Topological phases and edge modes of an uneven ladder

doi:10.1088/1674-1056/ad50c0

中国物理B ›› 2024, Vol. 33 ›› Issue (8): 80202-080202.doi: 10.1088/1674-1056/ad50c0

Topological phases and edge modes of an uneven ladder

Wen-Chuang Shang(商文创)^1,2, Yi-Ning Han(韩熠宁)^1,2, Shimpei Endo³, and Chao Gao(高超)^1,2,†

1 Department of Physics, Zhejiang Normal University, Jinhua 321004, China;
2 Key Laboratory of Optical Information Detection and Display Technology of Zhejiang, Zhejiang Normal University, Jinhua 321004, China;
3 Department of Engineering Science, The University of Electro-Communications, Tokyo 182-8585, Japan

收稿日期:2024-03-28 修回日期:2024-05-27 出版日期:2024-08-15 发布日期:2024-07-23
通讯作者: Chao Gao E-mail:gaochao@zjnu.edu.cn
基金资助:
This work was supported by the Natural Science Foundation of Zhejiang Province, China (Grant Nos. LR22A040001 and LY21A040004) and the National Natural Science Foundation of China (Grant Nos. 12074342 and 11835011).

Topological phases and edge modes of an uneven ladder

Wen-Chuang Shang(商文创)^1,2, Yi-Ning Han(韩熠宁)^1,2, Shimpei Endo³, and Chao Gao(高超)^1,2,†

1 Department of Physics, Zhejiang Normal University, Jinhua 321004, China;
2 Key Laboratory of Optical Information Detection and Display Technology of Zhejiang, Zhejiang Normal University, Jinhua 321004, China;
3 Department of Engineering Science, The University of Electro-Communications, Tokyo 182-8585, Japan

Received:2024-03-28 Revised:2024-05-27 Online:2024-08-15 Published:2024-07-23
Contact: Chao Gao E-mail:gaochao@zjnu.edu.cn
Supported by:
This work was supported by the Natural Science Foundation of Zhejiang Province, China (Grant Nos. LR22A040001 and LY21A040004) and the National Natural Science Foundation of China (Grant Nos. 12074342 and 11835011).

摘要/Abstract

摘要： We investigate the topological properties of a two-chain quantum ladder with uneven legs, i.e., the two chains differ in their periods by a factor of 2. Such an uneven ladder presents rich band structures classified by the closure of either direct or indirect bandgaps. It also provides opportunities to explore fundamental concepts concerning band topology and edge modes, including the difference of intracellular and intercellular Zak phases, and the role of the inversion symmetry (IS). We calculate the Zak phases of the two kinds and find excellent agreement with the dipole moment and extra charge accumulation. We also find that configurations with IS feature a pair of degenerate two-side edge modes emerging as the closure of the direct bandgap, while configurations without IS feature one-side edge modes emerging as not only the closure of both direct and indirect bandgaps but also within the band continuum. Furthermore, by projecting to the two sublattices, we find that the effective Bloch Hamiltonian corresponds to that of a generalized Su-Schrieffer-Heeger model or the Rice-Mele model whose hopping amplitudes depend on the quasimomentum. In this way, the topological phases can be efficiently extracted through winding numbers. We propose that uneven ladders can be realized by spin-dependent optical lattices and their rich topological characteristics can be examined by near future experiments.

关键词: ladder model, symmetry-protected topological phase, topological invariant, bulk-boundary correspondence

Abstract: We investigate the topological properties of a two-chain quantum ladder with uneven legs, i.e., the two chains differ in their periods by a factor of 2. Such an uneven ladder presents rich band structures classified by the closure of either direct or indirect bandgaps. It also provides opportunities to explore fundamental concepts concerning band topology and edge modes, including the difference of intracellular and intercellular Zak phases, and the role of the inversion symmetry (IS). We calculate the Zak phases of the two kinds and find excellent agreement with the dipole moment and extra charge accumulation. We also find that configurations with IS feature a pair of degenerate two-side edge modes emerging as the closure of the direct bandgap, while configurations without IS feature one-side edge modes emerging as not only the closure of both direct and indirect bandgaps but also within the band continuum. Furthermore, by projecting to the two sublattices, we find that the effective Bloch Hamiltonian corresponds to that of a generalized Su-Schrieffer-Heeger model or the Rice-Mele model whose hopping amplitudes depend on the quasimomentum. In this way, the topological phases can be efficiently extracted through winding numbers. We propose that uneven ladders can be realized by spin-dependent optical lattices and their rich topological characteristics can be examined by near future experiments.

Key words: ladder model, symmetry-protected topological phase, topological invariant, bulk-boundary correspondence

中图分类号: (Geometry, differential geometry, and topology)

02.40.-k

03.65.-w (Quantum mechanics) 03.65.Vf (Phases: geometric; dynamic or topological) 37.10.Jk (Atoms in optical lattices)

Wen-Chuang Shang(商文创), Yi-Ning Han(韩熠宁), Shimpei Endo, and Chao Gao(高超). Topological phases and edge modes of an uneven ladder[J]. 中国物理B, 2024, 33(8): 80202-080202.

Wen-Chuang Shang(商文创), Yi-Ning Han(韩熠宁), Shimpei Endo, and Chao Gao(高超). Topological phases and edge modes of an uneven ladder[J]. Chin. Phys. B, 2024, 33(8): 80202-080202.

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Topological phases and edge modes of an uneven ladder

Topological phases and edge modes of an uneven ladder

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