Abstract
The Klebanov-Tarnopolsky tensor model is a quantum field theory for rank-three tensor scalar fields with certain quartic potential. The theory possesses an unusual large N limit known as the melonic limit that is strongly coupled yet solvable, producing at large distance a rare example of non-perturbative non-supersymmetric conformal field theory that admits analytic solutions. We study the dynamics of defects in the tensor model defined by localized magnetic field couplings on a p-dimensional subspace in the d-dimensional spacetime. While we work with general p and d, the physically interesting cases include line defects in d = 2, 3 and surface defects in d = 3. By identifying a novel large N limit that generalizes the melonic limit in the presence of defects, we prove that the defect one-point function of the scalar field only receives contributions from a subset of the Feynman diagrams in the shape of melonic trees. These diagrams can be resummed using a closed Schwinger-Dyson equation which enables us to determine non-perturbatively this defect one-point function. At large distance, the solutions we find describe nontrivial conformal defects and we discuss their defect renormalization group (RG) flows. In particular, for line defects, we solve the exact RG flow between the trivial and the conformal lines in d = 4 − ϵ. We also compute the exact line defect entropy and verify the g-theorem. Furthermore we analyze the defect two-point function of the scalar field and its decomposition via the operator-product-expansion, providing explicit formulae for one-point functions of bilinear operators and the stress-energy tensor.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
K.G. Wilson, Confinement of quarks, Phys. Rev. D 10 (1974) 2445 [INSPIRE].
J. Kondo, Resistance minimum in dilute magnetic alloys, Prog. Theor. Phys. 32 (1964) 37 [INSPIRE].
I. Affleck and A.W.W. Ludwig, Universal noninteger ‘ground state degeneracy’ in critical quantum systems, Phys. Rev. Lett. 67 (1991) 161 [INSPIRE].
I. Affleck, Conformal field theory approach to the Kondo effect, Acta Phys. Polon. B 26 (1995) 1869 [cond-mat/9512099] [INSPIRE].
S. Yamaguchi, Holographic RG flow on the defect and g theorem, JHEP 10 (2002) 002 [hep-th/0207171] [INSPIRE].
D. Friedan and A. Konechny, On the boundary entropy of one-dimensional quantum systems at low temperature, Phys. Rev. Lett. 93 (2004) 030402 [hep-th/0312197] [INSPIRE].
D. Gaiotto, Boundary F-maximization, arXiv:1403.8052 [INSPIRE].
K. Jensen and A. O’Bannon, Constraint on defect and boundary renormalization group flows, Phys. Rev. Lett. 116 (2016) 091601 [arXiv:1509.02160] [INSPIRE].
H. Casini, I. Salazar Landea and G. Torroba, The g-theorem and quantum information theory, JHEP 10 (2016) 140 [arXiv:1607.00390] [INSPIRE].
H. Casini, I. Salazar Landea and G. Torroba, Irreversibility in quantum field theories with boundaries, JHEP 04 (2019) 166 [arXiv:1812.08183] [INSPIRE].
N. Kobayashi, T. Nishioka, Y. Sato and K. Watanabe, Towards a C-theorem in defect CFT, JHEP 01 (2019) 039 [arXiv:1810.06995] [INSPIRE].
Y. Wang, Defect a-theorem and a-maximization, JHEP 02 (2022) 061 [arXiv:2101.12648] [INSPIRE].
G. Cuomo, Z. Komargodski and A. Raviv-Moshe, Renormalization group flows on line defects, Phys. Rev. Lett. 128 (2022) 021603 [arXiv:2108.01117] [INSPIRE].
H.W. Diehl and S. Dietrich, Field-theoretical approach to static critical phenomena in semi-infinite systems, Z. Phys. B 42 (1981) 65 [INSPIRE].
H.W. Diehl, S. Dietrich and E. Eisenriegler, Universality, irrelevant surface operators, and corrections to scaling in systems with free surfaces and defect planes, Phys. Rev. B 27 (1983) 2937 [INSPIRE].
