Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

(2+1)D topological phases with RT symmetry: many-body invariant, classification, and higher order edge modes (2403.18887v1)

Published 27 Mar 2024 in cond-mat.str-el, cond-mat.mes-hall, hep-th, and quant-ph

Abstract: It is common in condensed matter systems for reflection ($R$) and time-reversal ($T$) symmetry to both be broken while the combination $RT$ is preserved. In this paper we study invariants that arise due to $RT$ symmetry. We consider many-body systems of interacting fermions with fermionic symmetry groups $G_f = \mathbb{Z}_2f \times \mathbb{Z}_2{RT}$, $U(1)f \rtimes \mathbb{Z}_2{RT}$, and $U(1)f \times \mathbb{Z}_2{RT}$. We show that (2+1)D invertible fermionic topological phases with these symmetries have a $\mathbb{Z} \times \mathbb{Z}_8$, $\mathbb{Z}2 \times \mathbb{Z}_2$, and $\mathbb{Z}2 \times \mathbb{Z}_4$ classification, respectively, which we compute using the framework of $G$-crossed braided tensor categories. We provide a many-body $RT$ invariant in terms of a tripartite entanglement measure, and which we show can be understood using an edge conformal field theory computation in terms of vertex states. For $G_f = U(1)f \rtimes \mathbb{Z}_2{RT}$, which applies to charged fermions in a magnetic field, the non-trivial value of the $\mathbb{Z}_2$ invariant requires strong interactions. For symmetry-preserving boundaries, the phases are distinguished by zero modes at the intersection of the reflection axis and the boundary. Additional invariants arise in the presence of translation or rotation symmetry.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (63)
  1. 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, 405–408 (1982).
  2. Xiao-Gang Wen, Quantum Field Theory of Many-Body Systems (Oxford Univ. Press, Oxford, 2004).
  3. B Andrei Bernevig, Topological insulators and topological superconductors (Princeton university press, 2013).
  4. M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
  5. Xiao-Liang Qi and Shou-Cheng Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83, 1057–1110 (2011).
  6. Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman,  and Sankar Das Sarma, “Non-abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083 (2008).
  7. T. Senthil, “Symmetry Protected Topological phases of Quantum Matter,” Ann. Rev. Condensed Matter Phys. 6, 299 (2015), arXiv:1405.4015 [cond-mat.str-el] .
  8. Maissam Barkeshli, Parsa Bonderson, Meng Cheng,  and Zhenghan Wang, “Symmetry fractionalization, defects, and gauging of topological phases,” Phys. Rev. B 100, 115147 (2019), arXiv:1410.4540 .
  9. Maissam Barkeshli, Yu-An Chen, Po-Shen Hsin,  and Naren Manjunath, “Classification of (2+1)21(2+1)( 2 + 1 )d invertible fermionic topological phases with symmetry,” Phys. Rev. B 105, 235143 (2022).
  10. Qing-Rui Wang and Zheng-Cheng Gu, “Construction and classification of symmetry-protected topological phases in interacting fermion systems,” Phys. Rev. X 10, 031055 (2020).
  11. David Aasen, Parsa Bonderson,  and Christina Knapp, “Characterization and classification of fermionic symmetry enriched topological phases,”  (2021), arXiv:2109.10911 [cond-mat.str-el] .
  12. Daniel Bulmash and Maissam Barkeshli, “Fermionic symmetry fractionalization in (2+1) dimensions,” Physical Review B 105 (2022), 10.1103/physrevb.105.125114, arXiv:2109.10913 [cond-mat.str-el] .
  13. Hao Song, Sheng-Jie Huang, Liang Fu,  and Michael Hermele, “Topological phases protected by point group symmetry,” Physical Review X 7, 011020 (2017).
  14. Sheng-Jie Huang, Hao Song, Yi-Ping Huang,  and Michael Hermele, “Building crystalline topological phases from lower-dimensional states,” Phys. Rev. B 96, 205106 (2017a).
  15. Ken Shiozaki, Hassan Shapourian,  and Shinsei Ryu, “Many-body topological invariants in fermionic symmetry-protected topological phases: Cases of point group symmetries,” Phys. Rev. B 95, 205139 (2017).
  16. Naren Manjunath and Maissam Barkeshli, “Crystalline gauge fields and quantized discrete geometric response for abelian topological phases with lattice symmetry,” Phys. Rev. Research 3, 013040 (2021).
  17. Naren Manjunath and Maissam Barkeshli, “Classification of fractional quantum hall states with spatial symmetries,”  (2020), arXiv:2012.11603 [cond-mat.str-el] .
  18. Yuxuan Zhang, Naren Manjunath, Gautam Nambiar,  and Maissam Barkeshli, “Fractional disclination charge and discrete shift in the hofstadter butterfly,” Phys. Rev. Lett. 129, 275301 (2022a).
