Bosonization based on Clifford algebras and its gauge theoretic interpretation

Abstract

We study the properties of a bosonization procedure based on Clifford algebra valued degrees of freedom, valid for spaces of any dimension. We present its interpretation in terms of fermions in presence of ℤ2 gauge fields satisfying a modified Gauss’ law, resembling Chern-Simons-like theories. Our bosonization prescription involves constraints, which are interpreted as a flatness condition for the gauge field. Solution of the constraints is presented for toroidal geometries of dimension two. Duality between our model and (d − 1)- form ℤ2 gauge theory is derived, which elucidates the relation between the approach taken here with another bosonization map proposed recently.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    P. Jordan and E.P. Wigner, Über das Paulische Äquivalenzverbot, Z. Phys. 47 (1928) 631 [INSPIRE].

    ADS  Article  Google Scholar 

  2. [2]

    E. Witten, Non-abelian bosonization in two dimensions, Commun. Math. Phys. 92 (1984) 455 [INSPIRE].

    ADS  Article  Google Scholar 

  3. [3]

    D. Sénéchal, An Introduction to Bosonization, in Theoretical Methods for Strongly Correlated Electrons, D. Sénéchal, A.M. Tremblay and C. Bourbonnais eds., CRM Series in Mathematical Physics, Springer (2004).

    Book  Google Scholar 

  4. [4]

    T.D. Schultz, D.C. Mattis and E.H. Lieb, Two-Dimensional Ising Model as a Soluble Problem of Many Fermions, Rev. Mod. Phys. 36 (1964) 856 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  5. [5]

    S. Mandal and N. Surendran, Exactly solvable Kitaev model in three dimensions, Phys. Rev. B 79 (2009) 024426.

  6. [6]

    A.O. Gogolin, A.A. Nersesyan and A.M. Tsvelik, Bosonization and Strongly Correlated Systems, Cambridge University Press (1998).

  7. [7]

    A. Kapustin and R. Thorngren, Fermionic SPT phases in higher dimensions and bosonization, JHEP 10 (2017) 080 [arXiv:1701.08264] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  8. [8]

    J. Condella and C.E. Detar, Potts flux tube model at nonzero chemical potential, Phys. Rev. D 61 (2000) 074023 [hep-lat/9910028] [INSPIRE].

    ADS  Article  Google Scholar 

  9. [9]

    Y. Delgado, C. Gattringer and A. Schmidt, Solving the sign problem of two flavor scalar electrodynamics at finite chemical potential, PoS LATTICE2013 (2014) 147 [arXiv:1311.1966] [INSPIRE].

  10. [10]

    C. Gattringer, T. Kloiber and V. Sazanov, Solving the sign problems of the massless lattice Schwinger model with a dual formulation, Nucl. Phys. B 879 (2015) 732.

    ADS  Article  Google Scholar 

  11. [11]

    A.Yu. Kitaev, Fault-tolerant quantum computation by anyons, Annals Phys. 303 (2003) 2 [quant-ph/9707021] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  12. [12]

    A. Kitaev and C. Laumann, Topological phases and quantum computation, arXiv:0904.2771.

  13. [13]

    Y.-A. Chen, A. Kapustin and Ð. Radičević, Exact bosonization in two spatial dimensions and a new class of lattice gauge theories, Annals Phys. 393 (2018) 234 [arXiv:1711.00515] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  14. [14]

    Y.-A. Chen and A. Kapustin, Bosonization in three spatial dimensions and a 2-form gauge theory, Phys. Rev. B 100 (2019) 245127 [arXiv:1807.07081] [INSPIRE].

    ADS  Article  Google Scholar 

  15. [15]

    J. Wosiek, A local representation for fermions on a lattice, Acta Phys. Polon. B 13 (1982) 543 [INSPIRE].

    Google Scholar 

  16. [16]

    C.P. Burgess, C.A. Lütken and F. Quevedo, Bosonization in higher dimensions, Phys. Lett. B 336 (1994) 18 [hep-th/9407078] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  17. [17]

    P. Kopietz, Bosonization of Interacting Fermions in Arbitrary Dimensions, Springer (1997).

  18. [18]

    S.B. Bravyi and A.Yu. Kitaev, Fermionic Quantum Computation, Annals Phys. 298 (2002) 210.

    ADS  MathSciNet  Article  Google Scholar 

  19. [19]

    R.C. Ball, Fermions without Fermion Fields, Phys. Rev. Lett. 95 (2005) 176407 [cond-mat/0409485] [INSPIRE].

    ADS  Article  Google Scholar 

  20. [20]

    F. Verstraete and J.I. Cirac, Mapping local Hamiltonians of fermions to local Hamiltonians of spins, J. Stat. Mech. 2005 (2005) P09012.

  21. [21]

    E. Fradkin, Jordan-Wigner transformation for quantum-spin systems in two dimensions and fractional statistics, Phys. Rev. B 63 (1989) 322.

    ADS  MathSciNet  Google Scholar 

  22. [22]

    A. Karch and D. Tong, Particle-Vortex Duality from 3D Bosonization, Phys. Rev. X 6 (2016) 031043 [arXiv:1606.01893] [INSPIRE].

    Google Scholar 

  23. [23]

    E. Zohar and J.I. Cirac, Eliminating fermionic matter fields in lattice gauge theories, Phys. Rev. B 98 (2018) 075119 [arXiv:1805.05347] [INSPIRE].