J.L. Cardy, Conformal invariance and surface critical behavior, Nucl. Phys. B 240 (1984) 514 [INSPIRE].
E. Eisenriegler, Universal amplitude ratios for the surface tension of polymer solutions, J. Chem. Phys. 81 (1984) 4666.
H.W. Diehl, The theory of boundary critical phenomena, Int. J. Mod. Phys. B 11 (1997) 3503 [cond-mat/9610143] [INSPIRE].
B.M. Law, Wetting, adsorption and surface critical phenomena, Prog. Surf. Sci. 66 (2001) 159.
M. Henningson and K. Skenderis, Weyl anomaly for Wilson surfaces, JHEP 06 (1999) 012 [hep-th/9905163] [INSPIRE].
C.R. Graham and E. Witten, Conformal anomaly of submanifold observables in AdS/CFT correspondence, Nucl. Phys. B 546 (1999) 52 [hep-th/9901021] [INSPIRE].
A. Schwimmer and S. Theisen, Entanglement entropy, trace anomalies and holography, Nucl. Phys. B 801 (2008) 1 [arXiv:0802.1017] [INSPIRE].
G.Y. Cho, K. Shiozaki, S. Ryu and A.W.W. Ludwig, Relationship between symmetry protected topological phases and boundary conformal field theories via the entanglement spectrum, J. Phys. A 50 (2017) 304002 [arXiv:1606.06402] [INSPIRE].
T. Dimofte, D. Gaiotto and N.M. Paquette, Dual boundary conditions in 3d SCFT’s, JHEP 05 (2018) 060 [arXiv:1712.07654] [INSPIRE].
J. Estes, D. Krym, A. O’Bannon, B. Robinson and R. Rodgers, Wilson surface central charge from holographic entanglement entropy, JHEP 05 (2019) 032 [arXiv:1812.00923] [INSPIRE].
C.P. Herzog and I. Shamir, How a-type anomalies can depend on marginal couplings, Phys. Rev. Lett. 124 (2020) 011601 [arXiv:1907.04952] [INSPIRE].
L. Bianchi, Marginal deformations and defect anomalies, Phys. Rev. D 100 (2019) 126018 [arXiv:1907.06193] [INSPIRE].
N. Drukker, S. Giombi, A.A. Tseytlin and X. Zhou, Defect CFT in the 6d (2, 0) theory from M2 brane dynamics in AdS7 × S4, JHEP 07 (2020) 101 [arXiv:2004.04562] [INSPIRE].
C.P. Herzog and I. Shamir, Anomalies from correlation functions in defect conformal field theory, JHEP 07 (2021) 091 [arXiv:2103.06311] [INSPIRE].
A. Chalabi, A. O’Bannon, B. Robinson and J. Sisti, Central charges of 2d superconformal defects, JHEP 05 (2020) 095 [arXiv:2003.02857] [INSPIRE].
N. Drukker, M. Probst and M. Trépanier, Surface operators in the 6d N = (2, 0) theory, J. Phys. A 53 (2020) 365401 [arXiv:2003.12372] [INSPIRE].
Y. Wang, Surface defect, anomalies and b-extremization, JHEP 11 (2021) 122 [arXiv:2012.06574] [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized global symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, time reversal, and temperature, JHEP 05 (2017) 091 [arXiv:1703.00501] [INSPIRE].
P. Liendo, L. Rastelli and B.C. van Rees, The bootstrap program for boundary CFTd, JHEP 07 (2013) 113 [arXiv:1210.4258] [INSPIRE].
D. Gaiotto, D. Mazac and M.F. Paulos, Bootstrapping the 3d Ising twist defect, JHEP 03 (2014) 100 [arXiv:1310.5078] [INSPIRE].
F. Gliozzi, P. Liendo, M. Meineri and A. Rago, Boundary and interface CFTs from the conformal bootstrap, JHEP 05 (2015) 036 [arXiv:1502.07217] [INSPIRE].
