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Quantum quench of the Sachdev-Ye-Kitaev model

Andreas Eberlein, Valentin Kasper, Subir Sachdev, and Julia Steinberg
Phys. Rev. B 96, 205123 – Published 14 November 2017
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  • INTRODUCTION
  • KADANOFF-BAYM EQUATIONS FROM THE PATH…
  • KADANOFF-BAYM EQUATIONS: NUMERICAL STUDY
  • KADANOFF-BAYM EQUATIONS: LARGE-q LIMIT
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We describe the nonequilibrium quench dynamics of the Sachdev-Ye-Kitaev models of fermions with random all-to-all interactions. These provide tractable models of the dynamics of quantum systems without quasiparticle excitations. The Kadanoff-Baym equations show that, at long times, the fermion two-point function has a thermal form at a final temperature determined by energy conservation, and the numerical analysis is consistent with a thermalization rate proportional to this temperature. We also obtain an exact analytic solution of the quench dynamics in the large q limit of a model with q fermion interactions: in this limit, the thermalization of the two-point function is instantaneous.

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    • Received 7 July 2017
    • Revised 25 September 2017

    DOI:https://doi.org/10.1103/PhysRevB.96.205123

    ©2017 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    FermionsNonequilibrium statistical mechanics
    1. Techniques
    Non-Fermi-liquid theoryNumerical techniquesString theory techniques in condensed matter
    Particles & FieldsGravitation, Cosmology & AstrophysicsStatistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Andreas Eberlein1, Valentin Kasper1, Subir Sachdev1,2, and Julia Steinberg1

    • 1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
    • 2Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5

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    Issue

    Vol. 96, Iss. 20 — 15 November 2017

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    Images

    • Figure 1
      Figure 1

      The ratio of the Keldysh and spectral function is shown before and after a quench from a J2+J4 model to a purely J4 model. Fits to this data allow determination of the effective temperature as a function of time after the quench βeff(T) using Eq. (3.8).

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    • Figure 2
      Figure 2

      Fits to the values of the effective temperature βeff(T)=1/Teff(T) from results as in Fig. 1 to Eq. (1.8) to allow determination of the thermalization rate Γ for each quench.

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    • Figure 3
      Figure 3

      The values of the thermalization rate Γ obtained from Fig. 2 are shown as a function of the final temperature Tf for each quench. One of our main numerical results is the proportionality of Γ to Tf at small Tf≪J. At low temperatures the relaxation towards equilibrium is expected to be controlled by quantum and classical fluctuations, and hence we expect an equilibration rate proportional to Tf. At higher temperatures Γ is of order J4, and since at high temperatures the relaxation is essentially dominated by thermal fluctuations, the relaxation rate is dominated by the microscopic interaction energy scale of the fermions.

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    • Figure 4
      Figure 4

      The regions in the t1−t2 plane.

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    • Figure 5
      Figure 5

      The spectral function of the random-hopping model A long before (T=−14.7) and after (T=14.7) a parameter quench from J2,i=1 to J2,f=0.5 and 2.0. The width depends only on the value of Jf, becoming wider if Jf>Ji and narrower if Jf<Ji.

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    • Figure 6
      Figure 6

      The Keldysh component of the Green's function of the random-hopping model A long before (T=−14.7) and after (T=14.7) a parameter quench from J2,i=1 to J2,f=0.5 and 2.0.

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    • Figure 7
      Figure 7

      The spectral function for Majorana fermions is shown long after a quench starting with free Majorana fermions and switching on a q=4 SYK interaction term. Here J4,f=1.

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    • Figure 8
      Figure 8

      The propagation of the conjugacy property in the t1−t2 plane.

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