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Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry (2404.11650v2)

Published 17 Apr 2024 in cond-mat.str-el and cond-mat.mes-hall

Abstract: Topology is routinely used to understand the physics of electronic insulators. However, for strongly interacting electronic matter, such as Mott insulators, a comprehensive topological characterization is still lacking. When their ground state only contains short-range entanglement and does not break symmetries spontaneously, they generically realize crystalline fermionic symmetry-protected topological phases (cFSPTs), supporting gapless modes at the boundaries or at the lattice defects. Here, we provide an exhaustive classification of cFSPTs in two dimensions with $\mathrm{U}(1)$ charge-conservation and spinful time-reversal symmetries, namely, those generically present in spin-orbit coupled insulators, for any of the 17 wallpaper groups. It has been shown that the classification of cFSPTs can be understood from appropriate real-space decorations of lower-dimensional subspaces, and we expose how these relate to the Wyckoff positions of the lattice. We find that all nontrivial one-dimensional decorations require electronic interactions. Furthermore, we provide model Hamiltonians for various decorations, and discuss the signatures of cFSPTs. This classification paves the way to further explore topological interacting insulators, providing the backbone information in generic model systems and ultimately in experiments.

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References (82)
  1. A. Kitaev, Periodic table for topological insulators and superconductors, AIP Conference Proceedings 1134, 22 (2009), https://pubs.aip.org/aip/acp/article-pdf/1134/1/22/11584243/22_1_online.pdf .
  2. K. v. Klitzing, G. Dorda, and M. Pepper, New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance, Phys. Rev. Lett. 45, 494 (1980).
  3. D. S. Freed and M. J. Hopkins, Reflection positivity and invertible topological phases, Geometry & Topology 25, 1165 (2021).
  4. B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells, Science 314, 1757–1761 (2006).
  5. L. Fu and C. L. Kane, Topological insulators with inversion symmetry, Phys. Rev. B 76, 045302 (2007).
  6. L. Fu, Topological crystalline insulators, Phys. Rev. Lett. 106, 106802 (2011).
  7. F. Zhang, C. L. Kane, and E. J. Mele, Topological mirror superconductivity, Phys. Rev. Lett. 111, 056403 (2013).
  8. C.-K. Chiu, H. Yao, and S. Ryu, Classification of topological insulators and superconductors in the presence of reflection symmetry, Phys. Rev. B 88, 075142 (2013).
  9. T. Morimoto and A. Furusaki, Topological classification with additional symmetries from clifford algebras, Phys. Rev. B 88, 125129 (2013).
  10. C. Fang, M. J. Gilbert, and B. A. Bernevig, Entanglement spectrum classification of Cnsubscript𝐶𝑛{C}_{n}italic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT-invariant noninteracting topological insulators in two dimensions, Phys. Rev. B 87, 035119 (2013).
  11. M. Koshino, T. Morimoto, and M. Sato, Topological zero modes and dirac points protected by spatial symmetry and chiral symmetry, Phys. Rev. B 90, 115207 (2014).
  12. C.-X. Liu, R.-X. Zhang, and B. K. VanLeeuwen, Topological nonsymmorphic crystalline insulators, Phys. Rev. B 90, 085304 (2014).
  13. K. Shiozaki and M. Sato, Topology of crystalline insulators and superconductors, Phys. Rev. B 90, 165114 (2014).
  14. Y. Ando and L. Fu, Topological crystalline insulators and topological superconductors: From concepts to materials, Annual Review of Condensed Matter Physics 6, 361 (2015).
  15. L. Trifunovic and P. Brouwer, Bott periodicity for the topological classification of gapped states of matter with reflection symmetry, Phys. Rev. B 96, 195109 (2017).
  16. L. Trifunovic and P. W. Brouwer, Higher-order bulk-boundary correspondence for topological crystalline phases, Phys. Rev. X 9, 011012 (2019).
  17. C. Fang and L. Fu, New classes of topological crystalline insulators having surface rotation anomaly, Science Advances 5, 10.1126/sciadv.aat2374 (2019).
