Reentrant localization transitions in a topological Anderson insulator: A study of a generalized Su−Schrieffer−Heeger quasicrystal

Abstract

We study the topology and localization properties of a generalized Su−Schrieffer−Heeger (SSH) model with a quasi-periodic modulated hopping. It is found that the interplay of off-diagonal quasi-periodic modulations can induce topological Anderson insulator (TAI) phases and reentrant topological Anderson insulator (RTAI), and the topological phase boundaries can be uncovered by the divergence of the localization length of the zero-energy mode. In contrast to the conventional case that the TAI regime emerges in a finite range with the increase of disorder, the TAI and RTAI are robust against arbitrary modulation amplitude for our system. Furthermore, we find that the TAI and RTAI can induce the emergence of reentrant localization transitions. Such an interesting connection between the reentrant localization transition and the TAI/RTAI can be detected from the wave-packet dynamics in cold atom systems by adopting the technique of momentum-lattice engineering.

Graphical abstract

Keywords

topological Anderson insulator / localization / quasi-periodic systems

References

[1] P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109(5), 1492 (1958)

[2] P. A. Lee and T. V. Ramakrishnan, Disordered electronic systems, Rev. Mod. Phys. 57(2), 287 (1985)

[3] J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clément, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Direct observation of Anderson localization of matter waves in a controlled disorder, Nature 453(7197), 891 (2008)

[4] G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Anderson localization of a non-interacting Bose–Einstein condensate, Nature 453(7197), 895 (2008)

[5] A. A. Chabanov, M. Stoytchev, and A. Z. Genack, Statistical signatures of photon localization, Nature 404(6780), 850 (2000)

[6] P. Pradhan and S. Sridhar, Correlations due to localization in quantum eigenfunctions of disordered microwave cavities, Phys. Rev. Lett. 85(11), 2360 (2000)

[7] Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, Observation of a localization transition in quasiperiodic photonic lattices, Phys. Rev. Lett. 103(1), 013901 (2009)

[8] N. Mott, The mobility edge since 1967, J. Phys. C 20(21), 3075 (1987)

[9] S. Aubry and G. André, Analyticity breaking and Anderson localization in incommensurate lattices, Ann. Isr. Phys. Soc. 3, 133 (1980)

[10] P. G. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proc. Phys. Soc. A 68(10), 874 (1955)

[11] S. Longhi, Topological phase transition in non-Hermitian quasicrystals, Phys. Rev. Lett. 122(23), 237601 (2019)

[12] S. Longhi, Metal−insulator phase transition in a non-Hermitian Aubry−André−Harper model, Phys. Rev. B 100(12), 125157 (2019)

[13] Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Topological states and adiabatic pumping in quasicrystals, Phys. Rev. Lett. 109(10), 106402 (2012)

[14] M. Segev, Y. Silberberg, and D. N. Christodoulides, Anderson localization of light, Nat. Photonics 7(3), 197 (2013)

[15] J. Biddle and S. Das Sarma, Predicted mobility edges in one-dimensional incommensurate optical lattices: An exactly solvable model of Anderson localization, Phys. Rev. Lett. 104(7), 070601 (2010)

[16] S. Ganeshan, J. H. Pixley, and S. Das Sarma, Nearest neighbor tight binding models with an exact mobility edge in one dimension, Phys. Rev. Lett. 114(14), 146601 (2015)

[17] Y. Wang, L. Zhang, W. Sun, T. F. J. Poon, and X. J. Liu, Quantum phase with coexisting localized, extended, and critical zones, Phys. Rev. B 106(14), L140203 (2022)

[18] X.C. ZhouY. WangT.F. J. PoonQ.ZhouX.J. Liu, Exact new mobility edges between critical and localized states, arXiv: 2022)

[19] T. Liu, X. Xia, S. Longhi, and L. Sanchez-Palencia, Anomalous mobility edges in one-dimensional quasiperiodic models, SciPost Phys. 12, 027 (2022)

[20] W. Chen, S. Cheng, J. Lin, R. Asgari, and X. L. Gao, Breakdown of the correspondence between the real-complex and delocalization−localization transitions in non-Hermitian quasicrystals, Phys. Rev. B 106(14), 144208 (2022)

[21] D. Peng, S. Cheng, and X. L. Gao, Power law hopping of single particles in one-dimensional non-Hermitian quasicrystals, Phys. Rev. B 107(17), 174205 (2023)

[22] S. Roy, T. Mishra, B. Tanatar, and S. Basu, Reentrant localization transition in a quasiperiodic chain, Phys. Rev. Lett. 126(10), 106803 (2021)

