Abstract
Matrix models play an important role in studies of quantum gravity, being candidates for a formulation of M-theory, but are notoriously difficult to solve. In this work, we present a fresh approach by introducing a novel exact model, provably equivalent with a low-dimensional bosonic matrix model, which is on its own a well-known, unsolved model of quantum chaos. In our equivalent reformulation local structure becomes apparent, facilitating analytical and precise numerical study. We derive a substantial part of the low energy spectrum, find a conserved charge, and are able to derive numerically the Regge trajectories. To exemplify the usefulness of the approach, we address questions of equilibration starting from a non-equilibrium situation, building upon an intuition from quantum information. We finally discuss possible generalisations of the approach.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T. Banks, W. Fischler, S.H. Shenker and L. Susskind, M theory as a matrix model: a conjecture, Phys. Rev. D 55 (1997) 5112 [hep-th/9610043] [INSPIRE].
P. Hayden and J. Preskill, Black holes as mirrors: quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].
Y. Sekino and L. Susskind, Fast scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
M.R. Douglas, D.N. Kabat, P. Pouliot and S.H. Shenker, D-branes and short distances in string theory, Nucl. Phys. B 485 (1997) 85 [hep-th/9608024] [INSPIRE].
Y. Okawa and T. Yoneya, Multibody interactions of D particles in supergravity and matrix theory, Nucl. Phys. B 538 (1999) 67 [hep-th/9806108] [INSPIRE].
W. Taylor and M. Van Raamsdonk, Supergravity currents and linearized interactions for matrix theory configurations with fermionic backgrounds, JHEP 04 (1999) 013 [hep-th/9812239] [INSPIRE].
M. Hanada, J. Nishimura, Y. Sekino and T. Yoneya, Monte Carlo studies of matrix theory correlation functions, Phys. Rev. Lett. 104 (2010) 151601 [arXiv:0911.1623] [INSPIRE].
M. Hanada, J. Nishimura, Y. Sekino and T. Yoneya, Direct test of the gauge-gravity correspondence for matrix theory correlation functions, JHEP 12 (2011) 020 [arXiv:1108.5153] [INSPIRE].
Y. Sekino and T. Yoneya, Generalized AdS/CFT correspondence for matrix theory in the large-N limit, Nucl. Phys. B 570 (2000) 174 [hep-th/9907029] [INSPIRE].
Y. Sekino, Supercurrents in matrix theory and the generalized AdS/CFT correspondence, Nucl. Phys. B 602 (2001) 147 [hep-th/0011122] [INSPIRE].
G.K. Savvidy, Yang-Mills classical mechanics as a Kolmogorov K system, Phys. Lett. B 130 (1983) 303 [INSPIRE].
G.K. Savvidy, Classical and quantum mechanics of non-Abelian gauge fields, Nucl. Phys. B 246 (1984) 302 [INSPIRE].
T. Furusawa, Onset of chaos in the classical SU(2) Yang-Mills theory, Nucl. Phys. B 290 (1987) 469 [INSPIRE].
M.P. Joy and M. Sabir, Nonintegrability of SU(2) Yang-Mills and Yang-Mills Higgs systems, J. Phys. A 22 (1989) 5153 [INSPIRE].
T. Kunihiro et al., Chaotic behavior in classical Yang-Mills dynamics, Phys. Rev. D 82 (2010) 114015 [arXiv:1008.1156] [INSPIRE].
P. Olesen, Confinement and random fields, Nucl. Phys. B 200 (1982) 381 [INSPIRE].
N. Linden, S. Popescu, A.J. Short and A. Winter, Quantum mechanical evolution towards thermal equilibrium, Phys. Rev. E 79 (2009) 061103 [arXiv:0812.2385].
M. Cramer, C.M. Dawson, J. Eisert and T.J. Osborne, Exact relaxation in a class of nonequilibrium quantum lattice systems, Phys. Rev. Lett. 100 (2008) 030602 [INSPIRE].
A. Riera, C. Gogolin and J. Eisert, Thermalization in nature and on a quantum computer, Phys. Rev. Lett. 108 (2012) 080402 [arXiv:1102.2389].
C. Asplund, D. Berenstein and D. Trancanelli, Evidence for fast thermalization in the plane-wave matrix model, Phys. Rev. Lett. 107 (2011) 171602 [arXiv:1104.5469] [INSPIRE].
C.T. Asplund, D. Berenstein and E. Dzienkowski, Large-N classical dynamics of holographic matrix models, Phys. Rev. D 87 (2013) 084044 [arXiv:1211.3425] [INSPIRE].
N. Lashkari, D. Stanford, M. Hastings, T. Osborne and P. Hayden, Towards the fast scrambling conjecture, JHEP 04 (2013) 022 [arXiv:1111.6580] [INSPIRE].
P. Riggins and V. Sahakian, On black hole thermalization, D0 brane dynamics and emergent spacetime, Phys. Rev. D 86 (2012) 046005 [arXiv:1205.3847] [INSPIRE].
