Topological insulators are a new phase of matter with the distinctive characteristics of an insulating bulk and conducting edge states. Recent theories indicate there even exist topological edge states in the fractal-dimensional lattices, which are fundamentally different from the current studies that rely on the integer dimensions. Here, we propose and experimentally demonstrate the squeezed Chern insulator in a fractal-dimensional acoustic lattice. First, through calculating the topological invariant of our topological fractal system, we find the topological phase diagram is squeezed by about 0.54 times, compared with that of the original Haldane model. Then by introducing synthetic gauge flux into an acoustic fractal lattice, we experimentally observe the one-way edge states that are protected by a robust mobility gap within the squeezed topological regimes. Our work demonstrates the first example of acoustic topological fractal insulators and provides new directions for the advanced control of sound waves.
Higher-order topological insulators, which support lower-dimensional topological boundary states than the first-order topological insulators, have been intensely investigated in the integer dimensional systems. Here, we provide a new paradigm by presenting experimentally a higher-order topological phase in a fractal-dimensional system. Through applying the Benalcazar, Bernevig, and Hughes model into a Sierpinski carpet fractal lattice, we uncover a squeezed higher-order phase diagram featuring the abundant corner states, which consist of zero-dimensional outer corner states and 1.89-dimensional inner corner states. As a result, the codimension is now 1.89 and our model can be classified into the fractional-order topological insulators. The non-zero fractional charges at the outer/inner corners indicate that all corner states in the fractal system are indeed topologically nontrivial. Finally, in a fabricated acoustic fractal lattice, we experimentally observe the outer/inner corner states with local acoustic measurements. Our work demonstrates a higher-order topological phase in an acoustic fractal lattice and may pave the way to the fractional-order topological insulators.
We propose and experimentally demonstrate the first example of Chern fractal insulators in an acoustic Sierpinski lattice. Through introducing fractality into the well-known Haldane model, we find that the topological phase diagram is significantly squeezed by about 0.5 times, compared with that of the original Haldane model. The energy spectra for different generations of a fractal lattice exhibits a hierarchy of self-similar energy bands in the form of a devil's staircase. With local acoustic measurements, we fabricate the sonic sample based on the theoretical model and experimentally confirm the fractality-induced squeezing and observe the one-way edge states that are protected by a robust mobility gap. Our work demonstrates the fundamental interplay between the fractality and topological Haldane insulator, and may provide new directions for the advanced control of sound waves.
We demonstrate non-Hermitian topological phase transitions induced solely by imaginary disorder. Starting from the trivial phase in the well-known Benalcazar, Bernevig, and Hughes model, we find that adding sufficient imaginary disorder to the trivial bulk can lead to a higher-order topological insulator supporting topological corner-localized states. In experiments, we elaborately design a two-dimensional reconfigurable acoustic lattice with a loss configuration that can be randomly set. By increasing the strength of lossy disorder, we observe the transitions from a trivial to a higher-order topological phase and subsequently to a gapless phase. To further show that imaginary disorder offers an alternative degree of freedom for controlling topological phases, we propose a non-Hermitian topological disclination state induced by imaginary disorder. Our work demonstrates an example of a non-Hermitian higher-order topological Anderson insulator and offers a reconfigurable platform to study the interplay between non-Hermitian disorder and topological phases.
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