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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. Jordan and E.P. Wigner, Über das Paulische Äquivalenzverbot, Z. Phys. 47 (1928) 631 [INSPIRE].
E. Witten, Non-abelian bosonization in two dimensions, Commun. Math. Phys. 92 (1984) 455 [INSPIRE].
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).
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].
S. Mandal and N. Surendran, Exactly solvable Kitaev model in three dimensions, Phys. Rev. B 79 (2009) 024426.
A.O. Gogolin, A.A. Nersesyan and A.M. Tsvelik, Bosonization and Strongly Correlated Systems, Cambridge University Press (1998).
A. Kapustin and R. Thorngren, Fermionic SPT phases in higher dimensions and bosonization, JHEP 10 (2017) 080 [arXiv:1701.08264] [INSPIRE].
J. Condella and C.E. Detar, Potts flux tube model at nonzero chemical potential, Phys. Rev. D 61 (2000) 074023 [hep-lat/9910028] [INSPIRE].
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].
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.
A.Yu. Kitaev, Fault-tolerant quantum computation by anyons, Annals Phys. 303 (2003) 2 [quant-ph/9707021] [INSPIRE].
A. Kitaev and C. Laumann, Topological phases and quantum computation, arXiv:0904.2771.
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].
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].
J. Wosiek, A local representation for fermions on a lattice, Acta Phys. Polon. B 13 (1982) 543 [INSPIRE].
C.P. Burgess, C.A. Lütken and F. Quevedo, Bosonization in higher dimensions, Phys. Lett. B 336 (1994) 18 [hep-th/9407078] [INSPIRE].
P. Kopietz, Bosonization of Interacting Fermions in Arbitrary Dimensions, Springer (1997).
S.B. Bravyi and A.Yu. Kitaev, Fermionic Quantum Computation, Annals Phys. 298 (2002) 210.
R.C. Ball, Fermions without Fermion Fields, Phys. Rev. Lett. 95 (2005) 176407 [cond-mat/0409485] [INSPIRE].
F. Verstraete and J.I. Cirac, Mapping local Hamiltonians of fermions to local Hamiltonians of spins, J. Stat. Mech. 2005 (2005) P09012.
E. Fradkin, Jordan-Wigner transformation for quantum-spin systems in two dimensions and fractional statistics, Phys. Rev. B 63 (1989) 322.
A. Karch and D. Tong, Particle-Vortex Duality from 3D Bosonization, Phys. Rev. X 6 (2016) 031043 [arXiv:1606.01893] [INSPIRE].
E. Zohar and J.I. Cirac, Eliminating fermionic matter fields in lattice gauge theories, Phys. Rev. B 98 (2018) 075119 [arXiv:1805.05347] [INSPIRE].
A. Karch, D. Tong and C. Turner, A web of 2d dualities: ℤ2 gauge fields and Arf invariants, SciPost Phys. 7 (2019) 007 [arXiv:1902.05550] [INSPIRE].
R. Thorngren, Anomalies and Bosonization, Commun. Math. Phys. 378 (2020) 1775 [arXiv:1810.04414] [INSPIRE].
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].
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].
A.M. Szczerba, Spins and fermions on arbitrary lattices, Commun. Math. Phys. 98 (1985) 513 [INSPIRE].
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].
R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Commun. Math. Phys. 129 (1990) 393 [INSPIRE].
D.S. Freed and F. Quinn, Chern-Simons theory with finite gauge group, Commun. Math. Phys. 156 (1993) 435 [hep-th/9111004] [INSPIRE].
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].
F. Wilczek, Magnetic Flux, Angular Momentum, and Statistics, Phys. Rev. Lett. 48 (1982) 1144 [INSPIRE].
Y.-A. Chen, Exact bosonization in arbitrary dimensions, Phys. Rev. Res. 2 (2020) 033527 [arXiv:1911.00017] [INSPIRE].
H.A. Kramers and G.H. Wannier, Statistics of the Two-Dimensional Ferromagnet. Part I, Phys. Rev. 60 (1941) 252 [INSPIRE].
A. Hatcher, Algebraic Topology, Cambridge University Press (2002).
J.L. Gross and J. Yellen, Handbook of Graph Theory, CRC Press (2003).
J.A. Beachy, Introductory Lectures on Rings and Modules, Cambridge University Press (1999).
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized global symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
F.J. Wegner, Duality in Generalized Ising Models and Phase Transitions Without Local Order Parameters, J. Math. Phys. 12 (1971) 2259 [INSPIRE].
J.B. Kogut, An introduction to lattice gauge theory and spin systems, Rev. Mod. Phys. 51 (1979) 659 [INSPIRE].
S. Halperin and D. Toledo, Stiefel-Whitney homology classes, Annals Math. 96 (1972) 511.
D. Gaiotto and A. Kapustin, Spin TQFTs and fermionic phases of matter, Int. J. Mod. Phys. A 31 (2016) 1645044 [arXiv:1505.05856] [INSPIRE].
N.E. Steenrod, Products of Cocycles and Extensions of Mappings, Annals Math. 48 (1947) 290.
Ð. Radičević, Spin Structures and Exact Dualities in Low Dimensions, arXiv:1809.07757 [INSPIRE].
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].
A. Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964) 143.
L. Blasco, Paires duales réductives en caractéristique 2, Mém. Soc. Math. Fr. 52 (1993) 1.
W. Scharlau, Quadratic and Hermitian forms, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 270, Springer-Verlag, Berlin (1985) [DOI].
R.L. Griess Jr., Automorphisms of extraspecial groups and nonvanishing degree 2 cohomology, Pac. J. Math. 48 (1973) 403.
A.A. Kirillov, Elements of the theory of representations, Springer-Verlag (1976).
S.M. Bhattacharjee, M. Mj and A. Bandyopadhyay eds., Topology and Condensed Matter Physics, Springer Singapore (2017).
Author information
Authors and Affiliations
Corresponding author
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.
About this article
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
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2020)118