M. Billò, V. Gonçalves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP 04 (2016) 091 [arXiv:1601.02883] [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
V. Pestun, Localization of the four-dimensional N = 4 SYM to a two-sphere and 1/8 BPS Wilson loops, JHEP 12 (2012) 067 [arXiv:0906.0638] [INSPIRE].
S. Giombi and V. Pestun, The 1/2 BPS ’t Hooft loops in N = 4 SYM as instantons in 2d Yang-Mills, J. Phys. A 46 (2013) 095402 [arXiv:0909.4272] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].
N. Drukker, D. Gaiotto and J. Gomis, The virtue of defects in 4D gauge theories and 2D CFTs, JHEP 06 (2011) 025 [arXiv:1003.1112] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Exact results for supersymmetric Abelian vortex loops in 2 + 1 dimensions, JHEP 06 (2013) 099 [arXiv:1211.2861] [INSPIRE].
N. Drukker, T. Okuda and F. Passerini, Exact results for vortex loop operators in 3d supersymmetric theories, JHEP 07 (2014) 137 [arXiv:1211.3409] [INSPIRE].
H.-C. Kim, J. Kim and S. Kim, Instantons on the 5-sphere and M5-branes, arXiv:1211.0144 [INSPIRE].
B. Assel and J. Gomis, Mirror symmetry and loop operators, JHEP 11 (2015) 055 [arXiv:1506.01718] [INSPIRE].
H.-C. Kim, Line defects and 5d instanton partition functions, JHEP 03 (2016) 199 [arXiv:1601.06841] [INSPIRE].
Y. Wang, Taming defects in N = 4 super-Yang-Mills, JHEP 08 (2020) 021 [arXiv:2003.11016] [INSPIRE].
M. Beccaria, S. Giombi and A.A. Tseytlin, Correlators on non-supersymmetric Wilson line in N = 4 SYM and AdS2/CFT1, JHEP 05 (2019) 122 [arXiv:1903.04365] [INSPIRE].
S. Giombi and H. Khanchandani, CFT in AdS and boundary RG flows, JHEP 11 (2020) 118 [arXiv:2007.04955] [INSPIRE].
C.P. Herzog and N. Kobayashi, The O(N) model with ϕ6 potential in R2 × R+, JHEP 09 (2020) 126 [arXiv:2005.07863] [INSPIRE].
M.A. Metlitski, Boundary criticality of the O(N) model in d = 3 critically revisited, SciPost Phys. 12 (2022) 131 [arXiv:2009.05119] [INSPIRE].
S. Giombi, E. Helfenberger, Z. Ji and H. Khanchandani, Monodromy defects from hyperbolic space, JHEP 02 (2022) 041 [arXiv:2102.11815] [INSPIRE].
G. Cuomo, Z. Komargodski and M. Mezei, Localized magnetic field in the O(N) model, JHEP 02 (2022) 134 [arXiv:2112.10634] [INSPIRE].
G. Cuomo, M. Mezei and A. Raviv-Moshe, Boundary conformal field theory at large charge, JHEP 10 (2021) 143 [arXiv:2108.06579] [INSPIRE].
M. Beccaria, S. Giombi and A.A. Tseytlin, Wilson loop in general representation and RG flow in 1D defect QFT, J. Phys. A 55 (2022) 255401 [arXiv:2202.00028] [INSPIRE].
G. Cuomo, Z. Komargodski, M. Mezei and A. Raviv-Moshe, Spin impurities, Wilson lines and semiclassics, JHEP 06 (2022) 112 [arXiv:2202.00040] [INSPIRE].
D. Rodriguez-Gomez, A scaling limit for line and surface defects, JHEP 06 (2022) 071 [arXiv:2202.03471] [INSPIRE].
I. Buhl-Mortensen, M. de Leeuw, A.C. Ipsen, C. Kristjansen and M. Wilhelm, Asymptotic one-point functions in gauge-string duality with defects, Phys. Rev. Lett. 119 (2017) 261604 [arXiv:1704.07386] [INSPIRE].
