Papers
Topics
Authors
Recent
Search
2000 character limit reached

Generalized Kramers-Wanier Duality from Bilinear Phase Map

Published 24 Mar 2024 in cond-mat.str-el, cond-mat.stat-mech, math-ph, and math.MP | (2403.16017v2)

Abstract: We present the Bilinear Phase Map (BPM), a concept that extends the Kramers-Wannier (KW) transformation to investigate unconventional gapped phases, their dualities, and phase transitions. Defined by a matrix of $\mathbb{Z}_2$ elements, the BPM not only encapsulates the essence of KW duality but also enables exploration of a broader spectrum of generalized quantum phases and dualities. By analyzing the BPM's linear algebraic properties, we elucidate the loss of unitarity in duality transformations and derive general non-invertible fusion rules. Applying this framework to (1+1)D systems yields the discovery of new dualities, shedding light on the interplay between various Symmetry Protected Topological (SPT) and Spontaneous Symmetry Breaking (SSB) phases. Additionally, we construct a duality web that interconnects these phases and their transitions, offering valuable insights into relations between different quantum phases.

Authors (2)
Definition Search Book Streamline Icon: https://streamlinehq.com
References (64)
  1. Z.-C. Gu and X.-G. Wen, Phys. Rev. B 80, 155131 (2009).
  2. X.-G. Wen, Advances in Physics 44, 405 (1995), https://doi.org/10.1080/00018739500101566 .
  3. A. Y. Kitaev, Annals of physics 303, 2 (2003).
  4. M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006).
  5. A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006).
  6. C. Chamon, Phys. Rev. Lett. 94, 040402 (2005).
  7. S. Vijay, J. Haah, and L. Fu, Phys. Rev. B 92, 235136 (2015).
  8. S. Vijay, J. Haah, and L. Fu, Phys. Rev. B 94, 235157 (2016).
  9. E. Cobanera, G. Ortiz, and Z. Nussinov, Advances in physics 60, 679 (2011).
  10. J. B. Kogut, Rev. Mod. Phys. 51, 659 (1979).
  11. L. Bhardwaj and Y. Tachikawa, JHEP 03, 189, arXiv:1704.02330 [hep-th] .
  12. D. Aasen, R. S. K. Mong, and P. Fendley, J. Phys. A 49, 354001 (2016), arXiv:1601.07185 [cond-mat.stat-mech] .
  13. D. Aasen, P. Fendley, and R. S. K. Mong, Topological Defects on the Lattice: Dualities and Degeneracies (2020), arXiv:2008.08598 [cond-mat.stat-mech] .
  14. C.-T. Hsieh, Y. Nakayama, and Y. Tachikawa, Phys. Rev. Lett. 126, 195701 (2021).
  15. Y. Fukusumi and S. Iino, Phys. Rev. B 104, 125418 (2021).
  16. Y. Fukusumi, Y. Tachikawa, and Y. Zheng, SciPost Phys. 11, 082 (2021).
  17. L. Li, M. Oshikawa, and Y. Zheng, Phys. Rev. B 108, 214429 (2023a).
  18. H. Moradi, S. F. Moosavian, and A. Tiwari, Topological Holography: Towards a Unification of Landau and Beyond-Landau Physics (2022), arXiv:2207.10712 [cond-mat.str-el] .
  19. L. Lootens, C. Delcamp, and F. Verstraete, Dualities in one-dimensional quantum lattice models: topological sectors (2022), arXiv:2211.03777 [quant-ph] .
  20. H. Watanabe, Phys. Rev. B 98, 155137 (2018).
  21. Y. Yao and M. Oshikawa, Phys. Rev. Lett. 126, 217201 (2021).
  22. J. Kaidi, K. Ohmori, and Y. Zheng, Phys. Rev. Lett. 128, 111601 (2022a).
  23. J. Kaidi, K. Ohmori, and Y. Zheng, Commun. Math. Phys. 404, 1021 (2023), arXiv:2209.11062 [hep-th] .
  24. J. Kaidi, G. Zafrir, and Y. Zheng, JHEP 08, 053, arXiv:2205.01104 [hep-th] .
  25. Y. Choi, H. T. Lam, and S.-H. Shao, Noninvertible Global Symmetries in the Standard Model (2022b), arXiv:2205.05086 [hep-th] .
  26. Y. Choi, H. T. Lam, and S.-H. Shao, Phys. Rev. Lett. 130, 131602 (2023a).
  27. C. Cordova and K. Ohmori, Phys. Rev. X 13, 011034 (2023), arXiv:2205.06243 [hep-th] .
  28. L. Bhardwaj, S. Schafer-Nameki, and J. Wu, Fortsch. Phys. 70, 2200143 (2022), arXiv:2208.05973 [hep-th] .
  29. L. Bhardwaj, S. Schafer-Nameki, and A. Tiwari, SciPost Phys. 15, 122 (2023b), arXiv:2212.06159 [hep-th] .
  30. K. Inamura and X.-G. Wen, 2+1D symmetry-topological-order from local symmetric operators in 1+1D (2023), arXiv:2310.05790 [cond-mat.str-el] .
  31. A. Chatterjee, W. Ji, and X.-G. Wen, Emergent generalized symmetry and maximal symmetry-topological-order (2022), arXiv:2212.14432 [cond-mat.str-el] .
  32. C.-M. Chang, J. Chen, and F. Xu, SciPost Phys. 15, 216 (2023), arXiv:2208.02757 [hep-th] .
  33. C. Delcamp and A. Tiwari, Higher categorical symmetries and gauging in two-dimensional spin systems (2023), arXiv:2301.01259 [hep-th] .
  34. P. Putrov and J. Wang, Categorical Symmetry of the Standard Model from Gravitational Anomaly (2023), arXiv:2302.14862 [hep-th] .
