Spectrum of SYK Model

Abstract

This is the first part of a series of papers on the spectrum of the SYK model, which is a simple model of the black hole in physics literature. In this paper, we will give a rigorous proof of the almost sure convergence of the global density of the eigenvalues. We also discuss the largest eigenvalue of the SYK model.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Anderson, G.W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, vol. 118. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  2. 2.

    Cotler, J.S., Gur-Ari, G., Hanada, M., Polchinski, J., Saad, P., Shenker, S.H., Stanford, D., Streicher, A., Tezuka, M.: Black holes and random matrices. J. High Energy Phys. 2017, 118 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Erdős, L., Schröder, D.: Phase transitions in the density of quantum spin glasses. Math. Phys. Anal. Geom. 17(3–4), 441–464 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Feng, R., Tian, G., Wei, D.: Spectrum of SYK model II: central limit theorem (in preparation)

  5. 5.

    Feng, R., Tian, G., Wei, D.: Spectrum of SYK model III: large deviations and concentration of measures (in preparation)

  6. 6.

    García-García, A.M., Jia, Y., Verbaarschot, J.J.M.: Exact moments of the Sachdev–Ye–Kitaev model up to order 1/ N 2. High Energy Phys. 2018, 146 (2018)

    Article  MATH  Google Scholar 

  7. 7.

    García-García, A.M., Jia, Y., Verbaarschot, J.J.M.: Universality and Thouless energy in the supersymmetric Sachdev–Ye–Kitaev model (2018). arXiv:1801.01071

  8. 8.

    García-García, A.M., Verbaarschot, J.J.M.: Spectral and thermodynamic properties of the Sachdev–Ye–Kitaev model. Phys. Rev. D 94, 126010 (2016)

    Article  Google Scholar 

  9. 9.

    García-García, A.M., Verbaarschot, J.J.M.: Analytical spectral density of the Sachdev–Ye–Kitaev model at finite N. Phys. Rev. D 96, 066012 (2017)

    Article  Google Scholar 

  10. 10.

    Ismail, M.E.H., Stanton, D., Viennot, G.: The combinatorics of q-Hermite polynomials and the Askey–Wilson integral. Eur. J. Comb. 8(4), 379–392 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Keating, J.P., Linden, N., Wells, H.J.: Random matrices and quantum spin chains. Markov Processes Relat. Fields 231, 537–555 (2015)

    MathSciNet  Google Scholar 

  12. 12.

    Keating, J.P., Linden, N., Wells, H.J.: Spectra and eigenstates of spin chain Hamiltonians. Commun. Math. Phys. 338(1), 81–102 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Kitaev, A.: Hidden correlations in the Hawking radiation and thermal noise, KITP seminar (2015). http://online.kitp.ucsb.edu/online/joint98/kitaev/

  14. 14.

    Lawson, H.B., Michelsohn, M.-L.: Spin Geometry. Princeton Mathematical Series, vol. 38. Princeton University Press, Princeton (1990)

    MATH  Google Scholar 

  15. 15.

    Liu, Y., Nowak, M., Zahed, I.: Disorder in the Sachdev–Ye–Kitaev model. Phys. Lett. B 773, 647–653 (2017)

    Article  Google Scholar 

  16. 16.

    Maldacena, J., Stanford, D.: Remarks on the Sachdev–Ye–Kitaev model. Phys. Rev. D 94, 106002 (2016)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Mehta, M.: Random Matrices, 3rd edn. Academic Press, Boston (2004)

    MATH  Google Scholar 

  18. 18.

    Polchinski, J., Rosenhaus, V.: The spectrum in the Sachdev–Ye–Kitaev model. J. High Energy Phys. 2016, 1 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Sachdev, S.: private communications

  20. 20.

    Sachdev, S., Ye, J.: Gapless spin-fluid ground state in a random quantum Heisenberg magnet. Phys. Rev. Lett. 70, 3339–3342 (1993)

    Article  Google Scholar 

  21. 21.

    Sinai, Y., Soshnikov, A.: Central limit theorem for traces of large random symmetric matrices with independent matrix elements. Bol. Soc. Bras. Mat. 29(1), 1–24 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Talagrand, M.: The Parisi formula. Ann. Math. (2) 163(1), 221–263 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Tao, T.: Topics in Random Matrix Theory, Graduate Studies in Mathematics, vol. 132. American Mathematical Society, Providence (2012)

    Book  Google Scholar 

Download references

Acknowledgements

We thank Subir Sachdev for bringing our attention to the SYK model and sending us a short note on the SYK model, which is very helpful for us to start this project. We also thank Gerard Ben Arous, Zhi-Dong Bai, Peter J. Forrester, Dang-Zheng Liu, and Douglas Stanford for many helpful discussions.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Renjie Feng.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Feng, R., Tian, G. & Wei, D. Spectrum of SYK Model. Peking Math J 2, 41–70 (2019). https://doi.org/10.1007/s42543-018-0007-1

Download citation

Keywords

  • SYK model
  • Global density
  • Random matrices