D. Grabner, N. Gromov and J. Julius, Excited states of one-dimensional defect CFTs from the quantum spectral curve, JHEP 07 (2020) 042 [arXiv:2001.11039] [INSPIRE].
S. Komatsu and Y. Wang, Non-perturbative defect one-point functions in planar N = 4 super-Yang-Mills, Nucl. Phys. B 958 (2020) 115120 [arXiv:2004.09514] [INSPIRE].
K. Lang and W. Rühl, The critical O(N ) sigma model at dimensions 2 < d < 4: fusion coefficients and anomalous dimensions, Nucl. Phys. B 400 (1993) 597 [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].
J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a higher spin symmetry, J. Phys. A 46 (2013) 214011 [arXiv:1112.1016] [INSPIRE].
J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a slightly broken higher spin symmetry, Class. Quant. Grav. 30 (2013) 104003 [arXiv:1204.3882] [INSPIRE].
S. Giombi and V. Kirilin, Anomalous dimensions in CFT with weakly broken higher spin symmetry, JHEP 11 (2016) 068 [arXiv:1601.01310] [INSPIRE].
I.R. Klebanov, F. Popov and G. Tarnopolsky, TASI lectures on large N tensor models, PoS TASI2017 (2018) 004 [arXiv:1808.09434] [INSPIRE].
G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72 (1974) 461 [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].
R. Gurau, Colored group field theory, Commun. Math. Phys. 304 (2011) 69 [arXiv:0907.2582] [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].
S. Carrozza and A. Tanasa, O(N) random tensor models, Lett. Math. Phys. 106 (2016) 1531 [arXiv:1512.06718] [INSPIRE].
E. Witten, An SYK-like model without disorder, J. Phys. A 52 (2019) 474002 [arXiv:1610.09758] [INSPIRE].
I.R. Klebanov, P.N. Pallegar and F.K. Popov, Majorana fermion quantum mechanics for higher rank tensors, Phys. Rev. D 100 (2019) 086003 [arXiv:1905.06264] [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].
E. Lauria, P. Liendo, B.C. Van Rees and X. Zhao, Line and surface defects for the free scalar field, JHEP 01 (2021) 060 [arXiv:2005.02413] [INSPIRE].
C. Behan, L. Di Pietro, E. Lauria and B.C. Van Rees, Bootstrapping boundary-localized interactions, JHEP 12 (2020) 182 [arXiv:2009.03336] [INSPIRE].
L. Di Pietro, E. Lauria and P. Niro, 3d large N vector models at the boundary, SciPost Phys. 11 (2021) 050 [arXiv:2012.07733] [INSPIRE].
C. Behan, L. Di Pietro, E. Lauria and B.C. van Rees, Bootstrapping boundary-localized interactions II. Minimal models at the boundary, JHEP 03 (2022) 146 [arXiv:2111.04747] [INSPIRE].
L. Bianchi, A. Chalabi, V. Procházka, B. Robinson and J. Sisti, Monodromy defects in free field theories, JHEP 08 (2021) 013 [arXiv:2104.01220] [INSPIRE].
F. Parisen Toldin, F.F. Assaad and S. Wessel, Critical behavior in the presence of an order-parameter pinning field, Phys. Rev. B 95 (2017) 014401 [arXiv:1607.04270] [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].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
A. Patashinskii and V. Pokrovskii, Second order phase transitions in a Bose fluid, Sov. Phys. JETP 19 (1964) 677.
D. Benedetti, R. Gurau and K. Suzuki, Conformal symmetry and composite operators in the O(N)3 tensor field theory, JHEP 06 (2020) 113 [arXiv:2002.07652] [INSPIRE].
J. Kim, I.R. Klebanov, G. Tarnopolsky and W. Zhao, Symmetry breaking in coupled SYK or tensor models, Phys. Rev. X 9 (2019) 021043 [arXiv:1902.02287] [INSPIRE].
