Topological transition enabled by composite symmetry-breaking paths in trefoil-knot honeycomb lattices

doi:10.1007/s10483-026-3364-6

Abstract

Abstract:

Topological phases are governed by lattice symmetries, yet how different symmetry-breaking paths (SBPs) affect topological transitions remains insufficiently understood. Most existing studies rely on a single SBP, and address only one bandgap, limiting independent control of multiple gaps. Here, we investigate multiple isolated Dirac points in a trefoil-knot-modified honeycomb lattice, and show that a single SBP generally inverts all relevant Dirac points simultaneously, whereas the tailored combinations of SBPs enable selective and programmable band inversion at targeted gaps. The excitation-dependent responses reveal strong modal selectivity. This capability is exploited to realize independently controllable multi-channel signal splitting, which is unattainable with a single SBP. The results enable SBPs as an effective design degree of freedom for programmable and reconfigurable topological elastic devices.

Key words: symmetry-breaking path (SBP), honeycomb-like metastructure, topological metamaterial

2010 MSC Number:

O347.4

Tai REN, Xiuhui HOU, Tingting WANG, Zhiwei ZHU, Kai ZHANG, Zichen DENG. Topological transition enabled by composite symmetry-breaking paths in trefoil-knot honeycomb lattices. Applied Mathematics and Mechanics (English Edition), 2026, 47(3): 497-508.