  19. Yuxuan Zhang, Naren Manjunath, Gautam Nambiar,  and Maissam Barkeshli, “Quantized charge polarization as a many-body invariant in (2+1)⁢D21D(2+1)\mathrm{D}( 2 + 1 ) roman_D crystalline topological states and hofstadter butterflies,” Phys. Rev. X 13, 031005 (2023a).
  20. Jian-Hao Zhang, Shuo Yang, Yang Qi,  and Zheng-Cheng Gu, “Real-space construction of crystalline topological superconductors and insulators in 2d interacting fermionic systems,” Physical Review Research 4, 033081 (2022b).
  21. Naren Manjunath, Vladimir Calvera,  and Maissam Barkeshli, “Nonperturbative constraints from symmetry and chirality on majorana zero modes and defect quantum numbers in (2+1) dimensions,” Phys. Rev. B 107, 165126 (2023).
  22. Yuxuan Zhang, Naren Manjunath, Ryohei Kobayashi,  and Maissam Barkeshli, “Complete crystalline topological invariants from partial rotations in (2+ 1) d invertible fermionic states and hofstadter’s butterfly,” Physical Review Letters 131, 176501 (2023b).
  23. Naren Manjunath, Vladimir Calvera,  and Maissam Barkeshli, “Characterization and classification of interacting (2+ 1)-dimensional topological crystalline insulators with orientation-preserving wallpaper groups,” Physical Review B 109, 035168 (2024).
  24. Jonah Herzog-Arbeitman, B Andrei Bernevig,  and Zhi-Da Song, “Interacting topological quantum chemistry in 2d with many-body real space invariants,” Nature Communications 15, 1171 (2024).
  25. Hong Yao and Shinsei Ryu, “Interaction effect on topological classification of superconductors in two dimensions,” Physical Review B 88, 064507 (2013).
  26. Michael Levin and Xiao-Gang Wen, “Detecting topological order in a ground state wave function,” Phys. Rev. Lett. 96, 110405 (2006).
  27. Alexei Kitaev and John Preskill, “Topological entanglement entropy,” Physical review letters 96, 110404 (2006).
  28. Hossein Dehghani, Ze-Pei Cian, Mohammad Hafezi,  and Maissam Barkeshli, “Extraction of the many-body chern number from a single wave function,” Physical Review B 103, 075102 (2021).
  29. Ze-Pei Cian, Hossein Dehghani, Andreas Elben, Benoît Vermersch, Guanyu Zhu, Maissam Barkeshli, Peter Zoller,  and Mohammad Hafezi, “Many-body chern number from statistical correlations of randomized measurements,” Physical Review Letters 126, 050501 (2021).
  30. Ze-Pei Cian, Mohammad Hafezi,  and Maissam Barkeshli, “Extracting wilson loop operators and fractional statistics from a single bulk ground state,”  (2022), arXiv:2209.14302 [cond-mat.str-el] .
  31. Isaac H. Kim, Bowen Shi, Kohtaro Kato,  and Victor V. Albert, “Chiral central charge from a single bulk wave function,” Phys. Rev. Lett. 128, 176402 (2022).
  32. Ruihua Fan, Rahul Sahay,  and Ashvin Vishwanath, “Extracting the quantum hall conductance from a single bulk wavefunction,”  (2022), arXiv:2208.11710 .
  33. Hong-Hao Tu, Yi Zhang,  and Xiao-Liang Qi, “Momentum polarization: An entanglement measure of topological spin and chiral central charge,” Physical Review B 88 (2013), 10.1103/physrevb.88.195412, arXiv:1212.6951 .
  34. Michael P. Zaletel, Roger S. K. Mong,  and Frank Pollmann, “Topological characterization of fractional quantum hall ground states from microscopic hamiltonians,” Phys. Rev. Lett. 110, 236801 (2013).
  35. Michael P. Zaletel, “Detecting two-dimensional symmetry-protected topological order in a ground-state wave function,” Physical Review B 90 (2014), 10.1103/physrevb.90.235113.
  36. Ken Shiozaki, Hassan Shapourian, Kiyonori Gomi,  and Shinsei Ryu, “Many-body topological invariants for fermionic short-range entangled topological phases protected by antiunitary symmetries,” Physical Review B 98 (2018), 10.1103/physrevb.98.035151.
  37. Ryohei Kobayashi and Ken Shiozaki, “Anomaly indicator of rotation symmetry in (3+1)d topological order,”  (2019), arXiv:1901.06195 .
  38. Yizhi You, Julian Bibo,  and Frank Pollmann, “Higher-order entanglement and many-body invariants for higher-order topological phases,” Phys. Rev. Research 2, 033192 (2020).
  39. Ryohei Kobayashi, Taige Wang, Tomohiro Soejima, Roger S. K. Mong,  and Shinsei Ryu, “Extracting higher central charge from a single wave function,” Physical Review Letters 132 (2024), 10.1103/physrevlett.132.016602.
