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Calculations of Chern number: equivalence of real-space and twisted-boundary-condition formulae (2308.04164v4)

Published 8 Aug 2023 in quant-ph and physics.comp-ph

Abstract: Chern number is a crucial invariant for characterizing topological feature of two-dimensional quantum systems. Real-space Chern number allows us to extract topological properties of systems without involving translational symmetry, and hence plays an important role in investigating topological systems with disorder or impurity. On the other hand, the twisted boundary condition (TBC) can also be used to define the Chern number in the absence of translational symmetry. Based on the perturbative nature of the TBC under appropriate gauges, we derive the two real-space formulae of Chern number (namely the non-commutative Chern number and the Bott index formula), which are numerically confirmed for the Chern insulator and the quantum spin Hall insulator. Our results not only establish the equivalence between the real-space and TBC formula of the Chern number, but also provide concrete and instructive examples for deriving the real-space topological invariant through the twisted boundary condition.

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References (48)
  1. M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
  2. X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
  3. C. L. Kane, Topological band theory and the ℤ⁢2ℤ2\mathbb{Z}2blackboard_Z 2 invariant, Contemporary Concepts of Condensed Matter Science,  6, 3 (2013).
  4. Q. Niu and D. J. Thouless, Quantised adiabatic charge transport in the presence of substrate disorder and many-body interaction, Journal of Physics A: Mathematical and General 17, 2453 (1984).
  5. R. Tao and Y.-S. Wu, Gauge invariance and fractional quantum hall effect, Phys. Rev. B 30, 1097 (1984).
  6. Q. Niu, D. J. Thouless, and Y.-S. Wu, Quantized Hall conductance as a topological invariant, Phys. Rev. B 31, 3372 (1985).
  7. D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010).
  8. 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, 224208 (2021).
  9. R. B. Laughlin, Quantized hall conductivity in two dimensions, Phys. Rev. B 23, 5632 (1981).
  10. Y. Huo and R. N. Bhatt, Current carrying states in the lowest landau level, Phys. Rev. Lett. 68, 1375 (1992).
  11. J. E. Avron and R. Seiler, Quantization of the hall conductance for general, multiparticle schrödinger hamiltonians, Phys. Rev. Lett. 54, 259 (1985).
  12. Q. Niu, Theory of the quantized adiabatic particle transport, Modern Physics Letters B 5, 923 (1991).
  13. H. Watanabe and M. Oshikawa, Inequivalent berry phases for the bulk polarization, Phys. Rev. X 8, 021065 (2018).
  14. L. Lin, Y. Ke, and C. Lee, Topological invariants for interacting systems: From twisted boundary conditions to center-of-mass momentum, Phys. Rev. B 107, 125161 (2023).
  15. E. Prodan, T. L. Hughes, and B. A. Bernevig, Entanglement Spectrum of a Disordered Topological Chern Insulator, Phys. Rev. Lett. 105, 115501 (2010).
  16. E. Prodan, Robustness of the spin-chern number, Phys. Rev. B 80, 125327 (2009).
  17. E. Prodan, Non-commutative tools for topological insulators, New Journal of Physics 12, 065003 (2010).
  18. E. Prodan, Disordered topological insulators: a non-commutative geometry perspective, Journal of Physics A: Mathematical and Theoretical 44, 113001 (2011).
  19. E. Prodan, A computational non-commutative geometry program for disordered topological insulators, Vol. 23 (Springer, 2017).
  20. R. Exel and T. A. Loring, Invariants of almost commuting unitaries, Journal of Functional Analysis 95, 364 (1991).
  21. T. A. Loring and M. B. Hastings, Disordered topological insulators via C∗superscript𝐶∗C^{\ast}italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-algebras, EPL (Europhysics Letters) 92, 67004 (2010).
  22. M. B. Hastings and T. A. Loring, Topological insulators and C∗superscript𝐶∗C^{\ast}italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-algebras: Theory and numerical practice, Annals of Physics 326, 1699 (2011), july 2011 Special Issue.