L. Brady and V. Sahakian, Scrambling with matrix black holes, Phys. Rev. D 88 (2013) 046003 [arXiv:1306.5200] [INSPIRE].
S. Pramodh and V. Sahakian, From black hole to qubits: matrix theory is a fast scrambler, arXiv:1412.2396 [INSPIRE].
G. Mandal and T. Morita, Quantum quench in matrix models: dynamical phase transitions, selective equilibration and the generalized Gibbs ensemble, JHEP 10 (2013) 197 [arXiv:1302.0859] [INSPIRE].
N. Iizuka, D. Kabat, S. Roy and D. Sarkar, Black hole formation in fuzzy sphere collapse, Phys. Rev. D 88 (2013) 044019 [arXiv:1306.3256] [INSPIRE].
R.G. Leigh, D. Minic and A. Yelnikov, On the glueball spectrum of pure Yang-Mills theory in 2+1 dimensions,Phys. Rev. D 76 (2007) 065018 [hep-th/0604060] [INSPIRE].
M. Campostrini and J. Wosiek, High precision study of the structure of D = 4 supersymmetric Yang-Mills quantum mechanics, Nucl. Phys. B 703 (2004) 454 [hep-th/0407021] [INSPIRE].
E. Chalbaud, J.-P. Gallinar and G. Mata, The quantum harmonic oscillator on a lattice, J. Phys. A 19 (1986) L385.
M.B. Green, J.H. Schwarz and E. Witten, Superstring theory, vol. 1, Cambridge Univ. Pr., Cambridge U.K. (1987).
H.B. Meyer and M.J. Teper, Glueball Regge trajectories in (2 + 1)-dimensional gauge theories, Nucl. Phys. B 668 (2003) 111 [hep-lat/0306019] [INSPIRE].
D. Berenstein, private communication.
B. Collins, Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability, Int. Math. Res. Not. 17 (2003) 953 [math-ph/0205010].
B. Collins and P. Sniady, Integration with respect to the Haar measure on unitary, orthogonal and symplectic group, Commun. Math. Phys. 264 (2006) 773 [math-ph/0402073].
A.J. Short, Equilibration of quantum systems and subsystems, New J. Phys. 13 (2011) 053009 [arXiv:1012.4622].
A.J. Short and T.C. Farrelly, Quantum equilibration in finite time, New J. Phys. 14 (2012) 013063 [arXiv:1110.5759].
G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
H. Yukawa, Quantum theory of nonlocal fields. 1. Free fields, Phys. Rev. 77 (1950) 219 [INSPIRE].
M.A. Nielsen, A geometric approach to quantum circuit lower bounds, Quant. Informat. Comput. 6 (2006) 213 [quant-ph/0502070].
M.A. Nielsen and M.R. Dowling, The geometry of quantum computation, Quant. Informat. Comput. 8 (2008) 861 [quant-ph/0701004].
C.E. Mora and H.J. Briegel, Algorithmic complexity of quantum states, Int. J. Quant. Inform. 4 (2006) 715 [quant-ph/0412172].
D. Stanford and L. Susskind, Complexity and shock wave geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
L. Susskind, Entanglement is not enough, arXiv:1411.0690 [INSPIRE].
L. Masanes, A.J. Roncaglia and A. Acin, The complexity of energy eigenstates as a mechanism for equilibration, Phys. Rev. E 87 (2013) 032137 [arXiv:1108.0374].
Y. Ge and J. Eisert, Area laws and efficient descriptions of quantum many-body states, arXiv:1411.2995.
M. Müller, Quantum Kolmogorov complexity and the quantum Turing machine, Ph.D. thesis, Technical University of Berlin, Berlin Germany (2007) [arXiv:0712.4377].
A.S.L. Malabarba, L.P. Garcia-Pintos, N. Linden, T.C. Farrelly and A.J. Short, Quantum systems equilibrate rapidly for most observables, Phys. Rev. E 90 (2014) 012121 [arXiv:1402.1093].
D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from N =4 super Yang-Mills, JHEP 04 (2002) 013 [hep-th/0202021] [INSPIRE].
S. Sethi and M. Stern, D-brane bound states redux, Commun. Math. Phys. 194 (1998) 675 [hep-th/9705046] [INSPIRE].
Y.-H. Lin and X. Yin, On the ground state wave function of matrix theory, arXiv:1402.0055 [INSPIRE].
B. de Wit, M. Lüscher and H. Nicolai, The supermembrane is unstable, Nucl. Phys. B 320 (1989) 135 [INSPIRE].
M. Hanada, Y. Hyakutake, J. Nishimura and S. Takeuchi, Higher derivative corrections to black hole thermodynamics from supersymmetric matrix quantum mechanics, Phys. Rev. Lett. 102 (2009) 191602 [arXiv:0811.3102] [INSPIRE].
M. Hanada, Y. Hyakutake, G. Ishiki and J. Nishimura, Holographic description of quantum black hole on a computer, Science 344 (2014) 882 [arXiv:1311.5607] [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: 1403.1392
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
Hübener, R., Sekino, Y. & Eisert, J. Equilibration in low-dimensional quantum matrix models. J. High Energ. Phys. 2015, 166 (2015). https://doi.org/10.1007/JHEP04(2015)166
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2015)166