Abstract
I study the prospect of generating mass for symmetry-protected fermions without breaking the symmetry that forbids quadratic mass terms in the Lagrangian. I focus on 1+1 spacetime dimensions in the hope that this can provide guidance for interacting fermions in 3+1 dimensions. I first review the SO(8) Gross-Neveu model and emphasize a subtlety in the triality transformation. Then I focus on the “m = 0” manifold of the SO(7) Kitaev-Fidkowski model. I argue that this theory exhibits a phenomenon similar to “parity doubling” in hadronic physics, and this leads to the conclusion that the fermion propagator vanishes when pμ = 0. I also briefly explore a connection between this model and the two-channel, single-impurity Kondo effect. This paper may serve as an introduction to topological superconductors for high energy theorists, and perhaps as a taste of elementary particle physics for condensed matter theorists.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
ATLAS collaboration, Observation of a new particle in the search for the standard model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1 [arXiv:1207.7214] [INSPIRE].
CMS collaboration, Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys. Lett. B 716 (2012) 30 [arXiv:1207.7235] [INSPIRE].
L. Fidkowski and A. Kitaev, Effects of interactions on the topological classification of free fermion systems, Phys. Rev. B 81 (2010) 134509 [arXiv:0904.2197].
L. Fidkowski and A. Kitaev, Topological phases of fermions in one dimension, Phys. Rev. B 83 (2011) 075103 [arXiv:1008.4138].
S. Ryu, J.E. Moore and A.W.W. Ludwig, Electromagnetic and gravitational responses and anomalies in topological insulators and superconductors, Phys. Rev. B 85 (2012) 045104 [arXiv:1010.0936] [INSPIRE].
S. Ryu and S.-C. Zhang, Interacting topological phases and modular invariance, Phys. Rev. B 85 (2012) 245132 [arXiv:1202.4484] [INSPIRE].
X.-L. Qi, A new class of (2 + 1)-dimensional topological superconductors with \( {\mathbb{Z}}_8 \) topological classification, New J. Phys. 15 (2013) 065002 [arXiv:1202.3983].
L. Fidkowski, X. Chen and A. Vishwanath, Non-Abelian topological order on the surface of a 3D topological superconductor from an exactly solved model, Phys. Rev. X 3 (2013) 041016 [arXiv:1305.5851] [INSPIRE].
M.A. Metlitski, L. Fidkowski, X. Chen and A. Vishwanath, Interaction effects on 3D topological superconductors: surface topological order from vortex condensation, the 16 fold way and fermionic Kramers doublets, arXiv:1406.3032 [INSPIRE].
C. Wang and T. Senthil, Interacting fermionic topological insulators/superconductors in three dimensions, Phys. Rev. B 89 (2014) 195124 [Erratum ibid. B 91 (2015) 239902] [arXiv:1401.1142] [INSPIRE].
X.-G. Wen, A lattice non-perturbative definition of an SO(10) chiral gauge theory and its induced standard model, Chin. Phys. Lett. 30 (2013) 111101 [arXiv:1305.1045] [INSPIRE].
Y. You, Y. BenTov and C. Xu, Interacting topological superconductors and possible origin of 16n chiral fermions in the standard model, arXiv:1402.4151 [INSPIRE].
Y.-Z. You and C. Xu, Interacting topological insulator and emergent grand unified theory, Phys. Rev. B 91 (2015) 125147 [arXiv:1412.4784] [INSPIRE].
J. Wang and X.-G. Wen, A lattice non-perturbative Hamiltonian construction of 1 + 1D anomaly-free chiral fermions and bosons — on the equivalence of the anomaly matching conditions and the boundary fully gapping rules, arXiv:1307.7480 [INSPIRE].
K. Slagle, Y.-Z. You and C. Xu, Exotic quantum phase transitions of strongly interacting topological insulators, Phys. Rev. B 91 (2015) 115121 [arXiv:1409.7401] [INSPIRE].
