Abstract
We compute the effects of strong Hubbardlike local electronic interactions on three-dimensional four-component massless Dirac fermions, which in a noninteracting system possess a microscopic global U(1) ⊗ SU(2) chiral symmetry. A concrete lattice realization of such chiral Dirac excitations is presented, and the role of electron-electron interactions is studied by performing a field theoretic renormalization group (RG) analysis, controlled by a small parameter ϵ with ϵ = d−1, about the lower-critical one spatial dimension. Besides the noninteracting Gaussian fixed point, the system supports four quantum critical and four bicritical points at nonvanishing interaction couplings ∼ ϵ. Even though the chiral symmetry is absent in the interacting model, it gets restored (either partially or fully) at various RG fixed points as emergent phenomena. A representative cut of the global phase diagram displays a confluence of scalar and pseudoscalar excitonic and superconducting (such as the s-wave and p-wave) mass ordered phases, manifesting restoration of (a) chiral U(1) symmetry between two excitonic masses for repulsive interactions and (b) pseudospin SU(2) symmetry between scalar or pseudoscalar excitonic and superconducting masses for attractive interactions. Finally, we perturbatively study the effects of weak rotational symmetry breaking on the stability of various RG fixed points.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M.E. Peskin and D.V. Schroeder, An introduction to quantum field theory, Addison-Wesley, U.S.A. (1995).
S.Q. Shen, Topological insulators-Dirac equation in condensed matters, Springer Germany (2012).
B.A. Bernevig and T.L. Hughes, Topological insulators and topological superconductors, Princeton University Press, Princeton U.S.A. (2013).
T.O. Wehling, A.M. Black-Schaffer and A.V. Balatsky, Dirac materials, Adv. Phys. 63 (2014) 1 [arXiv:1405.5774] [INSPIRE].
N.P. Armitage, E.J. Mele and A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90 (2018) 015001 [arXiv:1705.01111].
A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov and A.K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81 (2009) 109 [arXiv:0709.1163] [INSPIRE].
S. Borisenko, Q. Gibson, D. Evtushinsky, V. Zabolotnyy, B. Buchner and R.J. Cava, Experimental realization of a three-dimensional dirac semimetal, Phys. Rev. Lett. 113 (2014) 027603.
Z.K. Liu et al., Discovery of a three-dimensional topological Dirac semimetal, Na3Bi, Science 343 (2014) 864.
P.W. Anderson, More is different, Science 177 (1972) 393 [INSPIRE].
J. Zinn-Justin, Quantum field theory and critical phenomena, Oxford Science, Oxford U.K. (2002).
M. Moshe and J. Zinn-Justin, Quantum field theory in the large N limit: a review, Phys. Rep. 385 (2003) 69.
R.D. Peccei and H.R. Quinn, CP conservation in the presence of instantons, Phys. Rev. Lett. 38 (1977) 1440 [INSPIRE].
S. Weinberg, A new light boson?, Phys. Rev. Lett. 40 (1978) 223 [INSPIRE].
F. Wilczek, Problem of strong P and T invariance in the presence of instantons, Phys. Rev. Lett. 40 (1978) 279 [INSPIRE].
B. Roy and M.S. Foster, Quantum multicriticality near the Dirac-semimetal to band-insulator critical point in two dimensions: a controlled ascent from one dimension, Phys. Rev. X 8 (2018) 011049 [arXiv:1705.10798] [INSPIRE].
T.R. Kirkpatrick and D. Belitz, Soft modes and nonanalyticities in a clean Dirac metal, Phys. Rev. B 99 (2019) 085109 [arXiv:1812.04592] [INSPIRE].
H.B. Nielsen and M. Ninomiya, No go theorem for regularizing chiral fermions, Phys. Lett. B 105 (1981) 219 [INSPIRE].
T.C. Lang and A.M. Läuchli, Quantum Monte Carlo simulation of the chiral Heisenberg Gross-Neveu-Yukawa phase transition with a single Dirac cone, Phys. Rev. Lett. 123 (2019) 137602 [arXiv:1808.01230] [INSPIRE].
Y. Huang, H. Guo, J. Maciejko, R.T. Scalettar and S. Feng, Antiferromagnetic transitions of Dirac fermions in three dimensions, Phys. Rev. B 102 (2020) 155152 [arXiv:2007.15175] [INSPIRE].
H.R. Quinn and M. WEinstein, Lattice theories of chiral fermions, Phys. Rev. D 34 (1986) 2440 [INSPIRE].
K.G. Wilson, Confinement of quarks, Phys. Rev. D 10 (1974) 2445 [INSPIRE].
J.B. Kogut and L. Susskind, Hamiltonian formulation of Wilson’s lattice gauge theories, Phys. Rev. D 11 (1975) 395 [INSPIRE].
B. Roy, P. Goswami and J.D. Sau, Continuous and discontinuous topological quantum phase transitions, Phys. Rev. B 94 (2016) 041101.
P. Nason, The lattice Schwinger model with SLAC fermions, Nucl. Phys. B 260 (1985) 269 [INSPIRE].
J.P. Costella, A new proposal for the fermion doubling problem, hep-lat/0207008 [INSPIRE].
