Abstract
We define and study various tensorial generalizations of the Gross-Neveu model in two dimensions, that is, models with four-fermion interactions and G3 symmetry, where we take either G = U(N) or G = O(N). Such models can also be viewed as two-dimensional generalizations of the Sachdev-Ye-Kitaev model, or more precisely of its tensorial counterpart introduced by Klebanov and Tarnopolsky, which is in part our motivation for studying them. Using the Schwinger-Dyson equations at large-N, we discuss the phenomenon of dynamical mass generation and possible combinations of couplings to avoid it. For the case G = U(N),we introduce an intermediate field representation and perform a stability analysis of the vacua. It turns out that the only apparently viable combination of couplings that avoids mass generation corresponds to an unstable vacuum. The stable vacuum breaks U(N)3 invariance, in contradiction with the Coleman-Mermin-Wagner theorem, but this is an artifact of the large-N expansion, similar to the breaking of continuous chiral symmetry in the chiral Gross-Neveu model.
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References
M. Moshe and J. Zinn-Justin, Quantum field theory in the large-N limit: a review, Phys. Rept. 385 (2003) 69 [hep-th/0306133] [INSPIRE].
P. Di Francesco, P.H. Ginsparg and J. Zinn-Justin, 2D gravity and random matrices, Phys. Rept. 254 (1995) 1 [hep-th/9306153] [INSPIRE].
J. Ambjørn, B. Durhuus and T. Jonsson, Three-dimensional simplicial quantum gravity and generalized matrix models, Mod. Phys. Lett. A 6 (1991) 1133 [INSPIRE].
N. Sasakura, Tensor model for gravity and orientability of manifold, Mod. Phys. Lett. A 6 (1991) 2613 [INSPIRE].
R. Gurau, The 1/N expansion of colored tensor models, Annales Henri Poincaré 12 (2011) 829 [arXiv:1011.2726] [INSPIRE].
R. Gurau and V. Rivasseau, The 1/N expansion of colored tensor models in arbitrary dimension, EPL 95 (2011) 50004 [arXiv:1101.4182] [INSPIRE].
R. Gurau, The complete 1/N expansion of colored tensor models in arbitrary dimension, Annales Henri Poincaré 13 (2012) 399 [arXiv:1102.5759] [INSPIRE].
V. Bonzom, R. Gurau, A. Riello and V. Rivasseau, Critical behavior of colored tensor models in the large-N limit, Nucl. Phys. B 853 (2011) 174 [arXiv:1105.3122] [INSPIRE].
V. Bonzom, R. Gurau and V. Rivasseau, Random tensor models in the large-N limit: uncoloring the colored tensor models, Phys. Rev. D 85 (2012) 084037 [arXiv:1202.3637] [INSPIRE].
D. Oriti, The microscopic dynamics of quantum space as a group field theory, arXiv:1110.5606 [INSPIRE].
J. Ben Geloun and V. Rivasseau, A renormalizable 4-dimensional tensor field theory, Commun. Math. Phys. 318 (2013) 69 [arXiv:1111.4997] [INSPIRE].
S. Carrozza, D. Oriti and V. Rivasseau, Renormalization of a SU(2) tensorial group field theory in three dimensions, Commun. Math. Phys. 330 (2014) 581 [arXiv:1303.6772] [INSPIRE].
S. Carrozza, Tensorial methods and renormalization in group field theories, arXiv:1310.3736 [INSPIRE].
D. Benedetti, J. Ben Geloun and D. Oriti, Functional renormalisation group approach for tensorial group field theory: a rank-3 model, JHEP 03 (2015) 084 [arXiv:1411.3180] [INSPIRE].
D. Benedetti and V. Lahoche, Functional renormalization group approach for tensorial group field theory: a rank-6 model with closure constraint, Class. Quant. Grav. 33 (2016) 095003 [arXiv:1508.06384] [INSPIRE].
S. Carrozza, Flowing in group field theory space: a review, SIGMA 12 (2016) 070 [arXiv:1603.01902] [INSPIRE].
S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
A. Kitaev, A simple model of quantum holography, talk given at KITP strings seminar and Entanglement program, February 12, April 7 and May 27, Santa Barbara, U.S.A. (2015), http://online.kitp.ucsb.edu/online/entangled15/.