V. Ayyar and S. Chandrasekharan, Massive fermions without fermion bilinear condensates, Phys. Rev. D 91 (2015) 065035 [arXiv:1410.6474] [INSPIRE].
X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83 (2011) 1057 [arXiv:1008.2026].
T. Senthil, Symmetry protected topological phases of quantum matter, Ann. Rev. Condensed Matter Phys. 6 (2015) 299 [arXiv:1405.4015] [INSPIRE].
A. Kitaev, Unpaired Majorana fermions in quantum wires, Phys. Usp. 44 (2001) 131 [cond-mat/0010440].
C.G. Callan Jr. and J.A. Harvey, Anomalies and fermion zero modes on strings and domain walls, Nucl. Phys. B 250 (1985) 427 [INSPIRE].
R. Jackiw and C. Rebbi, Solitons with fermion number 1/2, Phys. Rev. D 13 (1976) 3398 [INSPIRE].
A. Schnyder, S. Ryu, A. Furusaki and A. Ludwig, Classification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B 78 (2008) 195125 [INSPIRE].
A.P. Schnyder, S. Ryu, A. Furusaki and A.W.W. Ludwig, Classification of topological insulators and superconductors, AIP Conf. Proc. 1134 (2009) 10 [arXiv:0905.2029].
A. Kitaev, Periodic table for topological insulators and superconductors, AIP Conf. Proc. 1134 (2009) 22 [arXiv:0901.2686] [INSPIRE].
E. Cartan, Lecons sur la theorie des spineurs I & II (in French), Herman, Paris France (1938) translation by R. Streater, The theory of spinors, Herman, Paris France (1966).
T. Banks, D. Horn and H. Neuberger, Bosonization of the SU(N ) Thirring models, Nucl. Phys. B 108 (1976) 119 [INSPIRE].
E. Witten, Chiral symmetry, the 1/N expansion and the SU(N ) Thirring model, Nucl. Phys. B 145 (1978) 110 [INSPIRE].
S.R. Coleman, The quantum sine-Gordon equation as the massive Thirring model, Phys. Rev. D 11 (1975) 2088 [INSPIRE].
D.J. Gross and A. Neveu, Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D 10 (1974) 3235 [INSPIRE].
R. Shankar, Some novel features of the Gross-Neveu model, Phys. Lett. B 92 (1980) 333 [INSPIRE].
P. Forgács, F. Niedermayer and P. Weisz, The exact mass gap of the Gross-Neveu model. 1. The thermodynamic Bethe ansatz, Nucl. Phys. B 367 (1991) 123 [INSPIRE].
P. Forgács, F. Niedermayer and P. Weisz, The exact mass gap of the Gross-Neveu model. 2. The 1/N expansion, Nucl. Phys. B 367 (1991) 144 [INSPIRE].
B. Schroer and T.T. Truong, Z2 duality algebra in D = 2 quantum field theory, Nucl. Phys. B 154 (1979) 125 [INSPIRE].
A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [INSPIRE].
J.M. Maldacena and A.W.W. Ludwig, Majorana fermions, exact mapping between quantum impurity fixed points with four bulk fermion species and solution of the ‘unitarity puzzle’, Nucl. Phys. B 506 (1997) 565 [cond-mat/9502109] [INSPIRE].
H.-H. Lin, L. Balents and M.P.A. Fisher, Exact SO(8) symmetry in the weakly interacting two leg ladder, Phys. Rev. B 58 (1998) 1794 [cond-mat/9801285] [INSPIRE].
R. Konik and A.W.W. Ludwig, Exact zero temperature correlation functions for two leg Hubbard ladders and carbon nanotubes, Phys. Rev. B 64 (2001) 155112 [cond-mat/9810332] [INSPIRE].
