Abstract
Perturbing the standard Gross-Neveu model for N3 fermions by quartic interactions with the appropriate tensorial contraction patterns, we reduce the original U(N3) symmetry to either U(N) × U(N2) or U(N) × U(N) × U(N). In the large-N limit, we show that in three dimensions such models admit new ultraviolet fixed points with reduced symmetry, besides the well-known one with maximal symmetry. The phase diagram notably presents a new phase with spontaneous symmetry breaking of one U(N) component of the symmetry group.
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References
K.G. Wilson, Quantum field theory models in less than four-dimensions, Phys. Rev. D 7 (1973) 2911 [INSPIRE].
S.R. Coleman, R. Jackiw and H.D. Politzer, Spontaneous Symmetry Breaking in the O(N) Model for Large N ∗, Phys. Rev. D 10 (1974) 2491 [INSPIRE].
P. Di Francesco, P.H. Ginsparg and J. Zinn-Justin, 2-D Gravity and random matrices, Phys. Rept. 254 (1995) 1 [hep-th/9306153] [INSPIRE].
O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [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].
I.R. Klebanov and G. Tarnopolsky, On Large N Limit of Symmetric Traceless Tensor Models, JHEP 10 (2017) 037 [arXiv:1706.00839] [INSPIRE].
D. Benedetti, S. Carrozza, R. Gurau and M. Kolanowski, The 1/N expansion of the symmetric traceless and the antisymmetric tensor models in rank three, arXiv:1712.00249 [INSPIRE].
S. Carrozza, Large N limit of irreducible tensor models: O(N) rank-3 tensors with mixed permutation symmetry, JHEP 06 (2018) 039 [arXiv:1803.02496] [INSPIRE].
T. Azeyanagi, F. Ferrari, P. Gregori, L. Leduc and G. Valette, More on the New Large D Limit of Matrix Models, Annals Phys. 393 (2018) 308 [arXiv:1710.07263] [INSPIRE].
R. Gurau and J.P. Ryan, Colored Tensor Models — a review, SIGMA 8 (2012) 020 [arXiv:1109.4812] [INSPIRE].
R. Gurau, Random Tensors, Oxford University Press, Oxford (2016).
V. Bonzom, New 1/N expansions in random tensor models, JHEP 06 (2013) 062 [arXiv:1211.1657] [INSPIRE].
V. Bonzom, R. Gurau, J.P. Ryan and A. Tanasa, The double scaling limit of random tensor models, JHEP 09 (2014) 051 [arXiv:1404.7517] [INSPIRE].
S. Carrozza and A. Tanasa, O(N) Random Tensor Models, Lett. Math. Phys. 106 (2016) 1531 [arXiv:1512.06718] [INSPIRE].
A. Tanasa, The Multi-Orientable Random Tensor Model, a Review, SIGMA 12 (2016) 056 [arXiv:1512.02087] [INSPIRE].
V. Bonzom, Large N Limits in Tensor Models: Towards More Universality Classes of Colored Triangulations in Dimension d ≥ 2, SIGMA 12 (2016) 073 [arXiv:1603.03570] [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].
E. Witten, An SYK-Like Model Without Disorder, arXiv:1610.09758 [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, KITP strings seminar and Entanglement 2015 program, February 12, April 7, and May 27, 2015.
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].
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. Peng, M. Spradlin and A. Volovich, A Supersymmetric SYK-like Tensor Model, JHEP 05 (2017) 062 [arXiv:1612.03851] [INSPIRE].
C. Krishnan, S. Sanyal and P.N. Bala Subramanian, Quantum Chaos and Holographic Tensor Models, JHEP 03 (2017) 056 [arXiv:1612.06330] [INSPIRE].
C. Krishnan, K.V. Pavan Kumar and D. Rosa, Contrasting SYK-like Models, JHEP 01 (2018) 064 [arXiv:1709.06498] [INSPIRE].
K. Bulycheva, I.R. Klebanov, A. Milekhin and G. Tarnopolsky, Spectra of Operators in Large N Tensor Models, Phys. Rev. D 97 (2018) 026016 [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, JHEP 06 (2018) 094 [arXiv:1707.09352] [INSPIRE].
N. Halmagyi and S. Mondal, Tensor Models for Black Hole Probes, arXiv:1711.04385 [INSPIRE].
I.R. Klebanov, A. Milekhin, F. Popov and G. Tarnopolsky, Spectra of eigenstates in fermionic tensor quantum mechanics, Phys. Rev. D 97 (2018) 106023 [arXiv:1802.10263] [INSPIRE].
C.-M. Chang, S. Colin-Ellerin and M. Rangamani, On Melonic Supertensor Models, JHEP 10 (2018) 157 [arXiv:1806.09903] [INSPIRE].
S. Carrozza and V. Pozsgay, SYK-like tensor quantum mechanics with Sp(N ) symmetry, arXiv:1809.07753 [INSPIRE].
N. Delporte and V. Rivasseau, The Tensor Track V: Holographic Tensors, 2018, arXiv:1804.11101 [INSPIRE].
V. Rosenhaus, An introduction to the SYK model, arXiv:1807.03334 [INSPIRE].
I.R. Klebanov, F. Popov and G. Tarnopolsky, TASI Lectures on Large N Tensor Models, PoS(TASI2017)004 (2018) [arXiv:1808.09434] [INSPIRE].