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
J. Polchinski and V. Rosenhaus, The spectrum in the Sachdev-Ye-Kitaev model, JHEP 04 (2016) 001 [arXiv:1601.06768] [INSPIRE].
E. Witten, An SYK-like model without disorder, arXiv:1610.09758 [INSPIRE].
I.R. Klebanov and G. Tarnopolsky, Uncolored random tensors, melon diagrams and the Sachdev-Ye-Kitaev models, Phys. Rev. D 95 (2017) 046004 [arXiv:1611.08915] [INSPIRE].
C. Krishnan, K.V.P. Kumar and S. Sanyal, Random matrices and holographic tensor models, JHEP 06 (2017) 036 [arXiv:1703.08155] [INSPIRE].
V. Bonzom, L. Lionni and A. Tanasa, Diagrammatics of a colored SYK model and of an SYK-like tensor model, leading and next-to-leading orders, J. Math. Phys. 58 (2017) 052301 [arXiv:1702.06944] [INSPIRE].
K. Bulycheva, I.R. Klebanov, A. Milekhin and G. Tarnopolsky, Spectra of operators in large-N tensor models, arXiv:1707.09347 [INSPIRE].
S. Choudhury, A. Dey, I. Halder, L. Janagal, S. Minwalla and R. Poojary, Notes on melonic O(N)q−1 tensor models, arXiv:1707.09352 [INSPIRE].
C. Peng, M. Spradlin and A. Volovich, A supersymmetric SYK-like tensor model, JHEP 05 (2017) 062 [arXiv:1612.03851] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
S. Giombi, I.R. Klebanov and G. Tarnopolsky, Bosonic tensor models at large-N and small ϵ, Phys. Rev. D 96 (2017) 106014 [arXiv:1707.03866] [INSPIRE].
M. Berkooz, P. Narayan, M. Rozali and J. Simón, Comments on the random thirring model, JHEP 09 (2017) 057 [arXiv:1702.05105] [INSPIRE].
J. Murugan, D. Stanford and E. Witten, More on supersymmetric and 2d analogs of the SYK model, JHEP 08 (2017) 146 [arXiv:1706.05362] [INSPIRE].
F. Ferrari, The large D limit of planar diagrams, arXiv:1701.01171 [INSPIRE].
T. Azeyanagi, F. Ferrari and F.I. Schaposnik Massolo, Phase diagram of planar matrix quantum mechanics, tensor and SYK models, arXiv:1707.03431 [INSPIRE].
F. Ferrari, V. Rivasseau and G. Valette, A new large-N expansion for general matrix-tensor models, arXiv:1709.07366 [INSPIRE].
T. Azeyanagi, F. Ferrari, P. Gregori, L. Leduc and G. Valette, More on the new large D limit of matrix models, arXiv:1710.07263 [INSPIRE].
D.J. Gross and A. Neveu, Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D 10 (1974) 3235 [INSPIRE].
N.D. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one-dimensional or two-dimensional isotropic Heisenberg models, Phys. Rev. Lett. 17 (1966) 1133 [INSPIRE].
S.R. Coleman, There are no Goldstone bosons in two-dimensions, Commun. Math. Phys. 31 (1973) 259 [INSPIRE].
S. Prakash and R. Sinha, A complex fermionic tensor model in d dimensions, arXiv:1710.09357 [INSPIRE].
R.F. Dashen, B. Hasslacher and A. Neveu, Semiclassical bound states in an asymptotically free theory, Phys. Rev. D 12 (1975) 2443 [INSPIRE].
R. Gurau, Random tensors, Oxford University Press, Oxford U.K. (2016).
P.K. Mitter and P.H. Weisz, Asymptotic scale invariance in a massive thirring model with U(N) symmetry, Phys. Rev. D 8 (1973) 4410 [INSPIRE].
K.G. Klimenko, Generalization of Gross-Neveu model to the case of several coupling constants, Theor. Math. Phys. 66 (1986) 252 [Teor. Mat. Fiz. 66 (1986) 381] [INSPIRE].
A. Bondi, G. Curci, G. Paffuti and P. Rossi, Metric and central charge in the perturbative approach to two-dimensional fermionic models, Annals Phys. 199 (1990) 268 [INSPIRE].