R.I. Nepomechie, Non-Abelian bosonization, triality and superstring theory, Phys. Lett. B 178 (1986) 207 [Addendum ibid. B 180 (1986) 423] [Erratum ibid. B 180 (1986) 424] [INSPIRE].
E. Witten, Some properties of the (ψψ)2 model in two-dimensions, Nucl. Phys. B 142 (1978) 285 [INSPIRE].
R. Shankar, Solvable models with selftriality in statistical mechanics and field theory, Phys. Rev. Lett. 46 (1981) 379 [INSPIRE].
M. Karowski and H.J. Thun, Complete S matrix of the O(2N ) Gross-Neveu model, Nucl. Phys. B 190 (1981) 61 [INSPIRE].
J.B. Kogut, An introduction to lattice gauge theory and spin systems, Rev. Mod. Phys. 51 (1979) 659 [INSPIRE].
C. Itzykson, Ising fermions. 1. Two-dimensions, Nucl. Phys. B 210 (1982) 448 [INSPIRE].
S. Elitzur, Impossibility of spontaneously breaking local symmetries, Phys. Rev. D 12 (1975) 3978 [INSPIRE].
K. Rajagopal and F. Wilczek, The condensed matter physics of QCD, hep-ph/0011333 [INSPIRE].
R.F. Dashen, B. Hasslacher and A. Neveu, Semiclassical bound states in an asymptotically free theory, Phys. Rev. D 12 (1975) 2443 [INSPIRE].
S. Mandelstam, Soliton operators for the quantized sine-Gordon equation, Phys. Rev. D 11 (1975) 3026 [INSPIRE].
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].
J.B. Zuber and C. Itzykson, Quantum field theory and the two-dimensional Ising model, Phys. Rev. D 15 (1977) 2875 [INSPIRE].
B. Schroer and T.T. Truong, The order/disorder quantum field operators associated to the two-dimensional Ising model in the continuum limit, Nucl. Phys. B 144 (1978) 80 [INSPIRE].
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
D. Boyanovsky, Field theory of the two-dimensional Ising model: conformal invariance, order and disorder, and bosonization, Phys. Rev. B 39 (1989) 6744.
F. Wilczek and A. Zee, Families from spinors, Phys. Rev. D 25 (1982) 553 [INSPIRE].
C.S. Aulakh and A. Girdhar, SO(10) à la Pati-Salam, Int. J. Mod. Phys. A 20 (2005) 865 [hep-ph/0204097] [INSPIRE].
G. Källén, On the definition of the renormalization constants in quantum electrodynamics, Helv. Phys. Acta 25 (1952) 417 [INSPIRE].
H. Lehmann, On the properties of propagation functions and renormalization contants of quantized fields, Nuovo Cim. 11 (1954) 342 [INSPIRE].
R.L. Jaffe, D. Pirjol and A. Scardicchio, Parity doubling among the baryons, Phys. Rept. 435 (2006) 157 [hep-ph/0602010] [INSPIRE].
V. Gurarie, Single-particle Green’s functions and interacting topological insulators, Phys. Rev. B 83 (2011) 085426.
Y.-Z. You, Z. Wang, J. Oon and C. Xu, Topological number and fermion Green’s function for strongly interacting topological superconductors, Phys. Rev. B 90 (2014) 060502 [arXiv:1403.4938] [INSPIRE].
R.L. Jaffe, D. Pirjol and A. Scardicchio, Parity doubling and SU(2) L × SU(2) R restoration in the hadron spectrum, Phys. Rev. Lett. 96 (2006) 121601 [hep-ph/0511081] [INSPIRE].
E. Witten, Baryons in the 1/N expansion, Nucl. Phys. B 160 (1979) 57 [INSPIRE].
P. Fendley and H. Saleur, BPS kinks in the Gross-Neveu model, Phys. Rev. D 65 (2002) 025001 [hep-th/0105148] [INSPIRE].
N. Seiberg, Modifying the sum over topological sectors and constraints on supergravity, JHEP 07 (2010) 070 [arXiv:1005.0002] [INSPIRE].
