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Diffusive topological transport in spatiotemporal thermal lattices

Nature Physics volume 18, pages 450–456 (2022)Cite this article

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Abstract

Topological phases have been studied in photonic, acoustic and phononic metamaterials, promising a range of applications. Such topological modes usually stem from collective resonant effects in periodic lattices. One may, therefore, expect similar features to be forbidden for thermal diffusion that is purely dissipative and mostly incoherent, prohibiting collective resonances. Here we report the discovery of diffusion-based topological states supported by spatiotemporally modulated advections stacked over a fluidic surface. This arrangement imitates a periodic propagating potential in an effective thermal lattice. We observe edge states in topologically non-trivial and bulk states in topologically trivial lattices. Interface states form at boundaries between these two types of lattice, manifesting inhomogeneous thermal properties on the fluidic surface. Our findings establish a framework for topological diffusion and thermal edge or bulk states, and it may allow a distinct mechanism for the flexible manipulation of diffusive phenomena for robust heat and mass transfer.

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Fig. 1: Topological transport in thermal diffusion.
Fig. 2: Measured results for topologically non-trivial and trivial responses in the gapped phase.
Fig. 3: Measured temperature profiles and effective thermal resistances for Case I.
Fig. 4: Measured temperature profiles and effective thermal resistances for Case II.

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Source data are provided with this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

References

  1. Yang, Y. et al. Realization of a three-dimensional photonic topological insulator. Nature 565, 622–626 (2019).

    Article  ADS  Google Scholar 

  2. Lustig, E. et al. Photonic topological insulator in synthetic dimensions. Nature 567, 356–360 (2019).

    Article  ADS  Google Scholar 

  3. Zilberberg, O. et al. Photonic topological boundary pumping as a probe of 4D quantum Hall physics. Nature 553, 59–62 (2018).

    Article  ADS  Google Scholar 

  4. Liu, Y. et al. Bulk–disclination correspondence in topological crystalline insulators. Nature 589, 381–385 (2021).

    Article  ADS  Google Scholar 

  5. Stützer, S. et al. Photonic topological Anderson insulators. Nature 560, 461–465 (2018).

    Article  ADS  Google Scholar 

  6. Shen, H., Zhen, B. & Fu, L. Topological band theory for non-Hermitian Hamiltonians. Phys. Rev. Lett. 120, 146402 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  7. Ghatak, A. & Das, T. New topological invariants in non-Hermitian systems. J. Phys.: Condens. Matter 31, 263001 (2019).

    ADS  Google Scholar 

  8. Feng, L., El-Ganainy, R. & Ge, L. Non-Hermitian photonics based on parity–time symmetry. Nat. Photon. 11, 752–762 (2017).

    Article  ADS  Google Scholar 

  9. Wang, K. et al. Generating arbitrary topological windings of a non-Hermitian band. Science 371, 1240–1245 (2021).

    Article  ADS  Google Scholar 

  10. Khanikaev, A. B. et al. Photonic topological insulators. Nat. Mater. 12, 233–239 (2013).

    Article  Google Scholar 

  11. Chen, Z. & Segev, M. Highlighting photonics: looking into the next decade. eLight 1, 2 (2021).

    Article  Google Scholar 

  12. Hokmabadi, M. P., Schumer, A., Christodoulides, D. N. & Khajavikhan, M. Non-Hermitian ring laser gyroscopes with enhanced Sagnac sensitivity. Nature 576, 70–74 (2019).

    Article  ADS  Google Scholar 

  13. Hadad, Y., Soric, J. C., Khanikaev, A. B. & Alu, A. Self-induced topological protection in nonlinear circuit arrays. Nat. Electron. 1, 178–182 (2018).

    Article  Google Scholar 

  14. Leykam, D., Bliokh, K. Y., Huang, C., Chong, Y. D. & Nori, F. Edge modes, degeneracies, and topological numbers in non-Hermitian systems. Phys. Rev. Lett. 118, 040401 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  15. Xu, H., Mason, D., Jiang, L. & Harris, J. G. E. Topological energy transfer in an optomechanical system with exceptional points. Nature 537, 80–83 (2016).

    Article  ADS  Google Scholar 

  16. Doppler, J. et al. Dynamically encircling an exceptional point for asymmetric mode switching. Nature 537, 76–79 (2016).

    Article  ADS  Google Scholar 

  17. Lin, Z. et al. Unidirectional invisibility induced by \(\mathcal {PT}\)-symmetric periodic structures. Phys. Rev. Lett. 106, 213901 (2011).