  23. T. A. Loring, K-theory and pseudospectra for topological insulators, Annals of Physics 356, 383 (2015).
  24. J. E. Avron, R. Seiler, and B. Simon, Homotopy and quantization in condensed matter physics, Phys. Rev. Lett. 51, 51 (1983).
  25. M. Yamanaka, M. Oshikawa, and I. Affleck, Nonperturbative approach to luttinger’s theorem in one dimension, Phys. Rev. Lett. 79, 1110 (1997).
  26. H. Watanabe, Insensitivity of bulk properties to the twisted boundary condition, Phys. Rev. B 98, 155137 (2018).
  27. E. Prodan, B. Leung, and J. Bellissard, The non-commutative nth-chern number (n≥1𝑛1n\geq 1italic_n ≥ 1), Journal of Physics A: Mathematical and Theoretical 46, 485202 (2013).
  28. C. Bourne and E. Prodan, Non-commutative chern numbers for generic aperiodic discrete systems, Journal of Physics A: Mathematical and Theoretical 51, 235202 (2018).
  29. T. A. Loring, A guide to the bott index and localizer index, arXiv preprint arXiv:1907.11791 10.48550/arXiv.1907.11791 (2019).
  30. M. A. Bandres, M. C. Rechtsman, and M. Segev, Topological Photonic Quasicrystals: Fractal Topological Spectrum and Protected Transport, Phys. Rev. X 6, 011016 (2016).
  31. A. Agarwala and V. B. Shenoy, Topological insulators in amorphous systems, Phys. Rev. Lett. 118, 236402 (2017).
  32. H. Huang and F. Liu, Quantum spin hall effect and spin bott index in a quasicrystal lattice, Phys. Rev. Lett. 121, 126401 (2018a).
  33. H. Huang and F. Liu, Theory of spin bott index for quantum spin hall states in nonperiodic systems, Phys. Rev. B 98, 125130 (2018b).
  34. X. S. Wang, A. Brataas, and R. E. Troncoso, Bosonic bott index and disorder-induced topological transitions of magnons, Phys. Rev. Lett. 125, 217202 (2020).
  35. D. Toniolo, On the bott index of unitary matrices on a finite torus, Letters in Mathematical Physics 112, 126 (2022).
  36. B. Kang, W. Lee, and G. Y. Cho, Many-body invariants for chern and chiral hinge insulators, Phys. Rev. Lett. 126, 016402 (2021).
  37. C. L. Kane and E. J. Mele, Z2subscript𝑍2{Z}_{2}italic_Z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT topological order and the quantum spin hall effect, Phys. Rev. Lett. 95, 146802 (2005a).
  38. B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science 314, 1757 (2006).
  39. C. L. Kane and E. J. Mele, Quantum Spin Hall Effect in Graphene, Phys. Rev. Lett. 95, 226801 (2005b).
  40. X.-L. Qi, Y.-S. Wu, and S.-C. Zhang, General theorem relating the bulk topological number to edge states in two-dimensional insulators, Phys. Rev. B 74, 045125 (2006).
  41. D. N. Sheng, L. Balents, and Z. Wang, Phase diagram for quantum hall bilayers at ν=1𝜈1\nu=1italic_ν = 1, Phys. Rev. Lett. 91, 116802 (2003).
  42. T.-S. Zeng and W. Zhu, Chern-number matrix of the non-abelian spin-singlet fractional quantum hall effect, Phys. Rev. B 105, 125128 (2022).
  43. F. D. M. Haldane, Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the ”Parity Anomaly”, Phys. Rev. Lett. 61, 2015 (1988).
  44. T. Fukui, Y. Hatsugai, and H. Suzuki, Chern numbers in discretized brillouin zone: Efficient method of computing (spin) hall conductances, Journal of the Physical Society of Japan 74, 1674 (2005), https://doi.org/10.1143/JPSJ.74.1674 .
  45. H. Araki, T. Mizoguchi, and Y. Hatsugai, 𝕫Qsubscript𝕫𝑄{\mathbb{z}}_{Q}blackboard_z start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT berry phase for higher-order symmetry-protected topological phases, Phys. Rev. Res. 2, 012009 (2020).
  46. T. Mizoguchi, H. Araki, and Y. Hatsugai, Higher-order topological phase in a honeycomb-lattice model with anti-kekulé distortion, Journal of the Physical Society of Japan 88, 104703 (2019), https://doi.org/10.7566/JPSJ.88.104703 .
  47. W. A. Benalcazar and A. Cerjan, Chiral-symmetric higher-order topological phases of matter, Phys. Rev. Lett. 128, 127601 (2022).
  48. Y.-C. Tzeng, P.-Y. Chang, and M.-F. Yang, Interaction-induced metal to topological insulator transition, Phys. Rev. B 107, 155106 (2023).
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