    ADS  Article  Google Scholar 

  24. [24]

    A. Karch, D. Tong and C. Turner, A web of 2d dualities:2 gauge fields and Arf invariants, SciPost Phys. 7 (2019) 007 [arXiv:1902.05550] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  25. [25]

    R. Thorngren, Anomalies and Bosonization, Commun. Math. Phys. 378 (2020) 1775 [arXiv:1810.04414] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  26. [26]

    T. Senthil, D.T. Son, C. Wang and C. Xu, Duality between (2 + 1)d quantum critical points, Phys. Rept. 827 (2019) 1 [arXiv:1810.05174] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  27. [27]

    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].

    ADS  MathSciNet  Article  Google Scholar 

  28. [28]

    A.M. Szczerba, Spins and fermions on arbitrary lattices, Commun. Math. Phys. 98 (1985) 513 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  29. [29]

    A. Bochniak, B. Ruba, J. Wosiek and A. Wyrzykowski, Constraints of kinematic bosonization in two and higher dimensions, Phys. Rev. D 102 (2020) 114502 [arXiv:2004.00988] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  30. [30]

    R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Commun. Math. Phys. 129 (1990) 393 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  31. [31]

    D.S. Freed and F. Quinn, Chern-Simons theory with finite gauge group, Commun. Math. Phys. 156 (1993) 435 [hep-th/9111004] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  32. [32]

    Y. Wan, J.C. Wang and H. He, Twisted gauge theory model of topological phases in three dimensions, Phys. Rev. B 92 (2015) 045101 [arXiv:1409.3216] [INSPIRE].

    ADS  Article  Google Scholar 

  33. [33]

    F. Wilczek, Magnetic Flux, Angular Momentum, and Statistics, Phys. Rev. Lett. 48 (1982) 1144 [INSPIRE].

    ADS  Article  Google Scholar 

  34. [34]

    Y.-A. Chen, Exact bosonization in arbitrary dimensions, Phys. Rev. Res. 2 (2020) 033527 [arXiv:1911.00017] [INSPIRE].

    Article  Google Scholar 

  35. [35]

    H.A. Kramers and G.H. Wannier, Statistics of the Two-Dimensional Ferromagnet. Part I, Phys. Rev. 60 (1941) 252 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  36. [36]

    A. Hatcher, Algebraic Topology, Cambridge University Press (2002).

  37. [37]

    J.L. Gross and J. Yellen, Handbook of Graph Theory, CRC Press (2003).

  38. [38]

    J.A. Beachy, Introductory Lectures on Rings and Modules, Cambridge University Press (1999).

  39. [39]

    D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized global symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  40. [40]

    F.J. Wegner, Duality in Generalized Ising Models and Phase Transitions Without Local Order Parameters, J. Math. Phys. 12 (1971) 2259 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  41. [41]

    J.B. Kogut, An introduction to lattice gauge theory and spin systems, Rev. Mod. Phys. 51 (1979) 659 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  42. [42]

    S. Halperin and D. Toledo, Stiefel-Whitney homology classes, Annals Math. 96 (1972) 511.

    MathSciNet  Article  Google Scholar 

  43. [43]

    D. Gaiotto and A. Kapustin, Spin TQFTs and fermionic phases of matter, Int. J. Mod. Phys. A 31 (2016) 1645044 [arXiv:1505.05856] [INSPIRE].

    ADS  Article  Google Scholar 

  44. [44]

    N.E. Steenrod, Products of Cocycles and Extensions of Mappings, Annals Math. 48 (1947) 290.

    MathSciNet  Article  Google Scholar 

  45. [45]

    Ð. Radičević, Spin Structures and Exact Dualities in Low Dimensions, arXiv:1809.07757 [INSPIRE].

  46. [46]

    Z.-C. Gu and X.-G. Wen, Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory, Phys. Rev. B 90 (2014) 115141 [arXiv:1201.2648] [INSPIRE].

    ADS  Article  Google Scholar 

  47. [47]

    A. Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964) 143.

    MathSciNet  Article  Google Scholar 

  48. [48]

    L. Blasco, Paires duales réductives en caractéristique 2, Mém. Soc. Math. Fr. 52 (1993) 1.

    MATH  Google Scholar 

  49. [49]

    W. Scharlau, Quadratic and Hermitian forms, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 270, Springer-Verlag, Berlin (1985) [DOI].

  50. [50]

    R.L. Griess Jr., Automorphisms of extraspecial groups and nonvanishing degree 2 cohomology, Pac. J. Math. 48 (1973) 403.

    Article  Google Scholar 

  51. [51]

    A.A. Kirillov, Elements of the theory of representations, Springer-Verlag (1976).

  52. [52]

    S.M. Bhattacharjee, M. Mj and A. Bandyopadhyay eds., Topology and Condensed Matter Physics, Springer Singapore (2017).

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to A. Bochniak.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 2003.06905

Rights and permissions

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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bochniak, A., Ruba, B. Bosonization based on Clifford algebras and its gauge theoretic interpretation. J. High Energ. Phys. 2020, 118 (2020). https://doi.org/10.1007/JHEP12(2020)118

Download citation

Keywords

  • Gauge Symmetry
  • Lattice Quantum Field Theory
  • Topological States of Matter
  • Chern-Simons Theories