S. Hellerman and E. Sharpe, Sums over topological sectors and quantization of Fayet-Iliopoulos parameters, Adv. Theor. Math. Phys. 15 (2011) 1141 [arXiv:1012.5999] [INSPIRE].
T. Banks and N. Seiberg, Symmetries and strings in field theory and gravity, Phys. Rev. D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].
E. Sharpe, Decomposition in diverse dimensions, Phys. Rev. D 90 (2014) 025030 [arXiv:1404.3986] [INSPIRE].
I. Affleck and A.W.W. Ludwig, Critical theory of overscreened Kondo fixed points, Nucl. Phys. B 360 (1991) 641 [INSPIRE].
A.W.W. Ludwig and I. Affleck, Exact conformal field theory results on the multichannel Kondo effect: asymptotic three-dimensional space and time dependent multipoint and many particle Green’s functions, Nucl. Phys. B 428 (1994) 545 [INSPIRE].
V.J. Emery and S. Kivelson, Mapping of the two-channel Kondo problem to a resonant-level model, Phys. Rev. B 46 (1992) 10812.
I. Affleck and A.W.W. Ludwig, Exact critical theory of the two impurity Kondo model, Phys. Rev. Lett. 68 (1992) 1046 [INSPIRE].
I. Affleck, A.W.W. Ludwig and B.A. Jones, Conformal-field-theory approach to the two-impurity Kondo problem: comparison with numerical renormalization-group results, Phys. Rev. B 52 (1995) 9528.
V.A. Rubakov, Adler-Bell-Jackiw anomaly and fermion number breaking in the presence of a magnetic monopole, Nucl. Phys. B 203 (1982) 311 [INSPIRE].
C.G. Callan Jr., Monopole catalysis of baryon decay, Nucl. Phys. B 212 (1983) 391 [INSPIRE].
J. Polchinski, Monopole catalysis: the fermion rotor system, Nucl. Phys. B 242 (1984) 345 [INSPIRE].
S.D. Drell, A.C. Finn and A.C. Hearn, Bounds on propagators, coupling constants, and vertex functions, Phys. Rev. 136 (1964) B1439.
M.A. Metlitski, L. Fidkowski, X. Chen and A. Vishwanath, Interaction effects on 3D topological superconductors: surface topological order from vortex condensation, the 16 fold way and fermionic Kramers doublets, arXiv:1406.3032 [INSPIRE].
H. Fritzsch and P. Minkowski, Unified interactions of leptons and hadrons, Annals Phys. 93 (1975) 193 [INSPIRE].
H. Georgi, The state of the art — gauge theories, in Particles and fields — 1974, proceedings of the Meeting of the Division of Particles and Fields of the APS, Williamsburg, C.E. Carlson ed., AIP, New York U.S.A. (1975) [AIP Conf. Proc. 23 (1975) 575] [INSPIRE].
J.D. Lykken, T. Montroy and S. Willenbrock, Group theoretic evidence for SO(10) grand unification, Phys. Lett. B 418 (1998) 141 [hep-ph/9710492] [INSPIRE].
J. Bagger and S. Dimopoulos, O(18) revived: splitting the spinor, Nucl. Phys. B 244 (1984) 247 [INSPIRE].
G. Senjanović, F. Wilczek and A. Zee, Reflections on mirror fermions, Phys. Lett. B 141 (1984) 389 [INSPIRE].
C. Itzykson and J.B. Zuber, Quantum field theory, McGraw-Hill, U.S.A. (1980), reprint Dover Publications, U.S.A. (2006).
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: 1412.0154
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
BenTov, Y. Fermion masses without symmetry breaking in two spacetime dimensions. J. High Energ. Phys. 2015, 34 (2015). https://doi.org/10.1007/JHEP07(2015)034
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2015)034