Abstract
Based on the quantum renormalization group, we derive the bulk geometry that emerges in the holographic dual of the fermionic U(N ) vector model at a nonzero charge density. The obstruction that prohibits the metallic state from being smoothly deformable to the direct product state under the renormalization group flow gives rise to a horizon at a finite radial coordinate in the bulk. The region outside the horizon is described by the Lifshitz geometry with a higher-spin hair determined by microscopic details of the boundary theory. On the other hand, the interior of the horizon is not described by any Riemannian manifold, as it exhibits an algebraic non-locality. The non-local structure inside the horizon carries the information on the shape of the filled Fermi sea.
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References
S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
G. ’t Hooft, The black hole interpretation of string theory, Nucl. Phys. B 335 (1990) 138 [INSPIRE].
L. Susskind, L. Thorlacius and J. Uglum, The stretched horizon and black hole complementarity, Phys. Rev. D 48 (1993) 3743 [hep-th/9306069] [INSPIRE].
S.D. Mathur, The information paradox: a pedagogical introduction, Class. Quant. Grav. 26 (2009) 224001 [arXiv:0909.1038] [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
P. Kraus, H. Ooguri and S. Shenker, Inside the horizon with AdS/CFT, Phys. Rev. D 67 (2003) 124022 [hep-th/0212277] [INSPIRE].
A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Local bulk operators in AdS/CFT: a holographic description of the black hole interior, Phys. Rev. D 75 (2007) 106001 [Erratum ibid. D 75 (2007) 129902] [hep-th/0612053] [INSPIRE].
K. Papadodimas and S. Raju, An infalling observer in AdS/CFT, JHEP 10 (2013) 212 [arXiv:1211.6767] [INSPIRE].
D. Kabat and G. Lifschytz, Finite N and the failure of bulk locality: black holes in AdS/CFT, JHEP 09 (2014) 077 [arXiv:1405.6394] [INSPIRE].
K. Papadodimas and S. Raju, Black hole interior in the holographic correspondence and the information paradox, Phys. Rev. Lett. 112 (2014) 051301 [arXiv:1310.6334] [INSPIRE].
S.-S. Lee, Holographic matter: deconfined string at criticality, Nucl. Phys. B 862 (2012) 781 [arXiv:1108.2253] [INSPIRE].
S.-S. Lee, Background independent holographic description: from matrix field theory to quantum gravity, JHEP 10 (2012) 160 [arXiv:1204.1780] [INSPIRE].
S.-S. Lee, Quantum renormalization group and holography, JHEP 01 (2014) 076 [arXiv:1305.3908] [INSPIRE].
Y. Nakayama, Vector β-function, Int. J. Mod. Phys. A 28 (2013) 1350166 [arXiv:1310.0574] [INSPIRE].
G. Bednik, Construction of holographic duals for quantum field theories with global symmetries from quantum renormalization group, MSc thesis, McMaster University, Hamilton ON Canada, May 2014.
I. Heemskerk and J. Polchinski, Holographic and Wilsonian renormalization groups, JHEP 06 (2011) 031 [arXiv:1010.1264] [INSPIRE].
T. Faulkner, H. Liu and M. Rangamani, Integrating out geometry: holographic Wilsonian RG and the membrane paradigm, JHEP 08 (2011) 051 [arXiv:1010.4036] [INSPIRE].
E. Kiritsis, Lorentz violation, gravity, dissipation and holography, JHEP 01 (2013) 030 [arXiv:1207.2325] [INSPIRE].
D. Marolf, Emergent gravity requires kinematic nonlocality, Phys. Rev. Lett. 114 (2015) 031104 [arXiv:1409.2509] [INSPIRE].
S.-S. Lee, Horizon as critical phenomenon, JHEP 09 (2016) 044 [arXiv:1603.08509] [INSPIRE].
S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].
K. Balasubramanian and K. Narayan, Lifshitz spacetimes from AdS null and cosmological solutions, JHEP 08 (2010) 014 [arXiv:1005.3291] [INSPIRE].
A. Donos and J.P. Gauntlett, Lifshitz solutions of D = 10 and D = 11 supergravity, JHEP 12 (2010) 002 [arXiv:1008.2062] [INSPIRE].