  18. Z. Song, Z. Fang, and C. Fang, (d−2)𝑑2(d-2)( italic_d - 2 )-dimensional edge states of rotation symmetry protected topological states, Phys. Rev. Lett. 119, 246402 (2017a).
  19. H. C. Po, A. Vishwanath, and H. Watanabe, Symmetry-based indicators of band topology in the 230 space groups, Nature Communications 8, 10.1038/s41467-017-00133-2 (2017).
  20. H. C. Po, Symmetry indicators of band topology, Journal of Physics: Condensed Matter 32, 263001 (2020).
  21. J. Cano and B. Bradlyn, Band representations and topological quantum chemistry, Annual Review of Condensed Matter Physics 12, 225 (2021), https://doi.org/10.1146/annurev-conmatphys-041720-124134 .
  22. t. . P. Chuang-Han Hsu and Xiaoting Zhou and Qiong Ma and Nuh Gedik and Arun Bansil and Vitor M Pereira and Hsin Lin and Liang Fu and Su-Yang Xu and Tay-Rong Chang, 2D Materials 6, 031004 (2019).
  23. L. Fidkowski and A. Kitaev, Effects of interactions on the topological classification of free fermion systems, Phys. Rev. B 81, 134509 (2010).
  24. L. Fidkowski and A. Kitaev, Topological phases of fermions in one dimension, Phys. Rev. B 83, 075103 (2011).
  25. S. Ryu and S.-C. Zhang, Interacting topological phases and modular invariance, Phys. Rev. B 85, 245132 (2012).
  26. X.-L. Qi, A new class of (2 + 1)-dimensional topological superconductors with ℤ8subscriptℤ8\mathbb{Z}_{8}blackboard_Z start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT topological classification, New Journal of Physics 15, 065002 (2013).
  27. L. Fidkowski, X. Chen, and A. Vishwanath, Non-abelian topological order on the surface of a 3d topological superconductor from an exactly solved model, Phys. Rev. X 3, 041016 (2013).
  28. H. Yao and S. Ryu, Interaction effect on topological classification of superconductors in two dimensions, Phys. Rev. B 88, 064507 (2013).
  29. C. Wang and T. Senthil, Interacting fermionic topological insulators/superconductors in three dimensions, Phys. Rev. B 89, 195124 (2014).
  30. Y.-Z. You and C. Xu, Symmetry-protected topological states of interacting fermions and bosons, Phys. Rev. B 90, 245120 (2014).
  31. T. Morimoto, A. Furusaki, and C. Mudry, Breakdown of the topological classification ℤℤ\mathbb{Z}blackboard_Z for gapped phases of noninteracting fermions by quartic interactions, Phys. Rev. B 92, 125104 (2015).
  32. T. Yoshida and A. Furusaki, Correlation effects on topological crystalline insulators, Phys. Rev. B 92, 085114 (2015).
  33. X.-Y. Song and A. P. Schnyder, Interaction effects on the classification of crystalline topological insulators and superconductors, Phys. Rev. B 95, 195108 (2017).
  34. X. Chen, Z.-C. Gu, and X.-G. Wen, Classification of gapped symmetric phases in one-dimensional spin systems, Phys. Rev. B 83, 035107 (2011a).
  35. X. Chen, Z.-C. Gu, and X.-G. Wen, Complete classification of one-dimensional gapped quantum phases in interacting spin systems, Phys. Rev. B 84, 235128 (2011b).
  36. X. Chen, Z.-X. Liu, and X.-G. Wen, Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations, Phys. Rev. B 84, 235141 (2011c).
  37. X. Chen, Y.-M. Lu, and A. Vishwanath, Symmetry-protected topological phases from decorated domain walls, Nature Communications 5, 10.1038/ncomms4507 (2014).
  38. A. Kapustin, Bosonic topological insulators and paramagnets: a view from cobordisms (2014b), arXiv:1404.6659 [cond-mat.str-el] .
  39. Z.-C. Gu and X.-G. Wen, Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ𝜎\sigmaitalic_σ models and a special group supercohomology theory, Phys. Rev. B 90, 115141 (2014).
