Abstract
We use the tensor network living on the Bruhat-Tits tree to give a concrete realization of the recently proposed p-adic AdS/CFT correspondence (a holographic duality based on the p-adic number field ℚ p ). Instead of assuming the p-adic AdS/CFT correspondence, we show how important features of AdS/CFT such as the bulk operator reconstruction and the holographic computation of boundary correlators are automatically implemented in this tensor network.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. ’t Hooft, Dimensional reduction in quantum gravity, Conf. Proc. C 930308 (1993) 284 [gr-qc/9310026] [INSPIRE].
L. Susskind, The world as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [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. Bianchi and R.C. Myers, On the Architecture of Spacetime Geometry, Class. Quant. Grav. 31 (2014) 214002 [arXiv:1212.5183] [INSPIRE].
B. Swingle, Constructing holographic spacetimes using entanglement renormalization, arXiv:1209.3304 [INSPIRE].
B. Czech et al., Tensor network quotient takes the vacuum to the thermal state, Phys. Rev. B 94 (2016) 085101 [arXiv:1510.07637] [INSPIRE].
B. Czech, L. Lamprou, S. McCandlish and J. Sully, Tensor Networks from Kinematic Space, JHEP 07 (2016) 100 [arXiv:1512.01548] [INSPIRE].
M. Nozaki, S. Ryu and T. Takayanagi, Holographic Geometry of Entanglement Renormalization in Quantum Field Theories, JHEP 10 (2012) 193 [arXiv:1208.3469] [INSPIRE].
A. Mollabashi, M. Nozaki, S. Ryu and T. Takayanagi, Holographic Geometry of cMERA for Quantum Quenches and Finite Temperature, JHEP 03 (2014) 098 [arXiv:1311.6095] [INSPIRE].
M. Miyaji and T. Takayanagi, Surface/State Correspondence as a Generalized Holography, PTEP 2015 (2015) 073B03 [arXiv:1503.03542] [INSPIRE].
M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, Continuous Multiscale Entanglement Renormalization Ansatz as Holographic Surface-State Correspondence, Phys. Rev. Lett. 115 (2015) 171602 [arXiv:1506.01353] [INSPIRE].
N. Bao et al., Consistency conditions for an AdS multiscale entanglement renormalization ansatz correspondence, Phys. Rev. D 91 (2015) 125036 [arXiv:1504.06632] [INSPIRE].
A. Almheiri, X. Dong and D. Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].
E. Mintun, J. Polchinski and V. Rosenhaus, Bulk-Boundary Duality, Gauge Invariance and Quantum Error Corrections, Phys. Rev. Lett. 115 (2015) 151601 [arXiv:1501.06577] [INSPIRE].
E.M. Brehm and B. Richter, Classical Holographic Codes, Phys. Rev. D 96 (2017) 066005 [arXiv:1609.03560] [INSPIRE].
M. Miyaji, T. Takayanagi and K. Watanabe, From path integrals to tensor networks for the AdS/CFT correspondence, Phys. Rev. D 95 (2017) 066004 [arXiv:1609.04645] [INSPIRE].
B. Czech, P.H. Nguyen and S. Swaminathan, A defect in holographic interpretations of tensor networks, JHEP 03 (2017) 090 [arXiv:1612.05698] [INSPIRE].
A. Peach and S.F. Ross, Tensor Network Models of Multiboundary Wormholes, Class. Quant. Grav. 34 (2017) 105011 [arXiv:1702.05984] [INSPIRE].
F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, JHEP 06 (2015) 149 [arXiv:1503.06237] [INSPIRE].
P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter and Z. Yang, Holographic duality from random tensor networks, JHEP 11 (2016) 009 [arXiv:1601.01694] [INSPIRE].
A. Bhattacharyya, Z.-S. Gao, L.-Y. Hung and S.-N. Liu, Exploring the Tensor Networks/AdS Correspondence, JHEP 08 (2016) 086 [arXiv:1606.00621] [INSPIRE].
M. Han and L.-Y. Hung, Loop Quantum Gravity, Exact Holographic Mapping and Holographic Entanglement Entropy, Phys. Rev. D 95 (2017) 024011 [arXiv:1610.02134] [INSPIRE].
G. Chirco, D. Oriti and M. Zhang, Group Field theory and Tensor Networks: towards a Ryu-Takayanagi formula in full quantum gravity, arXiv:1701.01383 [INSPIRE].
C.H. Lee and X.-L. Qi, Exact holographic mapping in free fermion systems, Phys. Rev. B 93 (2016)035112 [arXiv:1503.08592] [INSPIRE].
S. Singh and G.K. Brennen, Holographic Construction of Quantum Field Theory using Wavelets, arXiv:1606.05068 [INSPIRE].
M. Heydeman, M. Marcolli, I. Saberi and B. Stoica, Tensor networks, p-adic fields and algebraic curves: arithmetic and the AdS 3 /CFT 2 correspondence, arXiv:1605.07639 [INSPIRE].
S.S. Gubser, J. Knaute, S. Parikh, A. Samberg and P. Witaszczyk, p-adic AdS/CFT, Commun. Math. Phys. 352 (2017) 1019 [arXiv:1605.01061] [INSPIRE].
S.S. Gubser et al., Edge length dynamics on graphs with applications to p-adic AdS/CFT, JHEP 06 (2017) 157 [arXiv:1612.09580] [INSPIRE].
R. Orus, A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States, Annals Phys. 349 (2014) 117 [arXiv:1306.2164] [INSPIRE].
R. Orus, Advances on Tensor Network Theory: Symmetries, Fermions, Entanglement and Holography, Eur. Phys. J. B 87 (2014) 280 [arXiv:1407.6552] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
S. Singh and G. Vidal, Symmetry protected entanglement renormalization, Phys. Rev. B 88 (2013)121108 [arXiv:1303.6716] [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].
A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Local bulk operators in AdS/CFT: A boundary view of horizons and locality, Phys. Rev. D 73 (2006) 086003 [hep-th/0506118] [INSPIRE].
D. Kabat, G. Lifschytz and D.A. Lowe, Constructing local bulk observables in interacting AdS/CFT, Phys. Rev. D 83 (2011) 106009 [arXiv:1102.2910] [INSPIRE].
J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press (1992).
N. Koblitz, p-adic Numbers, p-adic Analysis and Zeta-Functions, 2nd edition, Springer (1984).
F.Q. Gouvêa, p-adic Numbers: An Introduction, 2nd edition, Springer (1997).
P.G.O. Freund and M. Olson, Nonarchimedean strings, Phys. Lett. B 199 (1987) 186 [INSPIRE].
L. Brekke and P.G.O. Freund, p-adic numbers in physics, Phys. Rept. 233 (1993) 1 [INSPIRE].
L. Brekke, P.G.O. Freund, M. Olson and E. Witten, Nonarchimedean String Dynamics, Nucl. Phys. B 302 (1988) 365 [INSPIRE].
P.G.O. Freund and E. Witten, Adelic string amplitudes, Phys. Lett. B 199 (1987) 191 [INSPIRE].
B. Dragovich, Zeta strings, hep-th/0703008 [INSPIRE].
B. Dragovich, A.Yu. Khrennikov, S.V. Kozyrev and I.V. Volovich, On p-Adic Mathematical Physics, Anal. Appl. 1 (2009) 1 [arXiv:0904.4205] [INSPIRE].
Y.I. Manin and M. Marcolli, Holography principle and arithmetic of algebraic curves, Adv. Theor. Math. Phys. 5 (2002) 617 [hep-th/0201036] [INSPIRE].
A. Ostrowski, Über einige Lösungen der Funktionalgleichung Ψ(x) · Ψ(y) = Ψ(xy), Acta Math. 41 (1916) 271.
F. Bruhat and J. Tits, Groupes réductifs sur un corps local: I. Données radicielles valuées, Inst. Hautes Études Sci. Publ. Math. 41 (1972) 5.
A.V. Zabrodin, Nonarchimedean Strings and Bruhat-tits Trees, Commun. Math. Phys. 123 (1989) 463 [INSPIRE].
E. Melzer, Nonarchimedean conformal field theories, Int. J. Mod. Phys. A 4 (1989) 4877 [INSPIRE].
F.R.K. Chung, Spectral graph theory, American Mathematical Society (1997).
A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Holographic representation of local bulk operators, Phys. Rev. D 74 (2006) 066009 [hep-th/0606141] [INSPIRE].
G. Brattle, Wavelets and Renormalization, World Scientific (1998).
G. Evenbly and S.R. White, Entanglement renormalization and wavelets, Phys. Rev. Lett. 116 (2016) 140403 [arXiv:1602.01166] [INSPIRE].
D.K. Hammond, P. Vandergheynst and R. Gribonval, Wavelets on graphs via spectral graph theory, Appl. Comput. Harm. Anal. 30 129 [arXiv:0912.3848].
D. Harlow, S.H. Shenker, D. Stanford and L. Susskind, Tree-like structure of eternal inflation: A solvable model, Phys. Rev. D 85 (2012) 063516 [arXiv:1110.0496] [INSPIRE].
S. Albeverio and S.V. Kozyrev, Coincidence of the continuous and discrete p-adic wavelet transforms, math-ph/0702010.
R.N.C. Pfeifer, G. Evenbly and G. Vidal, Entanglement renormalization, scale invariance and quantum criticality, Phys. Rev. A 79 (2009) 040301 [arXiv:0810.0580] [INSPIRE].
S. Yang, Z.C. Gu and X.G. Wen, Loop optimization for tensor network renormalization, Phys. Rev. Lett. 118 (2017) 110504 [arXiv:1512.04938].
G. Evenbly and G. Vidal, Tensor Network Renormalization Yields the Multiscale Entanglement Renormalization Ansatz, Phys. Rev. Lett. 115 (2015) 200401 [arXiv:1502.05385].
S.S. Gubser, C. Jepsen, S. Parikh and B. Trundy, O(N) and O(N) and O(N), JHEP 11 (2017) 107 [arXiv:1703.04202] [INSPIRE].
S.S. Gubser et al., Signs of the time: Melonic theories over diverse number systems, arXiv:1707.01087 [INSPIRE].
R.-b. Zhang, Lagrangian Formulation of Open and Closed p-adic Strings, Phys. Lett. B 209 (1988) 229 [INSPIRE].
B.L. Spokoiny, Quantum Geometry of Nonarchimedean Particles and Strings, Phys. Lett. B 208 (1988) 401 [INSPIRE].
G. Parisi, On p-adic functional integrals, Mod. Phys. Lett. A 3 (1988) 639 [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, An AdS 3 Dual for Minimal Model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
P. Fleig, H.P.A. Gustafsson, A. Kleinschmidt and D. Persson, Eisenstein series and automorphic representations, arXiv:1511.04265 [INSPIRE].
M. Bal, M. Mariën, J. Haegeman and F. Verstraete, Renormalization group flows of Hamiltonians using tensor networks, Phys. Rev. Lett. 118 (2017) 250602 [arXiv:1703.00365] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1703.05445
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Bhattacharyya, A., Hung, LY., Lei, Y. et al. Tensor network and (p-adic) AdS/CFT. J. High Energ. Phys. 2018, 139 (2018). https://doi.org/10.1007/JHEP01(2018)139
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2018)139