Abstract
We derive an exact formula for a matrix product state (MPS) representation (or a PEPS in higher number of dimensions) of the ground state of translationally invariant bosonic lattice systems in terms of a single one-dimensional Euclidean quantum mechanical path integral with sources. We explicitly evaluate the general formula in the special case of the one-dimensional Klein-Gordon harmonic chain, being a spatial discretization of 1+1 dimensional free boson QFT, obtaining an exact MPS with an infinite dimensional bond space. We analytically diagonalize the transfer matrix obtaining two Fock spaces with continuous modes and check that the exact MPS construction reproduces the correct correlation functions. We also comment on possible holographic interpretations.
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References
U. Schollwock, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326 (2011) 96 [arXiv:1008.3477].
F. Verstraete, J.I. Cirac and V. Murg, Matrix Product States, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Adv. Phys. 57 (2008)143 [arXiv:0907.2796].
G. Vidal, Entanglement renormalization, Phys. Rev. Lett. 99 (2007) 220405 [cond-mat/0512165] [INSPIRE].
F. Verstraete and J.I. Cirac, Matrix product states represent ground states faithfully, Phys. Rev. B 73 (2006) 094423 [cond-mat/0505140].
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].
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].
B. Swingle, Entanglement renormalization and holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
J. Haegeman, T.J. Osborne, H. Verschelde and F. Verstraete, Entanglement renormalization for quantum fields in real space, Phys. Rev. Lett. 110 (2013) 100402 [arXiv:1102.5524] [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].
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. Caputa et al., Anti-de Sitter space from optimization of path integrals in conformal field theories, Phys. Rev. Lett. 119 (2017) 071602 [arXiv:1703.00456] [INSPIRE].
P. Hayden et al., Holographic duality from random tensor networks, JHEP 11 (2016) 009 [arXiv:1601.01694] [INSPIRE].
J.I. Cirac and G. Sierra, Infinite matrix product states, conformal field theory and the Haldane-Shastry model, Phys. Rev. B 81 (2010) 104431 [arXiv:0911.3029].
M.M. Rams et al., Truncating an exact Matrix Product State for the XY model: transfer matrix and its renormalisation, arXiv:1411.2607.
R. König and V.B. Scholz, Matrix product approximations to multipoint functions in two-dimensional conformal field theory, Phys. Rev. Lett. 117 (2016) 121601 [arXiv:1601.00470] [INSPIRE].
N. Schuch, J.I. Cirac and M.M. Wolf, Quantum states on harmonic lattices, Commun. Math. Phys. 267 (2006) 65 [quant-ph/0509166].
A. Bhattacharyya et al., Path-integral complexity for perturbed CFTs, JHEP 07 (2018) 086 [arXiv:1804.01999] [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. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Correlation functions in the CFT d /AdS d+1 correspondence, Nucl. Phys. B 546 (1999) 96 [hep-th/9804058] [INSPIRE].
T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, AdS dynamics from conformal field theory, hep-th/9808016 [INSPIRE].
A. Botero and B. Reznik, Spatial structures and localization of vacuum entanglement in the linear harmonic chain, Phys. Rev. A 70 (2004) 052329 [quant-ph/0403233].
G. Evenbly and G. Vidal, Tensor network renormalization yields the multi-scale entanglement renormalization ansatz, Phys. Rev. Lett. 115 (2015) 200401 [arXiv:1502.05385].
R.A. Janik, Towards holography for quantum mechanics, JHEP 09 (2018) 045 [arXiv:1805.03606] [INSPIRE].
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Janik, R.A. Exact bosonic Matrix Product States (and holography). J. High Energ. Phys. 2019, 225 (2019). https://doi.org/10.1007/JHEP01(2019)225
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DOI: https://doi.org/10.1007/JHEP01(2019)225