  40. Q.-R. Wang and Z.-C. Gu, Construction and classification of symmetry-protected topological phases in interacting fermion systems, Phys. Rev. X 10, 031055 (2020).
  41. C. Wang, C.-H. Lin, and Z.-C. Gu, Interacting fermionic symmetry-protected topological phases in two dimensions, Phys. Rev. B 95, 195147 (2017).
  42. M. Cheng, N. Tantivasadakarn, and C. Wang, Loop braiding statistics and interacting fermionic symmetry-protected topological phases in three dimensions, Phys. Rev. X 8, 011054 (2018).
  43. N. Tantivasadakarn and A. Vishwanath, Full commuting projector Hamiltonians of interacting symmetry-protected topological phases of fermions, Phys. Rev. B 98, 165104 (2018).
  44. J. Sullivan and M. Cheng, Interacting edge states of fermionic symmetry-protected topological phases in two dimensions, SciPost Phys. 9, 016 (2020).
  45. M. Cheng and C. Wang, Rotation symmetry-protected topological phases of fermions, Phys. Rev. B 105, 195154 (2022).
  46. R. Thorngren and D. V. Else, Gauging spatial symmetries and the classification of topological crystalline phases, Phys. Rev. X 8, 011040 (2018).
  47. D. V. Else and R. Thorngren, Crystalline topological phases as defect networks, Phys. Rev. B 99, 115116 (2019).
  48. D. S. Freed and M. J. Hopkins, Invertible phases of matter with spatial symmetry (2019), arXiv:1901.06419 [math-ph] .
  49. A. Debray, Invertible phases for mixed spatial symmetries and the fermionic crystalline equivalence principle (2021), arXiv:2102.02941 [math-ph] .
  50. J.-H. Zhang, Y. Qi, and Z.-C. Gu, Construction and classification of crystalline topological superconductor and insulators in three-dimensional interacting fermion systems, arXiv e-prints , arXiv:2204.13558 (2022), arXiv:2204.13558 [cond-mat.str-el] .
  51. A. Rasmussen and Y.-M. Lu, Intrinsically interacting topological crystalline insulators and superconductors (2018), arXiv:1810.12317 [cond-mat.str-el] .
  52. A. Rasmussen and Y.-M. Lu, Classification and construction of higher-order symmetry-protected topological phases of interacting bosons, Phys. Rev. B 101, 085137 (2020).
  53. N. Manjunath, V. Calvera, and M. Barkeshli, Characterization and classification of interacting (2+1)D topological crystalline insulators with orientation-preserving wallpaper groups (2023), arXiv:2309.12389 [cond-mat.str-el] .
  54. H. Yao and S. A. Kivelson, Fragile Mott Insulators, Physical Review Letters 105, 10.1103/physrevlett.105.166402 (2010).
  55. A. Turzillo and M. You, Fermionic matrix product states and one-dimensional short-range entangled phases with antiunitary symmetries, Phys. Rev. B 99, 035103 (2019).
  56. C. Bourne and Y. Ogata, The classification of symmetry protected topological phases of one-dimensional fermion systems, Forum of Mathematics, Sigma 9, e25 (2021).
  57. Ö. M. Aksoy and C. Mudry, Elementary derivation of the stacking rules of invertible fermionic topological phases in one dimension, Phys. Rev. B 106, 035117 (2022).
  58. D. J. Scalapino and S. A. Trugman, Local antiferromagnetic correlations and dx2−y2subscript𝑑superscript𝑥2superscript𝑦2d_{x^{2}-y^{2}}italic_d start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT pairing, Philosophical Magazine B 74, 607 (1996), https://doi.org/10.1080/01418639608240361 .
  59. R. Schumann, Thermodynamics of a 4-site Hubbard model by analytical diagonalization, Annalen der Physik 11, 49 (2002).
  60. J. Herzog-Arbeitman, A. Bernevig, and Z. Song, Interacting topological quantum chemistry in 2D with many-body real space invariants, Nature Communications https://doi.org/10.1038/s41467-024-45395-9 (2024).