D. Benedetti, Melonic CFTs, PoS CORFU2019 (2020) 168 [arXiv:2004.08616] [INSPIRE].
D. Benedetti, Instability of complex CFTs with operators in the principal series, JHEP 05 (2021) 004 [arXiv:2103.01813] [INSPIRE].
D. Benedetti, R. Gurau and S. Harribey, Line of fixed points in a bosonic tensor model, JHEP 06 (2019) 053 [arXiv:1903.03578] [INSPIRE].
D. Benedetti, R. Gurau, S. Harribey and D. Lettera, The F-theorem in the melonic limit, JHEP 02 (2022) 147 [arXiv:2111.11792] [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from conformal field theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
S. El-Showk and K. Papadodimas, Emergent spacetime and holographic CFTs, JHEP 10 (2012) 106 [arXiv:1101.4163] [INSPIRE].
V. Gorbenko, S. Rychkov and B. Zan, Walking, weak first-order transitions, and complex CFTs, JHEP 10 (2018) 108 [arXiv:1807.11512] [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].
F.K. Popov, Supersymmetric tensor model at large N and small ϵ, Phys. Rev. D 101 (2020) 026020 [arXiv:1907.02440] [INSPIRE].
I.R. Klebanov and G. Tarnopolsky, On large N limit of symmetric traceless tensor models, JHEP 10 (2017) 037 [arXiv:1706.00839] [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].
E. Witten, An SYK-like model without disorder, J. Phys. A 52 (2019) 474002 [arXiv:1610.09758] [INSPIRE].
J. Liu, E. Perlmutter, V. Rosenhaus and D. Simmons-Duffin, d-dimensional SYK, AdS loops, and 6j symbols, JHEP 03 (2019) 052 [arXiv:1808.00612] [INSPIRE].
D.A. Roberts, D. Stanford and A. Streicher, Operator growth in the SYK model, JHEP 06 (2018) 122 [arXiv:1802.02633] [INSPIRE].
C.P. Herzog and A. Shrestha, Two point functions in defect CFTs, JHEP 04 (2021) 226 [arXiv:2010.04995] [INSPIRE].
D.J. Gross and V. Rosenhaus, All point correlation functions in SYK, JHEP 12 (2017) 148 [arXiv:1710.08113] [INSPIRE].
I.R. Klebanov, F. Popov and G. Tarnopolsky, TASI lectures on large N tensor models, PoS TASI2017 (2018) 004 [arXiv:1808.09434] [INSPIRE].
J. Padayasi, A. Krishnan, M.A. Metlitski, I.A. Gruzberg and M. Meineri, The extraordinary boundary transition in the 3d O(N) model via conformal bootstrap, SciPost Phys. 12 (2022) 190 [arXiv:2111.03071] [INSPIRE].
C. Behan, Unitary subsector of generalized minimal models, Phys. Rev. D 97 (2018) 094020 [arXiv:1712.06622] [INSPIRE].
I.R. Klebanov and A.A. Tseytlin, Entropy of near extremal black p-branes, Nucl. Phys. B 475 (1996) 164 [hep-th/9604089] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, Gauge theories labelled by three-manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, 3-manifolds and 3d indices, Adv. Theor. Math. Phys. 17 (2013) 975 [arXiv:1112.5179] [INSPIRE].
J. Eckhard, S. Schäfer-Nameki and J.-M. Wong, An N = 1 3d-3d correspondence, JHEP 07 (2018) 052 [arXiv:1804.02368] [INSPIRE].
D. Gang, N. Kim and S. Lee, Holography of 3d-3d correspondence at large N, JHEP 04 (2015) 091 [arXiv:1409.6206] [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: 2206.14206
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
Popov, F.K., Wang, Y. Non-perturbative defects in tensor models from melonic trees. J. High Energ. Phys. 2022, 57 (2022). https://doi.org/10.1007/JHEP11(2022)057
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2022)057