topological protection against perturbations.

Consequently, both topological invariants and topological boundary states are irreparably affected by any bias that breaks the underlying symmetry. In this work, the international research team challenged this traditional common belief, thereby broadening the understanding of SPT boundary states. They found that even if the system no longer has quantized topological invariants and some types of global symmetry, topological boundary states can still exist in the corresponding subspaces, protected by so-called sub-symmetries.

“Our discovery challenges the common thinking of the symmetry-protected topological phase in topology and renews the correspondence between topological invariants and boundary states,” said Domenico Bongiovanni, a principal investigator, postdoctoral researcher at INRS-EMT. “Our idea has the potential to explain the topological origin of many unconventional states and can find application in different physical platforms and systems.”

By introducing and exploring the concept of sub-symmetry, the researchers found that global symmetry in the traditional sense is not entirely necessary for the protection of topological boundary states. In this regard, topological boundary states are preserved as long as the symmetries of specific subspaces are satisfied, even when the overall topological invariants no longer exist.

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

Furthermore, the team innovatively introduced the concept of long-range coupling symmetry into the Kagome lattice model, which resolves current controversies on 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 the research and application of topological states in different physical contexts. This impact of this work is expected to 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 sub-symmetry protected boundary states.

More information:
Ziteng Wang et al, Undersymmetry Protected Topological States, Physics of nature (2023). DOI: 10.1038/s41567-023-02011-9

About the magazine:
Physics of nature

Provided by Institut national de la recherche scientifique – INRS

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