Abstract
Compact nonlocal Abelian gauge theory in (2 + 1) dimensions, also known as loop model, is a massless theory with a critical line that is explicitly covariant under duality transformations. It corresponds to the large NF limit of self-dual electrodynamics in mixed three-four dimensions. It also provides a bosonic description for surface excitations of three-dimensional topological insulators. Upon mapping the model to a local gauge theory in (3 + 1) dimensions, we compute the spectrum of electric and magnetic solitonic excitations and the partition function on the three torus \( {\mathbbm{T}}_3 \). Analogous results for the S2 × S1 geometry show that the theory is conformal invariant and determine the manifestly self-dual spectrum of conformal fields, corresponding to order-disorder excitations with fractional statistics.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
H. Goldman and E. Fradkin, Loop Models, Modular Invariance and Three Dimensional Bosonization, Phys. Rev. B 97 (2018) 195112 [arXiv:1801.04936] [INSPIRE].
E.H. Fradkin, Field Theories of Condensed Matter Physics, second edition, Cambridge University Press, (2013).
B.A. Bernevig and T.L. Hughes, Topological Insulators and Topological Superconductors, Princeton University Press, Princeton, U.S.A., (2013).
X.-L. Qi, T. Hughes and S.-C. Zhang, Topological Field Theory of Time-Reversal Invariant Insulators, Phys. Rev. B 78 (2008) 195424 [arXiv:0802.3537] [INSPIRE].
G.Y. Cho and J.E. Moore, Topological BF field theory description of topological insulators, Annals Phys. 326 (2011) 1515 [arXiv:1011.3485] [INSPIRE].
A. Cappelli, E. Randellini and J. Sisti, Three-dimensional Topological Insulators and Bosonization, JHEP 05 (2017) 135 [arXiv:1612.05212] [INSPIRE].
A. Chan, T.L. Hughes, S. Ryu and E. Fradkin, Effective field theories for topological insulators by functional bosonization, Phys. Rev. B 87 (2013) 085132 [arXiv:1210.4305] [INSPIRE].
A.P.O. Chan, T. Kvorning, S. Ryu and E. Fradkin, Effective hydrodynamic field theory and condensation picture of topological insulators, Phys. Rev. B 93 (2016) 155122 [arXiv:1510.08975] [INSPIRE].
D.T. Son, Is the Composite Fermion a Dirac Particle?, Phys. Rev. X 5 (2015) 031027 [arXiv:1502.03446] [INSPIRE].
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].
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].
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].
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].
C. Turner, Dualities in 2+1 Dimensions, [arXiv:1905.12656] [INSPIRE].
W.-H. Hsiao and D.T. Son, Duality and universal transport in mixed-dimension electrodynamics, Phys. Rev. B 96 (2017) 075127 [arXiv:1705.01102] [INSPIRE].
P. Di Francesco, P. Mathieu and D. Sénéchal, Conformal Field Theory, Springer-Verlag, New York, U.S.A., (1997).
J.B. Kogut, An Introduction to Lattice Gauge Theory and Spin Systems, Rev. Mod. Phys. 51 (1979) 659 [INSPIRE].
A.M. Polyakov, Quark Confinement and Topology of Gauge Groups, Nucl. Phys. B 120 (1977) 429 [INSPIRE].
S.D. Geraedts and O.I. Motrunich, Line of continuous phase transitions in a three dimensional U(1) model with 1/r2 current-current interactions, Phys. Rev. B 85 (2012) 144303 [arXiv:1202.0838] [INSPIRE].
P. Ye, M. Cheng and E. Fradkin, Fractional S-duality, Classification of Fractional Topological Insulators and Surface Topological Order, Phys. Rev. B 96 (2017) 085125 [arXiv:1701.05559] [INSPIRE].
C.-T. Hsieh, G.Y. Cho and S. Ryu, Global anomalies on the surface of fermionic symmetry-protected topological phases in (3+1) dimensions, Phys. Rev. B 93 (2016) 075135 [arXiv:1503.01411] [INSPIRE].
X. Chen, A. Tiwari and S. Ryu, Bulk-boundary correspondence in (3+1)-dimensional topological phases, Phys. Rev. B 94 (2016) 045113 [arXiv:1509.04266] [INSPIRE].
A. Amoretti, A. Blasi, N. Maggiore and N. Magnoli, Three-dimensional dynamics of four-dimensional topological BF theory with boundary, New J. Phys. 14 (2012) 113014 [arXiv:1205.6156] [INSPIRE].
A.N. Redlich, Parity Violation and Gauge Noninvariance of the Effective Gauge Field Action in Three-Dimensions, Phys. Rev. D 29 (1984) 2366 [INSPIRE].
A.J. Niemi and G.W. Semenoff, Axial Anomaly Induced Fermion Fractionization and Effective Gauge Theory Actions in Odd Dimensional Space-Times, Phys. Rev. Lett. 51 (1983) 2077 [INSPIRE].
A. Cappelli and A. Coste, On the Stress Tensor of Conformal Field Theories in Higher Dimensions, Nucl. Phys. B 314 (1989) 707 [INSPIRE].
A. Cappelli and G. D’Appollonio, On the trace anomaly as a measure of degrees of freedom, Phys. Lett. B 487 (2000) 87 [hep-th/0005115] [INSPIRE].
A. Karch and D. Tong, Particle-Vortex Duality from 3d Bosonization, Phys. Rev. X 6 (2016) 031043 [arXiv:1606.01893] [INSPIRE].
A. Karch, B. Robinson and D. Tong, More Abelian Dualities in 2+1 Dimensions, JHEP 01 (2017) 017 [arXiv:1609.04012] [INSPIRE].
E. Witten, On S duality in Abelian gauge theory, Selecta Math. 1 (1995) 383.
W.-H. Hsiao and D.T. Son, Self-Dual ν = 1 Bosonic Quantum Hall State in Mixed Dimensional QED, Phys. Rev. B 100 (2020) 235150 [arXiv:1809.06886] [INSPIRE].
D. Dudal, A.J. Mizher and P. Pais, Exact quantum scale invariance of three-dimensional reduced QED theories, Phys. Rev. D 99 (2019) 045017 [arXiv:1808.04709] [INSPIRE].
E.C. Marino, Quantum Field Theory Approach to Condensed Matter Physics, Cambridge University Press, (2017).
A. Cappelli, G.V. Dunne, C.A. Trugenberger and G.R. Zemba, Conformal symmetry and universal properties of quantum Hall states, Nucl. Phys. B 398 (1993) 531 [hep-th/9211071] [INSPIRE].
L. Di Pietro, D. Gaiotto, E. Lauria and J. Wu, 3d Abelian Gauge Theories at the Boundary, JHEP 05 (2019) 091 [arXiv:1902.09567] [INSPIRE].
A. Messiah, Quantum Mechanics, Vol. I, Dover Publ., New York, U.S.A., (2014).
J.M. Blatt and V.F. Weisskopf, Theoretical Nuclear Physics, Dover Publ., New York, U.S.A., (2010).
H. Bateman, Higher Transcendental Functions, Vol. I, McGraw-Hill Book Company, New York, U.S.A., (1953).
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
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1912.04125
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
Andreucci, F., Cappelli, A. & Maffi, L. Quantization of a self-dual conformal theory in (2 + 1) dimensions. J. High Energ. Phys. 2020, 116 (2020). https://doi.org/10.1007/JHEP02(2020)116
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2020)116