[23] A. Padhan, M. K. Giri, S. Mondal, and T. Mishra, Emergence of multiple localization transitions in a one-dimensional quasiperiodic lattice, Phys. Rev. B 105(22), L220201 (2022)

[24] Z. W. Zuo and D. Kang, Reentrant localization transition in the Su-Schrieffer-Heeger model with random-dimer disorder, Phys. Rev. A 106(1), 013305 (2022)

[25] S. Aditya, K. Sengupta, and D. Sen, Periodically driven model with quasiperiodic potential and staggered hopping amplitudes: Engineering of mobility gaps and multifractal states, Phys. Rev. B 107(3), 035402 (2023)

[26] S. Roy, S. Chattopadhyay, T. Mishra, and S. Basu, Critical analysis of the reentrant localization transition in a one-dimensional dimerized quasiperiodic lattice, Phys. Rev. B 105(21), 214203 (2022)

[27] R. Qi, J. Cao, and X. P. Jiang, Multiple localization transitions and novel quantum phases induced by a staggered on-site potential, Phys. Rev. B 107(22), 224201 (2023)

[28] R.QiJ.Cao X.P. Jiang, Localization and mobility edges in non-Hermitian disorder-free lattices, arXiv: 2023)

[29] P.S. NairD. JoyS.Sanyal, Emergent scale and anomalous dynamics in certain quasi-periodic systems, arXiv: 2023)

[30] S.Z. LiZ. Li, The multiple re-entrant localization in a phase-shift quasiperiodic chain, arXiv: 2023)

[31] S.Z. LiZ. Li, Emergent recurrent extension phase transition in a quasiperiodic chain, arXiv: 2023)

[32] C. Wu, J. Fan, G. Chen, and S. Jia, Non-Hermiticity-induced reentrant localization in a quasiperiodic lattice, New J. Phys. 23(12), 123048 (2021)

[33] X. P. Jiang, Y. Qiao, and J. P. Cao, Mobility edges and reentrant localization in one-dimensional dimerized non-Hermitian quasiperiodic lattice, Chin. Phys. B 30(9), 097202 (2021)

[34] L. Zhou and W. Han, Driving-induced multiple PT-symmetry breaking transitions and reentrant localization transitions in non-Hermitian Floquet quasicrystals, Phys. Rev. B 106(5), 054307 (2022)

[35] W. Han and L. Zhou, Dimerization-induced mobility edges and multiple reentrant localization transitions in non-Hermitian quasicrystals, Phys. Rev. B 105(5), 054204 (2022)

[36] H. Wang, X. Zheng, J. Chen, L. Xiao, S. Jia, and L. Zhang, Fate of the reentrant localization phenomenon in the one-dimensional dimerized quasiperiodic chain with long-range hopping, Phys. Rev. B 107(7), 075128 (2023)

[37] S. Nakajima, N. Takei, K. Sakuma, Y. Kuno, P. Marra, and Y. Takahashi, Competition and interplay between topology and quasi-periodic disorder in Thouless pumping of ultracold atoms, Nat. Phys. 17(7), 844 (2021)

[38] S. Cheng, H. Yin, Z. Lu, C. He, P. Wang, and X. L. Gao, Predicting large-Chern-number phases in a shaken optical dice lattice, Phys. Rev. A 101(4), 043620 (2020)

[39] S. Cheng and X. L. Gao, Topological Floquet bands in a circularly shaken dice lattice, Phys. Rev. Res. 4(3), 033194 (2022)

[40] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49(6), 405 (1982)

[41] M. Z. Hasan and C. L. Kane, Topological insulators, Rev. Mod. Phys. 82(4), 3045 (2010)

[42] X. L. Qi and S. C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83(4), 1057 (2011)

[43] A. Bansil, H. Lin, and T. Das, Topological band theory, Rev. Mod. Phys. 88(2), 021004 (2016)

[44] C. K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88(3), 035005 (2016)

[45] N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90(1), 015001 (2018)

[46] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in polyacetylene, Phys. Rev. Lett. 42(25), 1698 (1979)

[47] F. Song, S. Yao, and Z. Wang, Non-Hermitian topological invariants in real space, Phys. Rev. Lett. 123(24), 246801 (2019)

[48] W. Song, W. Sun, C. Chen, Q. Song, S. Xiao, S. Zhu, and T. Li, Breakup and recovery of topological zero modes in finite non-Hermitian optical lattices, Phys. Rev. Lett. 123(16), 165701 (2019)

[49] Z. Xu, R. Zhang, S. Chen, L. Fu, and Y. Zhang, Fate of zero modes in a finite Su-Schrieffer-Heeger model with PT symmetry, Phys. Rev. A 101(1), 013635 (2020)

