Hidden multi-topological phases beyond conventional topological theory

Topological physics is a major research topic in modern science. The discovery of topological phases of matter has revolutionized how researchers think about electronic, photonic, and acoustic materials. In these materials, the global properties of the bulk band structure govern the existence of robust boundary states, including edge, surface, hinge, and corner states. Topologically-protected boundary states can find applications in a variety of photonic technologies and devices, particularly topological resonators and lasers. The intimate link between the bulk and the boundary is encoded in a quantized topological invariant and is known as the bulk–boundary correspondence (BBC). However, recent studies have revealed intriguing situations where conventional band topology is unable to predict certain boundary states. These scenarios have motivated the search for new theoretical frameworks that go beyond the standard topological band theory and explore new frontiers in topological phases.

 

In a new paper published in eLight, a team of scientists led by Professor Hrvoje Buljan and Professor Zhigang Chen from Nankai University and University of Zagreb has introduced a new class of topological phases of matter, termed multi-topological phases (MTPs), which offer an avenue for understanding physical phenomena that are not explicable with conventional band topology. Unlike conventional topological phases, which are characterized by a single invariant, an MTP is described by distinct, multiple topological invariants. Each invariant is quantized and linked to its own class of boundary states, so that a single material with MTP can host at the same time multiple sets of robust boundary modes. The BBC of an MTP is the association between each topological invariant and its corresponding set of boundary states.

 

Physically, MTPs emerge in lattice systems where the couplings across different unit cells obey a specific restriction, i.e., constrained inter-cell coupling. The researchers in this study prove that for a periodic lattice with constrained inter-cell coupling, one can associate one or more auxiliary Hamiltonians. These auxiliary Hamiltonians can have different symmetries and band structures compared with the original system, but they support the same boundary states as the original system. The topological protection of the boundary states thus arises from the topologies of the auxiliary Hamiltonians, which are otherwise hidden in the original system. The original system can even be topologically trivial according to conventional band topology; as long as the related auxiliary Hamiltonians are topological non-trivial, the boundary states can exist and enjoy topological protection. Because several auxiliary Hamiltonians can be defined for a given lattice, an MTP can support multiple sets of boundary states, where every set of boundary states is associated with one distinct topological invariant characterizing only one of the auxiliary Hamiltonians.

 

As a proof of principle, the MTP framework is applied to three different classes of materials: a one-dimensional (1D) topological insulator, a two-dimensional (2D) higher-order topological insulator, and a 2D indirectly gapped Chern insulator. In the first two models, a conventional topological phase cannot be defined due to the absence of protecting symmetries. In the third case, although a quantized Chern number can be defined via the conventional theory, it still fails to correctly predict the existence of boundary states. The researchers show that, in all three examples, MTP theory accurately predicts boundary states, thereby establishing a new form of BBC beyond conventional topological band theory. In addition, they experimentally realize the MTPs of the former two models in laser-written photonic lattices and observe the related topological boundary states.

 

The discovery of the MTPs represents a significant advance in the fundamental understanding of topological physics and uncovers a new pathway for designing  topological photonic materials with unconventional properties, which may inspire wide-ranging applications across various disciplines.

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