The properties of topological systems are inherently tied to their dimensionality. Indeed, higher-dimensional periodic systems exhibit topological phases not shared by their lower-dimensional counterparts. On the other hand, aperiodic arrays in lower-dimensional systems (e.g., the Harper model) have been successfully employed to emulate higher-dimensional physics. This raises a general question on the possibility of extended topological classification in lower dimensions, and whether the topological invariants of higher-dimensional periodic systems may assume a different meaning in their lower-dimensional aperiodic counterparts. Here, we demonstrate that, indeed, for a topological system in higher dimensions one can construct a one-dimensional (1D) deterministic aperiodic counterpart which retains its spectrum and topological characteristics. We consider a four-dimensional (4D) quantized hexadecapole higher-order topological insulator (HOTI) which supports topological corner modes. We apply the Lanczos transformation and map it onto an equivalent deterministic aperiodic 1D array (DAA) emulating 4D HOTI in 1D. We observe topological zero-energy zero-dimensional (0D) states of the DAA—the direct counterparts of corner states in 4D HOTI and the hallmark of the multipole topological phase, which is meaningless in lower dimensions. To explain this paradox, we show that higher-dimension invariant, the multipole polarization, retains its quantization in the DAA, yet changes its meaning by becoming a nonlocal correlator in the 1D system. By introducing nonlocal topological phases of DAAs, our discovery opens a direction in topological physics. It also unveils opportunities to engineer topological states in aperiodic systems and paves the path to application of resonances associates with such states protected by nonlocal symmetries.
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Topological dissipation in a time-multiplexed photonic resonator network
Topological phases feature robust edge states that are protected against the effects of defects and disorder. These phases have largely been studied in conservatively coupled systems, in which non-trivial topological invariants arise in the energy or frequency bands of a system. Here we show that, in dissipatively coupled systems, non-trivial topological invariants can emerge purely in a system’s dissipation. Using a highly scalable and easily reconfigurable time-multiplexed photonic resonator network, we experimentally demonstrate one- and two-dimensional lattices that host robust topological edge states with isolated dissipation rates, measure a dissipation spectrum that possesses a non-trivial topological invariant, and demonst rate topological protection of the network’s quality factor. The topologically non-trivial dissipation of our system exposes new opportunities to engineer dissipation in both classical and quantum systems. Moreover, our experimental platform’s straightforward scaling to higher dimensions and its ability to implement inhomogeneous, non-reciprocal and long range couplings may enable future work in the study of synthetic dimensions.
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- NSF-PAR ID:
- 10320635
- Date Published:
- Journal Name:
- Nature Physics
- ISSN:
- 1745-2473
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1038/s41567-021-01492-w