S. Harrison, S. Kachru and H. Wang, Resolving Lifshitz horizons, JHEP 02 (2014) 085 [arXiv:1202.6635] [INSPIRE].
E. Sezgin and P. Sundell, Massless higher spins and holography, Nucl. Phys. B 644 (2002) 303 [Erratum ibid. B 660 (2003) 403] [hep-th/0205131] [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].
S.R. Das and A. Jevicki, Large-N collective fields and holography, Phys. Rev. D 68 (2003) 044011 [hep-th/0304093] [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].
R.G. Leigh and A.C. Petkou, Holography of the N = 1 higher spin theory on AdS 4, JHEP 06 (2003) 011 [hep-th/0304217] [INSPIRE].
R. de Mello Koch, A. Jevicki, K. Jin and J.P. Rodrigues, AdS 4 /CFT 3 construction from collective fields, Phys. Rev. D 83 (2011) 025006 [arXiv:1008.0633] [INSPIRE].
M.R. Douglas, L. Mazzucato and S.S. Razamat, Holographic dual of free field theory, Phys. Rev. D 83 (2011) 071701 [arXiv:1011.4926] [INSPIRE].
L.A. Pando Zayas and C. Peng, Toward a higher-spin dual of interacting field theories, JHEP 10 (2013) 023 [arXiv:1303.6641] [INSPIRE].
R.G. Leigh, O. Parrikar and A.B. Weiss, Holographic geometry of the renormalization group and higher spin symmetries, Phys. Rev. D 89 (2014) 106012 [arXiv:1402.1430] [INSPIRE].
R.G. Leigh, O. Parrikar and A.B. Weiss, Exact renormalization group and higher-spin holography, Phys. Rev. D 91 (2015) 026002 [arXiv:1407.4574] [INSPIRE].
E. Mintun and J. Polchinski, Higher spin holography, RG and the light cone, arXiv:1411.3151 [INSPIRE].
M.A. Vasiliev, Higher spin gauge theories in four-dimensions, three-dimensions and two-dimensions, Int. J. Mod. Phys. D 5 (1996) 763 [hep-th/9611024] [INSPIRE].
M.A. Vasiliev, Higher spin gauge theories: star product and AdS space, hep-th/9910096 [INSPIRE].
S. Giombi and X. Yin, Higher spin gauge theory and holography: the three-point functions, JHEP 09 (2010) 115 [arXiv:0912.3462] [INSPIRE].
M.A. Vasiliev, Nonlinear equations for symmetric massless higher spin fields in (A)dS d , Phys. Lett. B 567 (2003) 139 [hep-th/0304049] [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].
I. Sachs, Higher spin versus renormalization group equations, Phys. Rev. D 90 (2014) 085003 [arXiv:1306.6654] [INSPIRE].
P. Lunts, S. Bhattacharjee, J. Miller, E. Schnetter, Y.B. Kim and S.-S. Lee, Ab initio holography, JHEP 08 (2015) 107 [arXiv:1503.06474] [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from conformal field theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
Y. Nakayama, Local renormalization group functions from quantum renormalization group and holographic bulk locality, JHEP 06 (2015) 092 [arXiv:1502.07049] [INSPIRE].
V. Shyam, General covariance from the quantum renormalization group, Phys. Rev. D 95 (2017) 066003 [arXiv:1611.05315] [INSPIRE].
S.D. Mathur, The fuzzball proposal for black holes: an elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [INSPIRE].
D. Grumiller, A. Pérez, S. Prohazka, D. Tempo and R. Troncoso, Higher spin black holes with soft hair, JHEP 10 (2016) 119 [arXiv:1607.05360] [INSPIRE].
S.W. Hawking, M.J. Perry and A. Strominger, Soft hair on black holes, Phys. Rev. Lett. 116 (2016) 231301 [arXiv:1601.00921] [INSPIRE].
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Hu, Q., Lee, SS. Non-local geometry inside Lifshitz horizon. J. High Energ. Phys. 2017, 56 (2017). https://doi.org/10.1007/JHEP07(2017)056
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DOI: https://doi.org/10.1007/JHEP07(2017)056