TrendMD

Figures/Tables 8

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

References 33

[1]	THOULESS, D. J., KOHMOTO, M., NIGHTINGALE, M. P., and DEN, NIJS. M. Quantized Hall conductance in a two-dimensional periodic potential. Physical Review Letters, 49, 405–408 (1982)
[2]	KANE, C. L. and MELE, E. J. Z₂ topological order and the quantum spin Hall effect. Physical Review Letters, 95, 146802 (2005)
[3]	BERNEVIG, B. A., HUGHES, T. L., and ZHANG, S. C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science, 314, 1757–1761 (2006)
[4]	WANG, Z., CHONG, Y. D., JOANNOPOULOS, J. D., and SOLJACIC, M. Observation of unidirectional backscattering-immune topological electromagnetic states. nature, 461, 772–720 (2009)
[5]	MITTAL, S., FAN, J., FAEZ, S., MIGDALL, A., TAYLOR, J. M., and HAFEZI, M. Topologically robust transport of photons in a synthetic gauge field. Physical Review Letters, 113, 087403 (2014)
[6]	KHANIKAEV, A. B., FLEURY, R., MOUSAVI, S. H., and ALÙ, A. Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice. Nature Communications, 6, 8260 (2015)
[7]	LI, S. F., KEVREKIDIS, P. G., and YANG, J. Emergence of elastic chiral Landau levels and snake states. Physical Review B, 109, 184109 (2024)
[8]	NASH, L. M., KLECKNER, D., READ, A., VITELLI, V., TURNER, A. M., and IRVINE, W. T. M. Topological mechanics of gyroscopic metamaterials. Proceedings of the National Academy of Sciences of the United States of America, 112, 14495–14500 (2015)
[9]	CHAUNSALI, R., CHEN, C. W., and YANG, J. Y. Subwavelength and directional control of flexural waves in zone-folding induced topological plates. Physical Review B, 97, 054307 (2018)
[10]	MOUSAVI, S. H., KHANIKAEV, A. B., and WANG, Z. Topologically protected elastic waves in phononic metamaterials. Nature Communications, 6, 8682 (2015)
[11]	MINIACI, M., PAL, R. K., MORVAN, B., and RUZZENE, M. Experimental observation of topologically protected helical edge modes in patterned elastic plates. Physical Review X, 8, 031074 (2018)
[12]	YU, S. Y., HE, C., WANG, Z., LIU, F. K., SUN, X. C., LI, Z., LU, H. Z., LU, M. H., LIU, X. P., and CHEN, Y. F. Elastic pseudospin transport for integratable topological phononic circuits. Nature Communications, 9, 3072 (2018)
[13]	HE, C., NI, X., GE, H., SUN, X. C., CHEN, Y. B., LU, M. H., LIU, X. P., and CHEN, Y. F. Acoustic topological insulator and robust one-way sound transport. Nature Physics, 12, 1124–1129 (2016)
[14]	ZHANG, Q., CHEN, Y., ZHANG, K., and HU, G. K. Dirac degeneracy and elastic topological valley modes induced by local resonant states. Physical Review B, 101, 014101 (2020)
[15]	TIAN, Z. H., SHEN, C., LI, J. F., REIT, E., BACHMAN, H., SOCOLAR, J. E. S., CUMMER, S. A., and HUANG, T. J. Dispersion tuning and route reconfiguration of acoustic waves in valley topological phononic crystals. Nature Communications, 11, 762 (2020)
[16]	YAO, Y. X., MA, Y. S., HONG, F., ZHANG, K., WANG, T. T., PENG, H. J., and DENG, Z. C. On Klein tunneling of low-frequency elastic waves in hexagonal topological plates. Applied Mathematics and Mechanics (English Edition), 45(7), 1139–1154 (2024) https://doi.org/10.1007/s10483-024-3163-9
[17]	WANG, L. Y., JIAN, S. K., and YAO, H. Hourglass semimetals with nonsymmorphic symmetries in three dimensions. Physical Review B, 96, 075110 (2017)
[18]	LI, W. Q., LI, Z. H., PAN, B. R., ZHOU, P., and SUN, L. Z. Symmetry-enforced 2D hourglass phononic nodal net. Physica Status Solidi — Rapid Research Letters, 17, 2300048 (2023)
[19]	CHENG, H. B., SHA, Y. X., LIU, R. J., FANG, C., and LU, L. Discovering topological surface states of Dirac points. Physical Review Letters, 124, 104301 (2020)
[20]	XIE, B. Y., LIU, H., CHENG, H., LIU, Z. Y., TIAN, J. G., and CHEN, S. Q. Dirac points and the transition towards Weyl points in three-dimensional sonic crystals. Light: Science & Applications, 9, 201 (2020)
[21]	WEI, Q., ZHANG, X. W., DENG, W. Y., LU, J. Y., HUANG, X. Q., YAN, M., CHEN, G., LIU, Z. Y., and JIA, S. T. Higher-order topological semimetal in acoustic crystals. Nature Materials, 20, 812–817 (2021)
[22]	DENG, W. Y., LU, J. Y., LI, F., HUANG, X. Q., YAN, M., MA, J. H., and LIU, Z. Y. Nodal rings and drumhead surface states in phononic crystals. Nature Communications, 10, 1769 (2019)
[23]	LIU, F., KONG, P., WANG, X. Y., WU, J. E., HE, Z. J., and DENG, K. Valley Hall effect in a bilayer graphene phononic crystal without localized Berry curvature. Physical Review B, 111, 094303 (2025)
[24]	LU, J. Y., QIU, C. Y., YE, L. P., FAN, X. Y., KE, M. Z., ZHANG, F., and LIU, Z. Y. Observation of topological valley transport of sound in sonic crystals. Nature Physics, 13, 369–374 (2017)
[25]	HUANG, X. Q., DENG, W. Y., LI, F., LU, J. Y., and LIU, Z. Y. Ideal type-II Weyl phase and topological transition in phononic crystals. Physical Review Letters, 124, 206802 (2020)
[26]	NARANG, P., GARCIA, C. A. C., and FELSER, C. The topology of electronic band structures. Nature Materials, 20, 293–300 (2021)
[27]	FU, L. and KANE, C. L. Topological insulators with inversion symmetry. Physical Review B, 76, 045302 (2007)
[28]	LIU, H. Multiband pure topological states in elastic structures. Frontiers in Physics, 10, 909820 (2022)
[29]	FUKUI, T., HATSUGAI, Y., and SUZUKI, H. Chern numbers in discretized Brillouin zone: efficient method of computing (spin) Hall conductances. Journal of the Physical Society of Japan, 74, 1674–1677 (2005)
[30]	ZHAO, X. L., MA, F. J., GUO, P. J., and LU, Z. Y. Two-dimensional quadratic double Weyl semimetal. Physical Review Research, 4, 043183 (2022)
[31]	HONG, F., ZHANG, K., QI, L. Y., DING, B., and DENG, Z. C. High-frequency topological corner and edge states in elastic honeycomb plates. International Journal of Mechanical Sciences, 246, 108141 (2023)
[32]	FAN, H. Y., XIA, B. Z., TONG, L., ZHENG, S. J., and YU, D. J. Elastic higher-order topological insulator with topologically protected corner states. Physical Review Letters, 122, 204301 (2019)
[33]	CHENG, S. G., LIU, H. W., JIANG, H., SUN, Q. F., and XIE, X. C. Manipulation and characterization of the valley-polarized topological kink states in graphene-based interferometers. Physical Review Letters, 121, 156801 (2018)