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Duality web on a 3D Euclidean lattice and manifestation of hidden symmetries

  • Jun Ho SonEmail author
  • Jing-Yuan Chen
  • S. Raghu
Open Access
Regular Article - Theoretical Physics

Abstract

We generalize our previous lattice construction of the abelian bosonization duality in 2 + 1 dimensions to the entire web of dualities as well as the Nf = 2 self-duality, via the lattice implementation of a set of modular transformations in the theory space. The microscopic construction provides explicit operator mappings, and allows the manifestation of some hidden symmetries. It also exposes certain caveats and implicit assumptions beneath the usual application of the modular transformations to generate the web of dualities. Finally, we make brief comments on the non-relativistic limit of the dualities.

Keywords

Duality in Gauge Field Theories Lattice Quantum Field Theory Nonperturbative Effects 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    M.A. Metlitski and A. Vishwanath, Particle-vortex duality of two-dimensional Dirac fermion from electric-magnetic duality of three-dimensional topological insulators, Phys. Rev. B 93 (2016) 245151 [arXiv:1505.05142] [INSPIRE].CrossRefGoogle Scholar
  2. [2]
    C. Wang and T. Senthil, Dual Dirac Liquid on the Surface of the Electron Topological Insulator, Phys. Rev. X 5 (2015) 041031 [arXiv:1505.05141] [INSPIRE].CrossRefGoogle Scholar
  3. [3]
    M. Mulligan, Particle-vortex symmetric liquid, Phys. Rev. B 95 (2017) 045118 [arXiv:1605.08047] [INSPIRE].CrossRefGoogle Scholar
  4. [4]
    D.T. Son, Is the Composite Fermion a Dirac Particle?, Phys. Rev. X 5 (2015) 031027 [arXiv:1502.03446] [INSPIRE].CrossRefGoogle Scholar
  5. [5]
    A. Hui, M. Mulligan and E.-A. Kim, Non-Abelian Fermionization and Fractional Quantum Hall Transitions, Phys. Rev. B 97 (2018) 085112 [arXiv:1710.11137] [INSPIRE].CrossRefGoogle Scholar
  6. [6]
    A. Hui, E.-A. Kim and M. Mulligan, Non-Abelian bosonization and modular transformation approach to superuniversality, Phys. Rev. B 99 (2019) 125135 [arXiv:1712.04942] [INSPIRE].CrossRefGoogle Scholar
  7. [7]
    M.E. Peskin, Mandelstamt Hooft Duality in Abelian Lattice Models, Annals Phys. 113 (1978) 122 [INSPIRE].CrossRefGoogle Scholar
  8. [8]
    C. Dasgupta and B.I. Halperin, Phase Transition in a Lattice Model of Superconductivity, Phys. Rev. Lett. 47 (1981) 1556 [INSPIRE].CrossRefGoogle Scholar
  9. [9]
    W. Chen, M.P.A. Fisher and Y.-S. Wu, Mott transition in an anyon gas, Phys. Rev. B 48 (1993) 13749 [cond-mat/9301037] [INSPIRE].
  10. [10]
    C.P. Burgess and F. Quevedo, Bosonization as duality, Nucl. Phys. B 421 (1994) 373 [hep-th/9401105] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    E.H. Fradkin and F.A. Schaposnik, The fermion-boson mapping in three-dimensional quantum field theory, Phys. Lett. B 338 (1994) 253 [hep-th/9407182] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  13. [13]
    A. Kapustin and M.J. Strassler, On mirror symmetry in three-dimensional Abelian gauge theories, JHEP 04 (1999) 021 [hep-th/9902033] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    M. Barkeshli and J. McGreevy, Continuous transition between fractional quantum Hall and superfluid states, Phys. Rev. B 89 (2014) 235116 [arXiv:1201.4393] [INSPIRE].CrossRefGoogle Scholar
  15. [15]
    S. Kachru, M. Mulligan, G. Torroba and H. Wang, Mirror symmetry and the half-filled Landau level, Phys. Rev. B 92 (2015) 235105 [arXiv:1506.01376] [INSPIRE].CrossRefGoogle Scholar
  16. [16]
    S. Kachru, M. Mulligan, G. Torroba and H. Wang, Bosonization and Mirror Symmetry, Phys. Rev. D 94 (2016) 085009 [arXiv:1608.05077] [INSPIRE].MathSciNetGoogle Scholar
  17. [17]
    O. Aharony, Baryons, monopoles and dualities in Chern-Simons-matter theories, JHEP 02 (2016) 093 [arXiv:1512.00161] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    N. Seiberg, T. Senthil, C. Wang and E. Witten, A Duality Web in 2+1 Dimensions and Condensed Matter Physics, Annals Phys. 374 (2016) 395 [arXiv:1606.01989] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    A. Karch and D. Tong, Particle-Vortex Duality from 3d Bosonization, Phys. Rev. X 6 (2016) 031043 [arXiv:1606.01893] [INSPIRE].CrossRefGoogle Scholar
  20. [20]
    A. Karch, B. Robinson and D. Tong, More Abelian Dualities in 2+1 Dimensions, JHEP 01 (2017) 017 [arXiv:1609.04012] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    F. Benini and S. Benvenuti, \( \mathcal{N} \) = 1 dualities in 2+1 dimensions, JHEP 11 (2018) 197 [arXiv:1803.01784] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    T. Senthil, D.T. Son, C. Wang and C. Xu, Duality between (2 + 1)d Quantum Critical Points, arXiv:1810.05174 [INSPIRE].