  61. M. Suzuki, Relationship among Exactly Soluble Models of Critical Phenomena. I*): 2D Ising Model, Dimer Problem and the Generalized XY-Model, Progress of Theoretical Physics 46, 1337 (1971), https://academic.oup.com/ptp/article-pdf/46/5/1337/5268367/46-5-1337.pdf .
  62. R. Verresen, R. Moessner, and F. Pollmann, One-dimensional symmetry protected topological phases and their transitions, Phys. Rev. B 96, 165124 (2017).
  63. V. Gurarie, Single-particle Green’s functions and interacting topological insulators, Phys. Rev. B 83, 085426 (2011).
  64. Z. Wang, X.-L. Qi, and S.-C. Zhang, Topological invariants for interacting topological insulators with inversion symmetry, Phys. Rev. B 85, 165126 (2012).
  65. Z. Wang and B. Yan, Topological Hamiltonian as an exact tool for topological invariants, Journal of Physics: Condensed Matter 25, 155601 (2013).
  66. W. A. Benalcazar, T. Li, and T. L. Hughes, Quantization of fractional corner charge in Cnsubscript𝐶𝑛{C}_{n}italic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT-symmetric higher-order topological crystalline insulators, Phys. Rev. B 99, 245151 (2019).
  67. Y. Fang and J. Cano, Filling anomaly for general two- and three-dimensional C4subscript𝐶4{C}_{4}italic_C start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT symmetric lattices, Phys. Rev. B 103, 165109 (2021).
  68. F. Pollmann and A. M. Turner, Detection of symmetry-protected topological phases in one dimension, Phys. Rev. B 86, 125441 (2012).
  69. H. Shapourian, K. Shiozaki, and S. Ryu, Many-body topological invariants for fermionic symmetry-protected topological phases, Phys. Rev. Lett. 118, 216402 (2017).
  70. E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain, Annals of Physics 16, 407 (1961).
  71. M. B. Hastings, Lieb-schultz-mattis in higher dimensions, Phys. Rev. B 69, 104431 (2004).
  72. M. B. Hastings, Sufficient conditions for topological order in insulators, Europhysics Letters (EPL) 70, 824 (2005).
  73. G. Y. Cho, C.-T. Hsieh, and S. Ryu, Anomaly manifestation of lieb-schultz-mattis theorem and topological phases, Phys. Rev. B 96, 195105 (2017).
  74. H. Tasaki, Lieb–schultz–mattis theorem with a local twist for general one-dimensional quantum systems, Journal of Statistical Physics 170, 653 (2018).
  75. C.-M. Jian, Z. Bi, and C. Xu, Lieb-schultz-mattis theorem and its generalizations from the perspective of the symmetry-protected topological phase, Phys. Rev. B 97, 054412 (2018).
  76. Y. Ogata and H. Tasaki, Lieb–Schultz–Mattis type theorems for quantum spin chains without continuous symmetry, Communications in Mathematical Physics 372, 951 (2019).
  77. Y. Ogata, Y. Tachikawa, and H. Tasaki, General Lieb–Schultz–Mattis Type Theorems for Quantum Spin Chains, Communications in Mathematical Physics 385, 79 (2021).
  78. Y. Yao and M. Oshikawa, Twisted Boundary Condition and Lieb-Schultz-Mattis Ingappability for Discrete Symmetries, Phys. Rev. Lett. 126, 217201 (2021).
  79. Ö. M. Aksoy, A. Tiwari, and C. Mudry, Lieb-Schultz-Mattis type theorems for Majorana models with discrete symmetries, Phys. Rev. B 104, 075146 (2021b).
  80. H. C. Po, H. Watanabe, and A. Vishwanath, Fragile topology and wannier obstructions, Phys. Rev. Lett. 121, 126402 (2018).
  81. A. Bouhon, A. M. Black-Schaffer, and R.-J. Slager, Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time-reversal symmetry, Phys. Rev. B 100, 195135 (2019).
  82. D. V. Else, H. C. Po, and H. Watanabe, Fragile topological phases in interacting systems, Phys. Rev. B 99, 125122 (2019).

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