[50] T. Xiao, D. Xie, Z. Dong, T. Chen, W. Yi, and B. Yan, Observation of topological phase with critical localization in a quasi-periodic lattice, Sci. Bull. (Beijing) 66(21), 2175 (2021)

[51] E. Prodan, T. L. Hughes, and B. A. Bernevig, Entanglement spectrum of a disordered topological Chern insulator, Phys. Rev. Lett. 105(11), 115501 (2010)

[52] X. Cai, L. J. Lang, S. Chen, and Y. Wang, Topological superconductor to Anderson localization transition in one-dimensional incommensurate lattices, Phys. Rev. Lett. 110(17), 176403 (2013)

[53] J. Liu, A. C. Potter, K. T. Law, and P. A. Lee, Zero-bias peaks in the tunneling conductance of spin-orbit-coupled superconducting wires with and without Majorana end-states, Phys. Rev. Lett. 109(26), 267002 (2012)

[54] H. Pan and S. Das Sarma, Disorder effects on Majorana zero modes: Kitaev chain versus semiconductor nanowire, Phys. Rev. B 103(22), 224505 (2021)

[55] J. Li, R. L. Chu, J. K. Jain, and S. Q. Shen, Topological Anderson insulator, Phys. Rev. Lett. 102(13), 136806 (2009)

[56] H. M. Guo, G. Rosenberg, G. Refael, and M. Franz, Topological Anderson insulator in three dimensions, Phys. Rev. Lett. 105(21), 216601 (2010)

[57] Y. Y. Zhang, R. L. Chu, F. C. Zhang, and S. Q. Shen, Localization and mobility gap in the topological Anderson insulator, Phys. Rev. B 85(3), 035107 (2012)

[58] D. W. Zhang, L. Z. Tang, L. J. Lang, H. Yan, and S. L. Zhu, Non-Hermitian topological Anderson insulators, Sci. China Phys. Mech. Astron. 63(6), 267062 (2020)

[59] C. W. Groth, M. Wimmer, A. R. Akhmerov, J. Tworzydlo, and C. W. J. Beenakker, Theory of the topological Anderson insulator, Phys. Rev. Lett. 103(19), 196805 (2009)

[60] A. Girschik, F. Libisch, and S. Rotter, Topological insulator in the presence of spatially correlated disorder, Phys. Rev. B 88(1), 014201 (2013)

[61] L. Z. Tang, S. N. Liu, G. Q. Zhang, and D. W. Zhang, Topological Anderson insulators with different bulk states in quasiperiodic chains, Phys. Rev. A 105(6), 063327 (2022)

[62] Y. P. Wu, L. Z. Tang, G. Q. Zhang, and D. W. Zhang, Quantized topological Anderson-Thouless pump, Phys. Rev. A 106(5), L051301 (2022)

[63] S. N. Liu, G. Q. Zhang, L. Z. Tang, and D. W. Zhang, Topological Anderson insulators induced by random binary disorders, Phys. Lett. A 431, 128004 (2022)

[64] J. Song, H. Liu, H. Jiang, Q. F. Sun, and X. C. Xie, Dependence of topological Anderson insulator on the type of disorder, Phys. Rev. B 85(19), 195125 (2012)

[65] I. Mondragon-Shem, T. L. Hughes, J. Song, and E. Prodan, Topological criticality in the chiral-symmetric AIII class at strong disorder, Phys. Rev. Lett. 113(4), 046802 (2014)

[66] Z. Q. Zhang, B. L. Wu, J. Song, and H. Jiang, Topological Anderson insulator in electric circuits, Phys. Rev. B 100(18), 184202 (2019)

[67] L. Z. Tang, L. F. Zhang, G. Q. Zhang, and D. W. Zhang, Topological Anderson insulators in two-dimensional non-Hermitian disordered systems, Phys. Rev. A 101(6), 063612 (2020)

[68] J. Borchmann, A. Farrell, and T. Pereg-Barnea, Anderson topological superconductor, Phys. Rev. B 93(12), 125133 (2016)

[69] C. B. Hua, R. Chen, D. H. Xu, and B. Zhou, Disorder-induced Majorana zero modes in a dimerized Kitaev superconductor chain, Phys. Rev. B 100(20), 205302 (2019)

[70] G. Q. Zhang, L. Z. Tang, L. F. Zhang, D. W. Zhang, and S. L. Zhu, Connecting topological Anderson and Mott insulators in disordered interacting fermionic systems, Phys. Rev. B 104(16), L161118 (2021)

[71] S. Velury, B. Bradlyn, and T. L. Hughes, Topological crystalline phases in a disordered inversion-symmetric chain, Phys. Rev. B 103(2), 024205 (2021)