P. Goswami and S. Chakravarty, Quantum criticality between topological and band insulators in (3 + 1)-dimensions, Phys. Rev. Lett. 107 (2011) 196803 [arXiv:1101.2210] [INSPIRE].
H. Isobe and N. Nagaosa, Theory of quantum critical phenomenon in topological insulator: (3 + 1)D quantum electrodynamics in solids, Phys. Rev. B 86 (2012) 165127 [arXiv:1205.2427] [INSPIRE].
J. González, Phase diagram of the quantum electrodynamics of two-dimensional and three-dimensional Dirac semimetals, Phys. Rev. B 92 (2015) 125115 [arXiv:1502.07640] [INSPIRE].
I.S. Tupitsyn and N.V. Prokof’ev, Stability of Dirac liquids with strong Coulomb interaction, Phys. Rev. Lett. 118 (2017) 026403 [arXiv:1608.00133] [INSPIRE].
R.E. Throckmorton, J. Hofmann, E. Barnes and S. Das Sarma, Many-body effects and ultraviolet renormalization in three-dimensional Dirac materials, Phys. Rev. B 92 (2015) 115101.
V. Juričić, I.F. Herbut, and G.W. Semenoff, Coulomb interaction at the metal-insulator critical point in graphene, Phys. Rev. B 80 (2009) 081405.
J.E. Drut and T.A. Lahde, Is graphene in vacuum an insulator?, Phys. Rev. Lett. 102 (2009) 026802 [arXiv:0807.0834] [INSPIRE].
B. Roy and S. Das Sarma, Quantum phases of interacting electrons in three-dimensional dirty Dirac semimetals, Phys. Rev. B 94 (2016) 115137 [arXiv:1511.06367] [INSPIRE].
V.V. Braguta, M.I. Katsnelson, A.Y. Kotov and A.A. Nikolaev, Monte-Carlo study of Dirac semimetals phase diagram, Phys. Rev. B 94 (2016) 205147 [arXiv:1608.07162] [INSPIRE].
P.L. Zhao and A-M. Wang, Interplay between tilt, disorder, and Coulomb interaction in type-I Dirac fermions, Phys. Rev. B 100 (2019) 125138.
Z.-K. Yang, J.-R. Wang and G.-Z. Liu, Effects of Dirac cone tilt in a two-dimensional Dirac semimetal, Phys. Rev. B 98 (2018) 195123 [arXiv:1807.06536] [INSPIRE].
J. Maciejko and R. Nandkishore, Weyl semimetals with short-range interactions, Phys. Rev. B 90 (2014) 035126 [arXiv:1311.7133] [INSPIRE].
B. Roy, P. Goswami and V. Juričić, Interacting Weyl fermions: phases, phase transitions, and global phase diagram, Phys. Rev. B 95 (2017) 201102.
I.F. Herbut, V. Juricic and B. Roy, Theory of interacting electrons on the honeycomb lattice, Phys. Rev. B 79 (2009) 085116 [arXiv:0811.0610] [INSPIRE].
A.L. Szabó, R. Moessner and B. Roy, Interacting spin-3/2 fermions in a Luttinger (semi)metal: competing phases and their selection in the global phase diagram, arXiv:1811.12415 [INSPIRE].
B. Roy, S.A.A. Ghorashi, M.S. Foster and A.H. Nevidomskyy, Topological superconductivity of spin-3/2 carriers in a three-dimensional doped Luttinger semimetal, Phys. Rev. B 99 (2019) 054505 [arXiv:1708.07825] [INSPIRE].
C.N. Yang and S.C. Zhang, SO4 symmetry in a Hubbard model, Mod. Phys. Lett. B 04 (1990) 759.
A. Auerbach, Interacting electrons and quantum magnetism, Springer, Germany (1994).
M. Hermele, SU(2) gauge theory of the Hubbard model and application to the honeycomb lattice, Phys. Rev. B 76 (2007) 035125 [cond-mat/0701134] [INSPIRE].
T. Ohsaku, BCS and generalized BCS superconductivity in relativistic quantum field theory. 1. Formulation, Phys. Rev. B 65 (2002) 024512 [cond-mat/0112456] [INSPIRE].
L. Fu and E. Berg, Odd-parity topological superconductors: theory and application to CuxBi2Se3, Phys. Rev. Lett. 105 (2010) 097001 [arXiv:0912.3294] [INSPIRE].
K. Li, η-pairing in correlated fermion models with spin-orbit coupling, Phys. Rev. B 102 (2020) 165150 [INSPIRE].
B. Roy, Interacting nodal-line semimetal: proximity effect and spontaneous symmetry breaking, Phys. Rev. B 96 (2017) 041113.
B. Roy, P. Goswami and V. Juricic, Itinerant quantum multicriticality of two-dimensional Dirac fermions, Phys. Rev. B 97 (2018) 205117 [arXiv:1712.05400] [INSPIRE].
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: 2009.05055
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
Szabó, A.L., Roy, B. Emergent chiral symmetry in a three-dimensional interacting Dirac liquid. J. High Energ. Phys. 2021, 4 (2021). https://doi.org/10.1007/JHEP01(2021)004
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2021)004