  40. Hassan Shapourian, Ken Shiozaki,  and Shinsei Ryu, “Many-body topological invariants for fermionic symmetry-protected topological phases,” Phys. Rev. Lett. 118, 216402 (2017a).
  41. Alexander Altland and Martin R. Zirnbauer, “Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures,” Phys. Rev. B 55, 1142–1161 (1997).
  42. Alexei Kitaev, “Periodic table for topological insulators and superconductors,” AIP Conference Proceedings 1134, 22–30 (2009), https://aip.scitation.org/doi/pdf/10.1063/1.3149495 .
  43. Andreas P. Schnyder, Shinsei Ryu, Akira Furusaki,  and Andreas W. W. Ludwig, “Classification of topological insulators and superconductors in three spatial dimensions,” Phys. Rev. B 78, 195125 (2008).
  44. Lukasz Fidkowski and Alexei Kitaev, “Topological phases of fermions in one dimension,” Phys. Rev. B 83, 075103 (2011).
  45. Hui Li and F. D. M. Haldane, “Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-abelian fractional quantum hall effect states,” Phys. Rev. Lett. 101, 010504 (2008).
  46. Xiao-Liang Qi, Hosho Katsura,  and Andreas W. W. Ludwig, “General relationship between the entanglement spectrum and the edge state spectrum of topological quantum states,” Physical Review Letters 108 (2012), 10.1103/physrevlett.108.196402, arXiv:1103.5437 [cond-mat.mes-hall] .
  47. Pasquale Calabrese and John Cardy, “Quantum quenches in extended systems,” Journal of Statistical Mechanics: Theory and Experiment 2007, P06008–P06008 (2007).
  48. Nobuyuki Ishibashi, “The boundary and crosscap states in conformal field theories,” Modern Physics Letters A 04, 251–264 (1989).
  49. Yuhan Liu, Yuya Kusuki, Jonah Kudler-Flam, Ramanjit Sohal,  and Shinsei Ryu, “Multipartite entanglement in two-dimensional chiral topological liquids,”  (2023), arXiv:2301.07130 [cond-mat.str-el] .
  50. Itamar Hason, Zohar Komargodski,  and Ryan Thorngren, “Anomaly matching in the symmetry broken phase: Domain walls, cpt, and the smith isomorphism,” SciPost Physics 8 (2020), 10.21468/scipostphys.8.4.062.
  51. Zheng-Cheng Gu and Michael Levin, “Effect of interactions on two-dimensional fermionic symmetry-protected topological phases withz2symmetry,” Physical Review B 89 (2014), 10.1103/physrevb.89.201113.
  52. Sheng-Jie Huang, Hao Song, Yi-Ping Huang,  and Michael Hermele, “Building crystalline topological phases from lower-dimensional states,” Phys. Rev. B 96, 205106 (2017b).
  53. Ryan Thorngren and Dominic V. Else, “Gauging spatial symmetries and the classification of topological crystalline phases,” Phys. Rev. X 8, 011040 (2018).
  54. Dominic V. Else and Ryan Thorngren, “Crystalline topological phases as defect networks,” Phys. Rev. B 99, 115116 (2019).
  55. Arun Debray, “Invertible phases for mixed spatial symmetries and the fermionic crystalline equivalence principle,”  (2021), arXiv:2102.02941 [math-ph] .
  56. Wladimir A. Benalcazar, Tianhe Li,  and Taylor 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).
  57. Yan-Qi Wang and Joel E. Moore, “Boundary edge networks induced by bulk topology,” Phys. Rev. B 99, 155102 (2019).
  58. N. Read and Dmitry Green, “Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum hall effect,” Phys. Rev. B 61, 10267–10297 (2000).
  59. Siew-Ann Cheong and Christopher L. Henley, “Many-body density matrices for free fermions,”  (2003), arXiv:cond-mat/0206196 [cond-mat] .
  60. Jan Borchmann, Aaron Farrell, Shunji Matsuura,  and T. Pereg-Barnea, “Entanglement spectrum as a probe for the topology of a spin-orbit-coupled superconductor,” Phys. Rev. B 90, 235150 (2014).
  61. Ingo Peschel, “Calculation of reduced density matrices from correlation functions,” Journal of Physics A: Mathematical and General 36, L205–L208 (2003).
  62. Ingo Peschel and Viktor Eisler, “Reduced density matrices and entanglement entropy in free lattice models,” Journal of Physics A: Mathematical and Theoretical 42, 504003 (2009).
  63. Hassan Shapourian, Ken Shiozaki,  and Shinsei Ryu, “Partial time-reversal transformation and entanglement negativity in fermionic systems,” Physical Review B 95 (2017b), 10.1103/physrevb.95.165101.
Citations (6)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets