Abstract
We apply the quantum renormalization group to construct a holographic dual for the U(N) vector model for complex bosons defined on a lattice. The bulk geometry becomes dynamical as the hopping amplitudes which determine connectivity of space are promoted to quantum variables. In the large N limit, the full bulk equations of motion for the dynamical hopping fields are numerically solved for finite systems. From finite size scaling, we show that different phases exhibit distinct geometric features in the bulk. In the insulating phase, the space gets fragmented into isolated islands deep inside the bulk, exhibiting ultra-locality. In the superfluid phase, the bulk exhibits a horizon beyond which the geometry becomes non-local. Right at the horizon, the hopping fields decay with a universal power-law in coordinate distance between sites, while they decay in slower power-laws with continuously varying exponents inside the horizon. At the critical point, the bulk exhibits a local geometry whose characteristic length scale diverges asymptotically in the IR limit.
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References
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [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].
E.T. Akhmedov, A remark on the AdS/CFT correspondence and the renormalization group flow, Phys. Lett. B 442 (1998) 152 [hep-th/9806217] [INSPIRE].
J. de Boer, E.P. Verlinde and H.L. Verlinde, On the holographic renormalization group, JHEP 08 (2000) 003 [hep-th/9912012] [INSPIRE].
K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].
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].
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].
E. Kiritsis, Lorentz violation, gravity, dissipation and holography, JHEP 01 (2013) 030 [arXiv:1207.2325] [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from conformal field theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
T. Hartman, C.A. Keller and B. Stoica, Universal spectrum of 2d conformal field theory in the large c limit, JHEP 09 (2014) 118 [arXiv:1405.5137] [INSPIRE].
N. Benjamin, M.C.N. Cheng, S. Kachru, G.W. Moore and N.M. Paquette, Elliptic genera and 3d gravity, arXiv:1503.04800 [INSPIRE].
Y. Nakayama, a − c test of holography versus quantum renormalization group, Mod. Phys. Lett. A 29 (2014) 1450158 [arXiv:1401.5257] [INSPIRE].
Y. Nakayama, Local renormalization group functions from quantum renormalization group and holographic bulk locality, JHEP 06 (2015) 092 [arXiv:1502.07049] [INSPIRE].
S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Holographic superconductors, JHEP 12 (2008) 015 [arXiv:0810.1563] [INSPIRE].
S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [INSPIRE].
J. McGreevy, Holographic duality with a view toward many-body physics, Adv. High Energy Phys. 2010 (2010) 723105 [arXiv:0909.0518] [INSPIRE].
S. Sachdev, What can gauge-gravity duality teach us about condensed matter physics?, Ann. Rev. Cond. Mat. Phys. 3 (2012) 9 [arXiv:1108.1197] [INSPIRE].
S.R. Das and A. Jevicki, Large-N collective fields and holography, Phys. Rev. D 68 (2003) 044011 [hep-th/0304093] [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].
E. Mintun and J. Polchinski, Higher spin holography, RG and the light cone, arXiv:1411.3151 [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].
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].
I. Sachs, Higher spin versus renormalization group equations, Phys. Rev. D 90 (2014) 085003 [arXiv:1306.6654] [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].
J. Polchinski, Renormalization and effective Lagrangians, Nucl. Phys. B 231 (1984) 269 [INSPIRE].
J. Polonyi, Lectures on the functional renormalization group method, Cent. Eur. J. Phys. 1 (2003) 1 [hep-th/0110026] [INSPIRE].
H. Osborn, Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories, Nucl. Phys. B 363 (1991) 486 [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.-S. Lee, A non-Fermi liquid from a charged black hole: a critical Fermi ball, Phys. Rev. D 79 (2009) 086006 [arXiv:0809.3402] [INSPIRE].
H. Liu, J. McGreevy and D. Vegh, Non-Fermi liquids from holography, Phys. Rev. D 83 (2011) 065029 [arXiv:0903.2477] [INSPIRE].
M. Čubrović, J. Zaanen and K. Schalm, String theory, quantum phase transitions and the emergent Fermi-liquid, Science 325 (2009) 439 [arXiv:0904.1993] [INSPIRE].
S.A. Hartnoll and A. Tavanfar, Electron stars for holographic metallic criticality, Phys. Rev. D 83 (2011) 046003 [arXiv:1008.2828] [INSPIRE].
T. Faulkner and J. Polchinski, Semi-holographic Fermi liquids, JHEP 06 (2011) 012 [arXiv:1001.5049] [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].
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].
J.P. Boyd, Chebyshev and Fourier spectral methods, Courier Corporation (2001).
C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral methods: fundamentals in single domains, Springer-Verlag, Berlin New York (2006).
B. Fornberg, A practical guide to pseudospectral methods, volume 1, Cambridge University Press, Cambridge U.K. (1998).
W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical recipes: the art of scientific computing, 3rd edition, Cambridge University Press, Cambridge U.K. (2007).
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Lunts, P., Bhattacharjee, S., Miller, J. et al. Ab initio holography. J. High Energ. Phys. 2015, 107 (2015). https://doi.org/10.1007/JHEP08(2015)107
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DOI: https://doi.org/10.1007/JHEP08(2015)107