[72] W. J. Zhang, Y. P. Wu, L. Z. Tang, and G. Q. Zhang, Topological and dynamical phase transitions in the Su–Schrieffer–Heeger model with quasiperiodic and long-range hoppings, Commum. Theor. Phys. 74(7), 075702 (2022)

[73] H. Jiang, L. Wang, Q. F. Sun, and X. C. Xie, Numerical study of the topological Anderson insulator in HgTe/CdTe quantum wells, Phys. Rev. B 80(16), 165316 (2009)

[74] S. Stützer, Y. Plotnik, Y. Lumer, P. Titum, N. H. Lindner, M. Segev, M. C. Rechtsman, and A. Szameit, Photonic topological Anderson insulators, Nature 560(7719), 461 (2018)

[75] G. G. Liu, Y. Yang, X. Ren, H. Xue, X. Lin, Y. H. Hu, H. X. Sun, B. Peng, P. Zhou, Y. Chong, and B. Zhang, Topological Anderson insulator in disordered photonic crystals, Phys. Rev. Lett. 125(13), 133603 (2020)

[76] E. J. Meier, F. A. An, A. Dauphin, M. Maffei, P. Massignan, T. L. Hughes, and B. Gadway, Observation of the topological Anderson insulator in disordered atomic wires, Science 362(6417), 929 (2018)

[77] S. Longhi, Topological Anderson phase in quasi-periodic waveguide lattices, Opt. Lett. 45(14), 4036 (2020)

[78] Z. Lu, Z. Xu, and Y. Zhang, Exact mobility edges and topological Anderson insulating phase in a slowly varying quasiperiodic model, Ann. Phys. 534(8), 2200203 (2022)

[79] Z. H. Wang, F. Xu, L. Li, D. H. Xu, and B. Wang, Topological superconductors and exact mobility edges in non-Hermitian quasicrystals, Phys. Rev. B 105(2), 024514 (2022)

[80] A. MacKinnon and B. Kramer, The scaling theory of electrons in disordered solids: Additional numerical results, Z. Phys. B 53(1), 1 (1983)

[81] Y. Wang, L. Zhang, S. Niu, D. Yu, and X. J. Liu, Realization and detection of nonergodic critical phases in an optical Raman lattice, Phys. Rev. Lett. 125(7), 073204 (2020)

[82] Y. C. Zhang, Y. Y. Zhang, Lyapunov exponent, and mobility edges, and critical region in the generalized Aubry−André model with an unbounded quasiperiodic potential, Phys. Rev. B 105(17), 174206 (2022)

[83] Y. Liu, Y. Wang, Z. Zheng, and S. Chen, Exact non-Hermitian mobility edges in one-dimensional quasicrystal lattice with exponentially decaying hopping and its dual lattice, Phys. Rev. B 103(13), 134208 (2021)

[84] X. Li and S. Das Sarma, Mobility edge and intermediate phase in one-dimensional incommensurate lattice potentials, Phys. Rev. B 101(6), 064203 (2020)

[85] J.K. AsbóthL.OroszlányA. Pályi, A Short Course on Topological Insulators: Band Structure and Edge States in One and Two Dimensions, Springer International Publishing, Switzerland, 2016

[86] H. P. Lüschen, S. Scherg, T. Kohlert, M. Schreiber, P. Bordia, X. Li, S. Das Sarma, and I. Bloch, Single-particle mobility edge in a one-dimensional quasiperiodic optical lattice, Phys. Rev. Lett. 120(16), 160404 (2018)

[87] Z. Xu, H. Huangfu, Y. Zhang, and S. Chen, Dynamical observation of mobility edges in one-dimensional incommensurate optical lattices, New J. Phys. 22(1), 013036 (2020)

[88] H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77(3), 313 (1916)

[89] G. H. Choe, Ergodicity and irrational rotations, Proc. R. Ir. Acad. 93A, 193 (1993)

[90] A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321(1), 2 (2006)

[91] L. Lin, Y. Ke, and C. Lee, Real-space representation of the winding number for a one-dimensional chiral-symmetric topological insulator, Phys. Rev. B 103(22), 224208 (2021)

Declarations

The authors declare no competing interests and no conflicts.

Acknowledgements

Z.X. was supported by the National Natural Science Foundation of China (Grant No. 12375016), the Fundamental Research Program of Shanxi Province, China (Grant No. 20210302123442), and Beijing National Laboratory for Condensed Matter Physics (No. 2023BNLCMPKF001). Y. Zhang was supported by the National Natural Science Foundation of China (No. 12074340). This work was also supported by the Natural Science Foundation for Shanxi Province (Grant No. 1331KSC).

RIGHTS & PERMISSIONS

2024 Higher Education Press