    Article  ADS  Google Scholar 

  18. Peng, B. et al. Parity-time-symmetric whispering-gallery microcavities. Nat. Phys. 10, 394–398 (2014).

    Article  Google Scholar 

  19. Peng, B. et al. Loss-induced suppression and revival of lasing. Science 346, 328–332 (2014).

    Google Scholar 

  20. Zhao, H. et al. Topological hybrid silicon microlasers. Nat. Commun. 9, 981 (2018).

    Article  ADS  Google Scholar 

  21. Zhao, H. et al. Non-Hermitian topological light steering. Science 365, 1163–1166 (2019).

    Article  ADS  Google Scholar 

  22. Zeng, Y. et al. Electrically pumped topological laser with valley edge modes. Nature 578, 246–250 (2020).

    Article  ADS  Google Scholar 

  23. Bandres, M. A. et al. Topological insulator laser: experiments. Science 359, eaar4005 (2018).

    Article  Google Scholar 

  24. Harari, G. et al. Topological insulator laser: theory. Science 359, eaar4003 (2018).

    Article  Google Scholar 

  25. Choi, J. H. et al. Room temperature electrically pumped topological insulator lasers. Nat. Commun. 12, 3434 (2021).

    Article  ADS  Google Scholar 

  26. Bahari, B. et al. Nonreciprocal lasing in topological cavities of arbitrary geometries. Science 358, 636–640 (2017).

    Article  ADS  Google Scholar 

  27. Liang, G. Q. & Chong, Y. D. Optical resonator analog of a two-dimensional topological insulator. Phys. Rev. Lett. 110, 203904 (2013).

    Article  ADS  Google Scholar 

  28. Takata, K. & Notomi, M. Photonic topological insulating phase induced solely by gain and loss. Phys. Rev. Lett. 121, 213902 (2018).

    Article  ADS  Google Scholar 

  29. Li, M., Ni, X., Weiner, M., Alù, A. & Khanikaev, A. B. Topological phases and nonreciprocal edge states in non-Hermitian Floquet insulators. Phys. Rev. B 100, 45423 (2019).

    Article  ADS  Google Scholar 

  30. Liertzer, M. et al. Pump-induced exceptional points in lasers. Phys. Rev. Lett. 108, 173901 (2012).

    Article  ADS  Google Scholar 

  31. Feng, L., Wong, Z. J., Ma, R. M., Wang, Y. & Zhang, X. Single-mode laser by parity-time symmetry breaking. Science 346, 972–975 (2014).

    Article  Google Scholar 

  32. Yoon, J. W. et al. Time-asymmetric loop around an exceptional point over the full optical communications band. Nature 562, 86–90 (2018).

    Article  ADS  Google Scholar 

  33. Xia, S. et al. Nontrivial coupling of light into a defect: the interplay of nonlinearity and topology. Light Sci. Appl. 9, 147 (2020).

    Article  ADS  Google Scholar 

  34. Xia, S. et al. Nonlinear tuning of PT symmetry and non-Hermitian topological states. Science 372, 72–76 (2021).

    Article  ADS  Google Scholar 

  35. Liu, S. et al. Gain- and loss-induced topological insulating phase in a non-Hermitian electrical circuit. Phys. Rev. Appl. 13, 014047 (2020).

    Article  ADS  Google Scholar 

  36. Gao, H. et al. Non-Hermitian route to higher-order topology in an acoustic crystal. Nat. Commun. 12, 1888 (2021).

    Article  ADS  Google Scholar 

  37. Li, Y. et al. Anti–parity-time symmetry in diffusive systems. Science 364, 170–173 (2019).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Fan, C. Z., Gao, Y. & Huang, J. P. Shaped graded materials with an apparent negative thermal conductivity. Appl. Phys. Lett. 92, 251907 (2008).

    Article  ADS  Google Scholar 

  39. Han, T. et al. Experimental demonstration of a bilayer thermal cloak. Phys. Rev. Lett. 112, 054302 (2014).

    Article  ADS  Google Scholar 

  40. Zhou, X., Xu, G. & Zhang, H. Y. Binary masses manipulation with composite bilayer metamaterial. Compos. Struct. 267, 113866 (2021).

    Article  Google Scholar 

  41. Zhou, X. & Xu, G. Self-adaptive field manipulation with thermal logic material. Int. J. Heat Mass Trans. 172, 121147 (2021).

