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Equilibration in low-dimensional quantum matrix models

Open Access
Regular Article - Theoretical Physics

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.

Keywords

M(atrix) Theories Matrix Models Field Theories in Lower Dimensions Gauge Symmetry 

Notes

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.

References

  1. [1]
    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].ADSMathSciNetMATHGoogle Scholar
  2. [2]
    P. Hayden and J. Preskill, Black holes as mirrors: quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    Y. Sekino and L. Susskind, Fast scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    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].ADSCrossRefMATHGoogle Scholar
  6. [6]
    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].ADSMathSciNetMATHGoogle Scholar
  7. [7]
    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].ADSCrossRefGoogle Scholar
  8. [8]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    Y. Sekino, Supercurrents in matrix theory and the generalized AdS/CFT correspondence, Nucl. Phys. B 602 (2001) 147 [hep-th/0011122] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    G.K. Savvidy, Yang-Mills classical mechanics as a Kolmogorov K system, Phys. Lett. B 130 (1983) 303 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    G.K. Savvidy, Classical and quantum mechanics of non-Abelian gauge fields, Nucl. Phys. B 246 (1984) 302 [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    T. Furusawa, Onset of chaos in the classical SU(2) Yang-Mills theory, Nucl. Phys. B 290 (1987) 469 [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    M.P. Joy and M. Sabir, Nonintegrability of SU(2) Yang-Mills and Yang-Mills Higgs systems, J. Phys. A 22 (1989) 5153 [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  15. [15]
    T. Kunihiro et al., Chaotic behavior in classical Yang-Mills dynamics, Phys. Rev. D 82 (2010) 114015 [arXiv:1008.1156] [INSPIRE].ADSGoogle Scholar
  16. [16]
    P. Olesen, Confinement and random fields, Nucl. Phys. B 200 (1982) 381 [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    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].ADSMathSciNetGoogle Scholar
  18. [18]
    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].ADSCrossRefGoogle Scholar
  19. [19]
    A. Riera, C. Gogolin and J. Eisert, Thermalization in nature and on a quantum computer, Phys. Rev. Lett. 108 (2012) 080402 [arXiv:1102.2389].ADSCrossRefGoogle Scholar
  20. [20]
    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].ADSCrossRefGoogle Scholar
  21. [21]
    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].ADSGoogle Scholar
  22. [22]
    N. Lashkari, D. Stanford, M. Hastings, T. Osborne and P. Hayden, Towards the fast scrambling conjecture, JHEP 04 (2013) 022 [arXiv:1111.6580] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    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].ADSGoogle Scholar
  24. [24]
    L. Brady and V. Sahakian, Scrambling with matrix black holes, Phys. Rev. D 88 (2013) 046003 [arXiv:1306.5200] [INSPIRE].ADSGoogle Scholar
  25. [25]
    S. Pramodh and V. Sahakian, From black hole to qubits: matrix theory is a fast scrambler, arXiv:1412.2396 [INSPIRE].
  26. [26]
    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].ADSCrossRefGoogle Scholar
  27. [27]
    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].ADSGoogle Scholar
  28. [28]
    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].ADSMathSciNetGoogle Scholar
  29. [29]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    E. Chalbaud, J.-P. Gallinar and G. Mata, The quantum harmonic oscillator on a lattice, J. Phys. A 19 (1986) L385.MathSciNetGoogle Scholar
  31. [31]
    M.B. Green, J.H. Schwarz and E. Witten, Superstring theory, vol. 1, Cambridge Univ. Pr., Cambridge U.K. (1987).Google Scholar
  32. [32]
    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].ADSCrossRefMATHGoogle Scholar
  33. [33]
    D. Berenstein, private communication.Google Scholar
  34. [34]
    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].MathSciNetCrossRefMATHGoogle Scholar
  35. [35]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  36. [36]
    A.J. Short, Equilibration of quantum systems and subsystems, New J. Phys. 13 (2011) 053009 [arXiv:1012.4622].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    A.J. Short and T.C. Farrelly, Quantum equilibration in finite time, New J. Phys. 14 (2012) 013063 [arXiv:1110.5759].ADSCrossRefGoogle Scholar
  38. [38]
    G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].ADSGoogle Scholar
  39. [39]
    H. Yukawa, Quantum theory of nonlocal fields. 1. Free fields, Phys. Rev. 77 (1950) 219 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    M.A. Nielsen, A geometric approach to quantum circuit lower bounds, Quant. Informat. Comput. 6 (2006) 213 [quant-ph/0502070].MathSciNetMATHGoogle Scholar
  41. [41]
    M.A. Nielsen and M.R. Dowling, The geometry of quantum computation, Quant. Informat. Comput. 8 (2008) 861 [quant-ph/0701004].MathSciNetMATHGoogle Scholar
  42. [42]
    C.E. Mora and H.J. Briegel, Algorithmic complexity of quantum states, Int. J. Quant. Inform. 4 (2006) 715 [quant-ph/0412172].CrossRefMATHGoogle Scholar
  43. [43]
    D. Stanford and L. Susskind, Complexity and shock wave geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].ADSGoogle Scholar
  44. [44]
    L. Susskind, Entanglement is not enough, arXiv:1411.0690 [INSPIRE].
  45. [45]
    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].ADSGoogle Scholar
  46. [46]
    Y. Ge and J. Eisert, Area laws and efficient descriptions of quantum many-body states, arXiv:1411.2995.
  47. [47]
    M. Müller, Quantum Kolmogorov complexity and the quantum Turing machine, Ph.D. thesis, Technical University of Berlin, Berlin Germany (2007) [arXiv:0712.4377].
  48. [48]
    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].ADSGoogle Scholar
  49. [49]
    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].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    S. Sethi and M. Stern, D-brane bound states redux, Commun. Math. Phys. 194 (1998) 675 [hep-th/9705046] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    Y.-H. Lin and X. Yin, On the ground state wave function of matrix theory, arXiv:1402.0055 [INSPIRE].
  52. [52]
    B. de Wit, M. Lüscher and H. Nicolai, The supermembrane is unstable, Nucl. Phys. B 320 (1989) 135 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  53. [53]
    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].ADSCrossRefGoogle Scholar
  54. [54]
    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].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Dahlem Center for Complex Quantum SystemsFreie Universität BerlinBerlinGermany
  2. 2.KEK Theory Center, High Energy Accelerator Research OrganisationTsukubaJapan

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