Complex Sachdev-Ye-Kitaev model in the double scaling limit

Abstract

We solve for the exact energy spectrum, 2-point and 4-point functions of the complex SYK model, in the double scaling limit at all energy scales. This model has a U(1) global symmetry. The analysis shows how to incorporate a chemical potential in the chord diagram picture, and we present results for the various observables also at a given fixed charge sector. In addition to matching to the spectral asymmetry, we consider an analogous asymmetry measure of the 2-point function obeying a non-trivial dependence on the operator’s dimension. We also provide the chord diagram structure for an SYK-like model that has a U(M) global symmetry at any disorder realization. We then show how to exactly compute the effect of inserting very heavy operators, with formally infinite conformal dimension. The latter separate the gravitational spacetime into several parts connected by an interface, whose properties are exactly computable at all scales. In particular, light enough states can still go between the spaces. This behavior has a simple description in the chord diagram picture.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].

    ADS  Article  Google Scholar 

  2. [2]

    S. Sachdev, Holographic metals and the fractionalized Fermi liquid, Phys. Rev. Lett. 105 (2010) 151602 [arXiv:1006.3794] [INSPIRE].

    ADS  Article  Google Scholar 

  3. [3]

    A. Kitaev, A simple model of quantum holography, talks given at KITP, April 7 and May 27 (2015).

  4. [4]

    J. Polchinski and V. Rosenhaus, The spectrum in the Sachdev-Ye-Kitaev model, JHEP 04 (2016) 001 [arXiv:1601.06768] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  5. [5]

    J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  6. [6]

    J. Maldacena, S. H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  7. [7]

    D. Bagrets, A. Altland and A. Kamenev, Sachdev-Ye-Kitaev model as Liouville quantum mechanics, Nucl. Phys. B 911 (2016) 191 [arXiv:1607.00694] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  8. [8]

    D. Bagrets, A. Altland and A. Kamenev, Power-law out of time order correlation functions in the SYK model, Nucl. Phys. B 921 (2017) 727 [arXiv:1702.08902] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  9. [9]

    T. G. Mertens, G. J. Turiaci and H. L. Verlinde, Solving the Schwarzian via the Conformal Bootstrap, JHEP 08 (2017) 136 [arXiv:1705.08408] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  10. [10]

    H. T. Lam, T. G. Mertens, G. J. Turiaci and H. Verlinde, Shockwave S-matrix from Schwarzian Quantum Mechanics, JHEP 11 (2018) 182 [arXiv:1804.09834] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  11. [11]

    J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimensional nearly Anti-de-Sitter space, PTEP 2016 (2016) 12C104 [arXiv:1606.01857] [INSPIRE].

    MATH  Google Scholar 

  12. [12]

    K. Jensen, Chaos in AdS2 holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].

    ADS  Article  Google Scholar 

  13. [13]

    D. Stanford and E. Witten, Fermionic Localization of the Schwarzian Theory, JHEP 10 (2017) 008 [arXiv:1703.04612] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  14. [14]

    A. Kitaev and S. J. Suh, The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual, JHEP 05 (2018) 183 [arXiv:1711.08467] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  15. [15]

    A. Almheiri and J. Polchinski, Models of AdS2 backreaction and holography, JHEP 11 (2015) 014 [arXiv:1402.6334] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  16. [16]

    L. V. Iliesiu, S. S. Pufu, H. Verlinde and Y. Wang, An exact quantization of Jackiw-Teitelboim gravity, JHEP 11 (2019) 091 [arXiv:1905.02726] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  17. [17]

    P. Saad, S. H. Shenker and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115 [INSPIRE].

  18. [18]

    J. S. Cotler et al., Black Holes and Random Matrices, JHEP 05 (2017) 118 [Erratum ibid. 09 (2018) 002] [arXiv:1611.04650] [INSPIRE].

  19. [19]

    H. Gharibyan, M. Hanada, S. H. Shenker and M. Tezuka, Onset of random matrix behavior in scrambling systems, JHEP 07 (2018) 124 [Erratum ibid. 02 (2019) 197] [arXiv:1803.08050] [INSPIRE].

  20. [20]

    O. Parcollet and A. Georges, Non-Fermi-liquid regime of a doped Mott insulator, Phys. Rev. B 59 (1999) 5341.

    ADS  Article  Google Scholar 

  21. [21]

    M. Berkooz, P. Narayan, M. Rozali and J. Simón, Higher dimensional generalizations of the SYK model, JHEP 01 (2017) 138 [arXiv:1610.02422] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  22. [22]

    Y. Gu, X.-L. Qi and D. Stanford, Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models, JHEP 05 (2017) 125 [arXiv:1609.07832] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  23. [23]

    M. Berkooz, P. Narayan, M. Rozali and J. Simón, Comments on the random Thirring model, JHEP 09 (2017) 057 [arXiv:1702.05105] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  24. [24]

    J. Murugan, D. Stanford and E. Witten, More on supersymmetric and 2d analogs of the SYK model, JHEP 08 (2017) 146 [arXiv:1706.05362] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  25. [25]

    B. Lian, S. L. Sondhi and Z. Yang, The chiral SYK model, JHEP 09 (2019) 067 [arXiv:1906.03308] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  26. [26]

    V. Narovlansky and O. Aharony, Renormalization group in field theories with quantum quenched disorder, Phys. Rev. Lett. 121 (2018) 071601 [arXiv:1803.08529] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  27. [27]

    O. Aharony and V. Narovlansky, Renormalization group flow in field theories with quenched disorder, Phys. Rev. D 98 (2018) 045012 [arXiv:1803.08534] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  28. [28]

    D. J. Gross and V. Rosenhaus, The bulk dual of SYK: cubic couplings, JHEP 05 (2017) 092 [arXiv:1702.08016] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  29. [29]

    D. J. Gross and V. Rosenhaus, All point correlation functions in SYK, JHEP 12 (2017) 148 [arXiv:1710.08113] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  30. [30]

    A. M. García-García and J. J. M. Verbaarschot, Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 126010 [arXiv:1610.03816] [INSPIRE].

