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      Nonlocal topological insulators: Deterministic aperiodic arrays supporting localized topological states protected by nonlocal symmetries

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          Significance

          A concept of nonlocal topological phases of DAAs introduced here establishes a different direction in topological physics and offers approaches to emulate higher-dimensional topology in lower-dimensional systems. Our study also unveils opportunities to engineer topologically protected states in aperiodic systems and paves the path to application of resonances associated with such states, whose robustness is ensured by nonlocal symmetries of DAAs. In particular, the possibility to engineer multiple localized resonances via dimensional reduction and their unique features, such as precise spectral properties stemming from their topological nature, offers remarkable opportunities for practical applications, from robust resonators to sensors and aperiodic topological lasers.

          Abstract

          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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          Most cited references50

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          Colloquium: Topological insulators

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            An iteration method for the solution of the eigenvalue problem of linear differential and integral operators

            C. Lanczos (1950)
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              Topological insulators and superconductors

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                Author and article information

                Journal
                Proc Natl Acad Sci U S A
                Proc Natl Acad Sci U S A
                pnas
                PNAS
                Proceedings of the National Academy of Sciences of the United States of America
                National Academy of Sciences
                0027-8424
                1091-6490
                24 August 2021
                19 August 2021
                19 August 2021
                : 118
                : 34
                Affiliations
                [1] aDepartment of Electrical Engineering, Grove School of Engineering, City College of the City University of New York , New York, NY 10031;
                [2] bPhysics Program, Graduate Center of the City University of New York , New York, NY 10016;
                [3] cPhotonics Initiative, Advanced Science Research Center, City University of New York , New York, NY 10031
                Author notes
                1To whom correspondence may be addressed. Email: akhanikaev@ 123456ccny.cuny.edu .

                Edited by Dmitri Basov, Columbia University, New York, NY, and approved July 6, 2021 (received for review January 12, 2021)

                Author contributions: K.C., M.W., M.L., A.A., and A.B.K. designed research; K.C., M.W., M.L., X.N., and A.B.K. performed research; K.C., M.W., X.N., and A.B.K. analyzed data; and K.C., M.W., A.A., and A.B.K. wrote the paper.

                Article
                202100691
                10.1073/pnas.2100691118
                8403875
                34413190
                79112c7c-9bfd-4772-9658-92031d40ad99
                Copyright © 2021 the Author(s). Published by PNAS.

                This open access article is distributed under Creative Commons Attribution License 4.0 (CC BY).

                Page count
                Pages: 8
                Product
                Funding
                Funded by: Simons Foundation 100000893
                Award ID: 733690
                Award Recipient : Kai Chen Award Recipient : Matthew Weiner Award Recipient : Mengyao Li Award Recipient : Xiang Ni Award Recipient : Andrea Alu Award Recipient : Alexander B. Khanikaev
                Funded by: National Science Foundation (NSF) 100000001
                Award ID: DMR-1809915
                Award Recipient : Kai Chen Award Recipient : Matthew Weiner Award Recipient : Mengyao Li Award Recipient : Alexander B. Khanikaev
                Funded by: DOD | Defense Advanced Research Projects Agency (DARPA) 100000185
                Award ID: HR0011-18-2-0040
                Award Recipient : Kai Chen Award Recipient : Matthew Weiner Award Recipient : Mengyao Li Award Recipient : Xiang Ni Award Recipient : Andrea Alu Award Recipient : Alexander B. Khanikaev
                Categories
                426
                Physical Sciences
                Physics

                topological materials,aperiodic systems,higher-order topology

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