  23. [23]
    H. Goldman and E. Fradkin, Loop Models, Modular Invariance and Three Dimensional Bosonization, Phys. Rev. B 97 (2018) 195112 [arXiv:1801.04936] [INSPIRE].CrossRefGoogle Scholar
  24. [24]
    D. Arovas, J.R. Schrieffer and F. Wilczek, Fractional Statistics and the Quantum Hall Effect, Phys. Rev. Lett. 53 (1984) 722 [INSPIRE].CrossRefGoogle Scholar
  25. [25]
    A.M. Polyakov, Fermi-Bose Transmutations Induced by Gauge Fields, Mod. Phys. Lett. A 3 (1988) 325 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  26. [26]
    S.C. Zhang, T.H. Hansson and S. Kivelson, An effective field theory model for the fractional quantum hall effect, Phys. Rev. Lett. 62 (1988) 82 [INSPIRE].CrossRefGoogle Scholar
  27. [27]
    J.K. Jain, Composite fermion approach for the fractional quantum Hall effect, Phys. Rev. Lett. 63 (1989) 199 [INSPIRE].CrossRefGoogle Scholar
  28. [28]
    E. Fradkin and A. Lopez, Fractional Quantum Hall effect and Chern-Simons gauge theories, Phys. Rev. B 44 (1991) 5246 [INSPIRE].Google Scholar
  29. [29]
    J. Fröhlich and A. Zee, Large scale physics of the quantum Hall fluid, Nucl. Phys. B 364 (1991) 517 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  30. [30]
    X.-G. Wen, Topological orders and edge excitations in FQH states, Adv. Phys. 44 (1995) 405 [cond-mat/9506066] [INSPIRE].
  31. [31]
    S.H. Simon, The Chern-Simons Fermi Liquid Description of Fractional Quantum Hall States, in Composite Fermions, O. Heinonen ed., World Scientific, Singapore, (1998), [cond-mat/9812186].
  32. [32]
    S. Kivelson, D.-H. Lee and S.-C. Zhang, Global phase diagram in the quantum Hall effect, Phys. Rev. B 46 (1992) 2223 [INSPIRE].CrossRefGoogle Scholar
  33. [33]
    E. Witten, SL(2, ℤ) action on three-dimensional conformal field theories with Abelian symmetry, hep-th/0307041 [INSPIRE].
  34. [34]
    J.-Y. Chen, J.H. Son, C. Wang and S. Raghu, Exact Boson-Fermion Duality on a 3D Euclidean Lattice, Phys. Rev. Lett. 120 (2018) 016602 [arXiv:1705.05841] [INSPIRE].CrossRefGoogle Scholar
  35. [35]
    D.F. Mross, J. Alicea and O.I. Motrunich, Explicit derivation of duality between a free Dirac cone and quantum electrodynamics in (2+1) dimensions, Phys. Rev. Lett. 117 (2016) 016802 [arXiv:1510.08455] [INSPIRE].CrossRefGoogle Scholar
  36. [36]
    D.F. Mross, J. Alicea and O.I. Motrunich, Symmetry and duality in bosonization of two-dimensional Dirac fermions, Phys. Rev. X 7 (2017) 041016 [arXiv:1705.01106] [INSPIRE].CrossRefGoogle Scholar
  37. [37]
    J.-Y. Chen and M. Zimet, Strong-Weak Chern-Simons-Matter Dualities from a Lattice Construction, JHEP 08 (2018) 015 [arXiv:1806.04141] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    C. Xu and Y.-Z. You, Self-dual Quantum Electrodynamics as Boundary State of the three dimensional Bosonic Topological Insulator, Phys. Rev. B 92 (2015) 220416 [arXiv:1510.06032] [INSPIRE].CrossRefGoogle Scholar
  39. [39]
    F. Benini, P.-S. Hsin and N. Seiberg, Comments on global symmetries, anomalies and duality in (2 + 1)d, JHEP 04 (2017) 135 [arXiv:1702.07035] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    A. Vishwanath and T. Senthil, Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect, Phys. Rev. X 3 (2013) 011016 [arXiv:1209.3058] [INSPIRE].CrossRefGoogle Scholar
  41. [41]
    C. Wang, A. Nahum, M.A. Metlitski, C. Xu and T. Senthil, Deconfined quantum critical points: symmetries and dualities, Phys. Rev. X 7 (2017) 031051 [arXiv:1703.02426] [INSPIRE].CrossRefGoogle Scholar
  42. [42]
    M.P.A. Fisher and D.H. Lee, Correspondence between two-dimensional bosons and a bulk superconductor in a magnetic field, Phys. Rev. B 39 (1989) 2756.CrossRefGoogle Scholar
  43. [43]
    M.P.A. Fisher, P.B. Weichman, G. Grinstein and D.S. Fisher, Boson localization and the superfluid-insulator transition, Phys. Rev. B 40 (1989) 546 [INSPIRE].CrossRefGoogle Scholar
  44. [44]
    E.H. Fradkin and S. Kivelson, Modular invariance, selfduality and the phase transition between quantum Hall plateaus, Nucl. Phys. B 474 (1996) 543 [cond-mat/9603156] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Stanford Institute for Theoretical PhysicsStanford UniversityStanfordU.S.A.
  2. 2.SLAC National Accelerator LaboratoryMenlo ParkU.S.A.

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