R. Gurau and J.P. Ryan, Colored tensor models — A review, SIGMA 8 (2012) 020 [arXiv:1109.4812] [INSPIRE].
D. Benedetti and R. Gurau, Symmetry breaking in tensor models, Phys. Rev. D 92 (2015) 104041 [arXiv:1506.08542] [INSPIRE].
S. Carrozza and A. Tanasa, O(N) random tensor models, Lett. Math. Phys. 106 (2016) 1531 [arXiv:1512.06718] [INSPIRE].
R. Gurau, The complete 1/N expansion of a SYK-like tensor model, Nucl. Phys. B 916 (2017) 386 [arXiv:1611.04032] [INSPIRE].
S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph, J. Number Theor. 148 (2015) 328 [arXiv:1309.5865] [INSPIRE].
L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms, J. Math. Phys. 55 (2014) 102301 [arXiv:1405.5640] [INSPIRE].
E. Witten, Chiral symmetry, the 1/n expansion and the SU(N) Thirring model, Nucl. Phys. B 145 (1978) 110 [INSPIRE].
D. Anninos, S.A. Hartnoll and N. Iqbal, Holography and the Coleman-Mermin-Wagner theorem, Phys. Rev. D 82 (2010) 066008 [arXiv:1005.1973] [INSPIRE].
H. Eichenherr and M. Forger, On the dual symmetry of the nonlinear σ-models, Nucl. Phys B 155 (1979) 381 [INSPIRE].
E. Brezin, C. Itzykson, J. Zinn-Justin and J.B. Zuber, Remarks about the existence of nonlocal charges in two-dimensional models, Phys. Lett. B 82 (1979) 442.
S. Hikami and E. Brézin, Large order behavior of the 1/N expansion in zero-dimensions and one-dimensions, J. Phys. A 12 (1979) 759 [INSPIRE].
A. McKane and M. Stone, Nonlinear σ-models: a perturbative approach to symmetry restoration, Nucl. Phys. B 163 (1980) 169 [INSPIRE].
E. Abdalla, M.C.B. Abdalla and K.D. Rothe, Nonperturbative methods in two-dimensional quantum field theory, World Scientific, Singapore (1991).
D. Bombardelli et al., An integrability primer for the gauge-gravity correspondence: An introduction, J. Phys. A 49 (2016) 320301 [arXiv:1606.02945] [INSPIRE].
H.J. de Vega, H. Eichenherr and J.M. Maillet, Yang-Baxter algebras of monodromy matrices in integrable quantum field theories, Nucl. Phys. B 240 (1984) 377 [INSPIRE].
T. Hauer, Massive current algebra in the many flavor chiral Gross-Neveu model, Nucl. Phys. B 502 (1997) 436 [hep-th/9702016] [INSPIRE].
F. Loebbert, Lectures on Yangian symmetry, J. Phys. A 49 (2016) 323002 [arXiv:1606.02947] [INSPIRE].
M.R. Mehta, Euclidean continuation of the Dirac fermion, Phys. Rev. Lett. 65 (1990) 1983 [Erratum ibid. 66 (1991) 522] [INSPIRE].
P. van Nieuwenhuizen and A. Waldron, On euclidean spinors and Wick rotations, Phys. Lett. B 389 (1996) 29 [hep-th/9608174] [INSPIRE].
H. Nicolai, A Possible constructive approach to (SUPER \( \phi \) 3 ) in four-dimensions. 1. Euclidean formulation of the model, Nucl. Phys. B 140 (1978) 294 [INSPIRE].
D. Borthwick, Euclidean Majorana fermions, fermionic integrals and relative Pfaffians, J. Math. Phys. 34 (1993) 2691 [INSPIRE].
J. Ben Geloun and S. Ramgoolam, Counting tensor model observables and branched covers of the 2-sphere, Ann. Inst. Henri Poincaré D 1 (2014) 77 [arXiv:1307.6490] [INSPIRE].
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Benedetti, D., Carrozza, S., Gurau, R. et al. Tensorial Gross-Neveu models. J. High Energ. Phys. 2018, 3 (2018). https://doi.org/10.1007/JHEP01(2018)003
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DOI: https://doi.org/10.1007/JHEP01(2018)003