  35. Z. Sun and Y. Zheng, When are Duality Defects Group-Theoretical? (2023), arXiv:2307.14428 [hep-th] .
  36. C. Fechisin, N. Tantivasadakarn, and V. V. Albert, Non-invertible symmetry-protected topological order in a group-based cluster state (2023), arXiv:2312.09272 [cond-mat.str-el] .
  37. S.-H. Shao, What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetry (2023), arXiv:2308.00747 [hep-th] .
  38. Y. Choi, D.-C. Lu, and Z. Sun, Self-duality under gauging a non-invertible symmetry (2023c), arXiv:2310.19867 [hep-th] .
  39. S. D. Pace, Emergent generalized symmetries in ordered phases (2023), arXiv:2308.05730 [cond-mat.str-el] .
  40. K. Inamura and K. Ohmori, Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories (2023), arXiv:2305.05774 [cond-mat.str-el] .
  41. S. Chen and Y. Tanizaki, Solitonic symmetry as non-invertible symmetry: cohomology theories with TQFT coefficients (2023), arXiv:2307.00939 [hep-th] .
  42. Y. Fukusumi, Protected edge modes based on the bulk and boundary renormalization group: A relationship between duality and generalized symmetry (2023), arXiv:2312.12887 [hep-th] .
  43. R. Thorngren and Y. Wang, Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases (2019), arXiv:1912.02817 [hep-th] .
  44. R. Thorngren and Y. Wang, Fusion Category Symmetry II: Categoriosities at c𝑐citalic_c = 1 and Beyond (2021), arXiv:2106.12577 [hep-th] .
  45. C. Zhang and C. Córdova, Anomalies of (1+1)⁢D11𝐷(1+1)D( 1 + 1 ) italic_D categorical symmetries (2023), arXiv:2304.01262 [cond-mat.str-el] .
  46. C. Cordova, P.-S. Hsin, and C. Zhang, Anomalies of Non-Invertible Symmetries in (3+1)d (2023), arXiv:2308.11706 [hep-th] .
  47. A. Apte, C. Cordova, and H. T. Lam, Phys. Rev. B 108, 045134 (2023), arXiv:2212.14605 [hep-th] .
  48. N. Seiberg, S. Seifnashri, and S.-H. Shao, Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space (2024), arXiv:2401.12281 [cond-mat.str-el] .
  49. Y. Nagoya and S. Shimamori, JHEP 12, 062, arXiv:2309.05294 [hep-th] .
  50. T. Kennedy and H. Tasaki, Communications in Mathematical Physics 147, 431 (1992).
  51. T. Kennedy and H. Tasaki, Phys. Rev. B 45, 304 (1992).
  52. M. Oshikawa, Journal of Physics: Condensed Matter 4, 7469 (1992).
  53. M. Kohmoto and H. Tasaki, Phys. Rev. B 46, 3486 (1992).
  54. H.-H. Tu, G.-M. Zhang, and T. Xiang, Phys. Rev. B 78, 094404 (2008).
  55. D. V. Else, S. D. Bartlett, and A. C. Doherty, Phys. Rev. B 88, 085114 (2013).
  56. K. Okunishi, Phys. Rev. B 83, 104411 (2011).
  57. T. Devakul, D. J. Williamson, and Y. You, Classification of subsystem symmetry-protected topological phases (2018), arXiv:1808.05300 [cond-mat.str-el] .
  58. L. Li, M. Oshikawa, and Y. Zheng, Intrinsically/purely gapless-spt from non-invertible duality transformations (2023b), arXiv:2307.04788 [cond-mat.str-el] .
  59. Y. Yao and A. Furusaki, Phys. Rev. B 106, 045125 (2022).
  60. W. Cao, M. Yamazaki, and Y. Zheng, Phys. Rev. B 106, 075150 (2022).
  61. L. Li and Y. Yao, Phys. Rev. B 106, 224420 (2022).
  62. X. Chen, Y.-M. Lu, and A. Vishwanath, Nature communications 5, 3507 (2014).
  63. L. Li, M. Oshikawa, and Y. Zheng, Decorated Defect Construction of Gapless-SPT States (2022), arXiv:2204.03131 [cond-mat.str-el] .
  64. More precisely, the ground state is an eigenstate of Zi−1⁢Xi⁢Xi+1⁢Zi+2subscript𝑍𝑖1subscript𝑋𝑖subscript𝑋𝑖1subscript𝑍𝑖2Z_{i-1}X_{i}X_{i+1}Z_{i+2}italic_Z start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT italic_Z start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT with eigenvalue 1 and thus has the SPT feature. The reason why not choosing H=−\sum@⁢\slimits@i=1L⁢Zi−1⁢Xi⁢Xi+1⁢Zi+2𝐻\sum@subscriptsuperscript\slimits@𝐿𝑖1subscript𝑍𝑖1subscript𝑋𝑖subscript𝑋𝑖1subscript𝑍𝑖2H=-\sum@\slimits@^{L}_{i=1}Z_{i-1}X_{i}X_{i+1}Z_{i+2}italic_H = - start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT italic_Z start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT italic_Z start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT is that this Hamiltonian has an emergent symmetry \prod@⁢\slimits@i⁢Xi\prod@subscript\slimits@𝑖subscript𝑋𝑖\prod@\slimits@_{i}X_{i}start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and is in the corresponding SSB phase.
Citations (4)
View on Semantic Scholar

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Collections

Sign up for free to add this paper to one or more collections.

Sign Up

Tweets

Sign up for free to view the 2 tweets with 2 likes about this paper.

Sign Up for Free