D. Benedetti, S. Carrozza, R. Gurau and A. Sfondrini, Tensorial Gross-Neveu models, JHEP 01 (2018) 003 [arXiv:1710.10253] [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].
S. Prakash and R. Sinha, A Complex Fermionic Tensor Model in d Dimensions, JHEP 02 (2018) 086 [arXiv:1710.09357] [INSPIRE].
S. Giombi, I.R. Klebanov, F. Popov, S. Prakash and G. Tarnopolsky, Prismatic Large N Models for Bosonic Tensors, Phys. Rev. D 98 (2018) 105005 [arXiv:1808.04344] [INSPIRE].
E. Witten, Chiral Symmetry, the 1/N Expansion and the SU(N) Thirring Model, Nucl. Phys. B 145 (1978) 110 [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].
B. Rosenstein, B.J. Warr and S.H. Park, The Four Fermi Theory Is Renormalizable in (2 + 1)-Dimensions, Phys. Rev. Lett. 62 (1989) 1433 [INSPIRE].
C. de Calan, P.A. Faria da Veiga, J. Magnen and R. Seneor, Constructing the three-dimensional Gross-Neveu model with a large number of flavor components, Phys. Rev. Lett. 66 (1991) 3233 [INSPIRE].
B. Rosenstein, B. Warr and S.H. Park, Dynamical symmetry breaking in four Fermi interaction models, Phys. Rept. 205 (1991) 59 [INSPIRE].
I.R. Klebanov and A.M. Polyakov, AdS dual of the critical O(N) vector model, Phys. Lett. B 550 (2002) 213 [hep-th/0210114] [INSPIRE].
E. Sezgin and P. Sundell, Holography in 4D (super) higher spin theories and a test via cubic scalar couplings, JHEP 07 (2005) 044 [hep-th/0305040] [INSPIRE].
S. Giombi, Higher Spin — CFT Duality, in Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015), Boulder, CO, U.S.A., June 1-26, 2015, pp. 137-214 (2017) [https://doi.org/10.1142/9789813149441_0003] [arXiv:1607.02967] [INSPIRE].
M.A. Vasiliev, From Coxeter Higher-Spin Theories to Strings and Tensor Models, JHEP 08 (2018) 051 [arXiv:1804.06520] [INSPIRE].
D.J. Gross and A. Neveu, Dynamical Symmetry Breaking in Asymptotically Free Field Theories, Phys. Rev. D 10 (1974) 3235 [INSPIRE].
G. Parisi, The Theory of Nonrenormalizable Interactions. 1. The Large N Expansion, Nucl. Phys. B 100 (1975) 368 [INSPIRE].
V.A. Nguyen, S. Dartois and B. Eynard, An analysis of the intermediate field theory of T 4 tensor model, JHEP 01 (2015) 013 [arXiv:1409.5751] [INSPIRE].
P. Diaz and J.A. Rosabal, Spontaneous Symmetry Breaking in Tensor Theories, JHEP 01 (2019) 094 [arXiv:1809.10153] [INSPIRE].
O. Aharony, G. Gur-Ari and R. Yacoby, d = 3 Bosonic Vector Models Coupled to Chern-Simons Gauge Theories, JHEP 03 (2012) 037 [arXiv:1110.4382] [INSPIRE].
S. Giombi, S. Minwalla, S. Prakash, S.P. Trivedi, S.R. Wadia and X. Yin, Chern-Simons Theory with Vector Fermion Matter, Eur. Phys. J. C 72 (2012) 2112 [arXiv:1110.4386] [INSPIRE].
J.-H. Park, Lecture Notes on Clifford Algebra, in Proceedings of the First Modave Summer School in Mathematical Physics, vol. 1, (2005), pg. 197.
D. Benedetti and R. Gurau, Symmetry breaking in tensor models, Phys. Rev. D 92 (2015) 104041 [arXiv:1506.08542] [INSPIRE].
R. Gurau, The Double Scaling Limit in Arbitrary Dimensions: A Toy Model, Phys. Rev. D 84 (2011) 124051 [arXiv:1110.2460] [INSPIRE].
V. Bonzom and F. Combes, Tensor models from the viewpoint of matrix models: the case of loop models on random surfaces, Ann. Inst. H. Poincaré Comb. Phys. Interact. 2 (2015) 1 [arXiv:1304.4152] [INSPIRE].
A. Anderson, R.C. Myers and V. Periwal, Complex random surfaces, Phys. Lett. B 254 (1991) 89 [INSPIRE].
P. Di Francesco, Rectangular matrix models and combinatorics of colored graphs, Nucl. Phys. B 648 (2003) 461 [cond-mat/0208037] [INSPIRE].
T.R. Morris, Checkered surfaces and complex matrices, Nucl. Phys. B 356 (1991) 703 [INSPIRE].
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Benedetti, D., Delporte, N. Phase diagram and fixed points of tensorial Gross-Neveu models in three dimensions. J. High Energ. Phys. 2019, 218 (2019). https://doi.org/10.1007/JHEP01(2019)218
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DOI: https://doi.org/10.1007/JHEP01(2019)218