    ADS  Article  Google Scholar 

  31. [31]

    A. M. García-García and J. J. M. Verbaarschot, Analytical spectral density of the Sachdev-Ye-Kitaev model at finite N, Phys. Rev. D 96 (2017) 066012 [arXiv:1701.06593] [INSPIRE].

    ADS  Article  Google Scholar 

  32. [32]

    A. M. García-García, Y. Jia and J. J. M. Verbaarschot, Exact moments of the Sachdev-Ye-Kitaev model up to order 1/N2, JHEP 04 (2018) 146 [arXiv:1801.02696] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  33. [33]

    L. Erdős and D. Schröder, Phase transition in the density of states of quantum spin glasses, Math. Phys. Anal. Geom. 17 (2014) 441 [arXiv:1407.1552] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  34. [34]

    M. Berkooz, P. Narayan and J. Simon, Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction, JHEP 08 (2018) 192 [arXiv:1806.04380] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  35. [35]

    M. Berkooz, M. Isachenkov, V. Narovlansky and G. Torrents, Towards a full solution of the large N double-scaled SYK model, JHEP 03 (2019) 079 [arXiv:1811.02584] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  36. [36]

    M. Berkooz, N. Brukner, V. Narovlansky and A. Raz, The double scaled limit of Super–Symmetric SYK models, JHEP 12 (2020) 110 [arXiv:2003.04405] [INSPIRE].

    ADS  Article  Google Scholar 

  37. [37]

    C. L. Baldwin and B. Swingle, Quenched vs. annealed: glassiness from SK to SYK, Phys. Rev. X 10 (2020) 031026 [arXiv:1911.11865] [INSPIRE].

    Google Scholar 

  38. [38]

    A. Goel, H. T. Lam, G. J. Turiaci and H. Verlinde, Expanding the black hole interior: partially entangled thermal states in SYK, JHEP 02 (2019) 156 [arXiv:1807.03916] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  39. [39]

    R. A. Davison, W. Fu, A. Georges, Y. Gu, K. Jensen and S. Sachdev, Thermoelectric transport in disordered metals without quasiparticles: The Sachdev-Ye-Kitaev models and holography, Phys. Rev. B 95 (2017) 155131 [arXiv:1612.00849] [INSPIRE].

    ADS  Article  Google Scholar 

  40. [40]

    S. Sachdev, Bekenstein-Hawking Entropy and Strange Metals, Phys. Rev. X 5 (2015) 041025 [arXiv:1506.05111] [INSPIRE].

    Google Scholar 

  41. [41]

    I. R. Klebanov and G. Tarnopolsky, Uncolored random tensors, melon diagrams, and the Sachdev-Ye-Kitaev models, Phys. Rev. D 95 (2017) 046004 [arXiv:1611.08915] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  42. [42]

    I. R. Klebanov, F. Popov and G. Tarnopolsky, TASI lectures on large N tensor models, PoS(TASI2017)004 [arXiv:1808.09434] [INSPIRE].

  43. [43]

    T. Azeyanagi, F. Ferrari and F. I. Schaposnik Massolo, Phase diagram of planar matrix quantum mechanics, tensor, and Sachdev-Ye-Kitaev models, Phys. Rev. Lett. 120 (2018) 061602 [arXiv:1707.03431] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  44. [44]

    F. Ferrari and F. I. Schaposnik Massolo, Phases of melonic quantum mechanics, Phys. Rev. D 100 (2019) 026007 [arXiv:1903.06633] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  45. [45]

    G. Tarnopolsky, private communications.

  46. [46]

    Y. Gu, A. Kitaev, S. Sachdev and G. Tarnopolsky, Notes on the complex Sachdev-Ye-Kitaev model, JHEP 02 (2020) 157 [arXiv:1910.14099] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  47. [47]

    J. Yoon, SYK models and SYK-like tensor models with global symmetry, JHEP 10 (2017) 183 [arXiv:1707.01740] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  48. [48]

    R. Bhattacharya, S. Chakrabarti, D. P. Jatkar and A. Kundu, SYK model, chaos and conserved charge, JHEP 11 (2017) 180 [arXiv:1709.07613] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  49. [49]

    I. Kourkoulou and J. Maldacena, Pure states in the SYK model and nearly-AdS2 gravity, arXiv:1707.02325 [INSPIRE].

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Correspondence to Vladimir Narovlansky.

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ArXiv ePrint: 2006.13983

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Berkooz, M., Narovlansky, V. & Raj, H. Complex Sachdev-Ye-Kitaev model in the double scaling limit. J. High Energ. Phys. 2021, 113 (2021). https://doi.org/10.1007/JHEP02(2021)113

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Keywords

  • AdS-CFT Correspondence
  • Holography and condensed matter physics (AdS/CMT)
  • Models of Quantum Gravity