    Article  Google Scholar 

  42. Xu, G. et al. Tunable analog thermal material. Nat. Commun. 11, 6028 (2020).

    Article  ADS  Google Scholar 

  43. Torrent, D., Poncelet, O. & Batsale, J.-C. Nonreciprocal thermal material by spatiotemporal modulation. Phys. Rev. Lett. 120, 125501 (2018).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

C.-W.Q. acknowledges financial support from the Ministry of Education, Republic of Singapore (grant no. R-263-000-E19-114). A.A. acknowledges financial support from the Office of Naval Research with grant no. N00014-19-1-2011, the Vannevar Bush Faculty Fellowship, the Air Force Office of Scientific Research with MURI grant no. FA9550-18-1-0379 and the Simons Foundation. Y.Y. and H.C. acknowledge the National Natural Science Foundation of China (NNSFC) under grant nos. 61625502, 61975176, 11961141010 and 62175215; the Top-Notch Young Talents Program of China; and the Fundamental Research Funds for the Central Universities. X.Z. acknowledges financial support from the Chongqing Natural Science Foundation (grant no. cstc2021jcyj-msxmX0627) and the Science and Technology Research Program of Chongqing Municipal Education Commission (grant no. KJQN202000829).

Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, National University of Singapore, Kent Ridge, Singapore

    Guoqiang Xu & Cheng-Wei Qiu

  2. Interdisciplinary Center for Quantum Information, State Key Laboratory of Modern Optical Instrumentation, College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou, China

    Yihao Yang & Hongsheng Chen

  3. ZJU-Hangzhou Global Science and Technology Innovation Center, Key Laboratory of Advanced Micro/Nano Electronic Devices and Smart Systems of Zhejiang, Zhejiang University, Hangzhou, China

    Yihao Yang & Hongsheng Chen

  4. International Joint Innovation Center, ZJU-UIUC Institute, The Electromagnetics Academy at Zhejiang University, Zhejiang University, Haining, China

    Yihao Yang & Hongsheng Chen

  5. School of Computer Science and Information Engineering, Chongqing Technology and Business University, Chongqing, China

    Xue Zhou

  6. Photonics Initiative, Advanced Science Research Center, City University of New York, New York, NY, USA

    Andrea Alù

  7. Physics Program, Graduate Center of the City University of New York, New York, NY, USA

    Andrea Alù

Authors
  1. Guoqiang Xu
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Contributions

G.X. and C.-W.Q. conceived the idea. G.X., X.Z. and C.-W.Q. proposed the methodology. G.X., Y.Y., H.C., A.A. and C.-W.Q. performed the theoretical derivations. G.X. and X.Z. implemented the experimental investigations. G.X., Y.Y., X.Z. and C.-W.Q. made the visualizations. G.X., Y.Y., H.C., A.A. and C.-W.Q. performed the theoretical analysis and wrote the manuscript. C.-W.Q. supervised the work. All the authors contributed to the discussion and to finalizing the manuscript.

Corresponding author

Correspondence to Cheng-Wei Qiu.

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Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks Muamer Kadic and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 The entire structure and experimental setups.

a indicates the entire structure for implementing advections (effective oscillations) on the fluid surface, a constraint shell is adopted outside the fluid surface to keep the concentric rotations for each unit and the internal gear. During the measurements, this constraint shell is removed to directly capture the temperature-related data of the fluid surface. b and c present the fabricated dynamic components and their combinations consisting of 40 internal gear sets for modulating the fluid surface. The azimuthal direction of the units is marked in (b). The black dashed border in (c) indicates the region for IR images when fluid is coated on these dynamic components, while the interval of two neighboring deriving gears on a transmission shaft is 4a. The black dot marked with ‘1’ indicates the first unit of the entire system at the initial location of z = 0. d provides the real experiment setups, and the left upper inset presents the general engagement of the internal gear set for one unit cell. The colored dots indicate the corresponding motors.

Supplementary information

Supplementary Information

Supplementary Notes 1–4 and Figs. 1–5.

Supplementary Video 1

Temperature profiles of the case with non-trivial lattices.

Supplementary Video 2

Temperature profiles of the case with trivial lattices.

Supplementary Video 3

Temperature profiles of manipulated Case I.

Supplementary Video 4

Temperature profiles of manipulated Case II.

Supplementary Video 5

Illustrations of the mechanism for dynamic control.

Source data

Source Data Fig. 1

Source data for Fig. 1d.

Source Data Fig. 2

Source data for Fig. 2a–h.

Source Data Fig. 3

Source data for Fig. 3b–e.

Source Data Fig. 4

Source data for Fig. 4b–e.

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Xu, G., Yang, Y., Zhou, X. et al. Diffusive topological transport in spatiotemporal thermal lattices. Nat. Phys. 18, 450–456 (2022). https://doi.org/10.1038/s41567-021-01493-9

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  • DOI: https://doi.org/10.1038/s41567-021-01493-9

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