nLab Sachdev-Ye-Kitaev model



Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT


physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics



From Maldacena-Stanford 16:

Studies of holography have been hampered by the lack of a simple solvable model that can capture features of Einstein gravity. The simplest model, which is a single matrix quantum mechanics, does not appear to lead to black holes. 𝒩=4\mathcal{N} = 4 super Yang Mills at strong ’t Hooft coupling certainly leads to black holes, and exact results are known at large N for many anomalous dimensions and some vacuum correlation functions, but at finite temperature the theory is difficult to study.

A system that reproduces some of the dynamics of black holes should be interacting, but we might hope for a model with interactions that are simple enough that it is still reasonable solvable.

Kitaev has proposed to study a quantum mechanical model of NN Majorana fermions interacting with random interactions (Kitaev 15). It is a simple variant of a model introduced by Sachdev and Ye (Sachdev-Ye 93), which was first discussed in relation to holography in (Sachdev 10).

From Maldacena 18:

The SYK model gives us a glimpse into the interior of an extremal black hole… That’s the feature of SYK that I find most interesting… It is a feature this model has, that I think no other model has


Let 𝒥 ijkl\mathcal{J}_{i j k l} be random variables with expectation values E[𝒥 ijkl]=0E[\mathcal{J}_{i j k l}]=0 and E[𝒥 ijkl 2]=6J 2N 3E[\mathcal{J}_{i j k l}^2]=\frac{6J^2}{N^3}.

The Lagrangian density defining the SYK model is this:

L=12 i=1 Nχ i tχ i14! i,j,k,l=1 N𝒥 ijklχ iχ jχ kχ l L = \frac{1}{2} \sum_{i=1}^N \chi_i \partial_t \chi_i - \frac{1}{4!} \sum_{i,j,k,l=1}^N \mathcal{J}_{i j k l} \chi_i \chi_j \chi_k \chi_l



The model is due to

Textbook accounts:

Further review:

See also

Further developments:

Relation to random matrix theory:

  • Yiyang Jia, Jacobus J. M. Verbaarschot, Spectral Fluctuations in the Sachdev-Ye-Kitaev Model (arXiv:1912.11923)

Discussion of possible realization of the SYK-model in condensed matter physics:

  • D. I. Pikulin, M. Franz, Black hole on a chip: proposal for a physical realization of the SYK model in a solid-state system, Phys. Rev. X 7, 031006 (2017) (arXiv:1702.04426)

Relation of the SYK-model to the strange metals via AdS/CMT:

SYK-model in AdS 2/CFT 1AdS_2/CFT_1

Discussion of the SYK-model as the AdS/CFT dual of JT-gravity in nearly AdS2/CFT1 and AdS-CFT in condensed matter physics:

Original articles:


Relation to black holes in terms of Majorana dimer states:

Relation to black holes in string theory and random matrix theory:

On non-perturbative effects and resurgence:

See also

In terms of chord diagrams

Discussion of (Lie algebra-)weight systems on chord diagrams encoding SYK model single trace observables:

(for more see at weight systems on chord diagrams in physics)

  • Antonio M. García-García, Yiyang Jia, Jacobus J. M. Verbaarschot, Exact moments of the Sachdev-Ye-Kitaev model up to order 1/N 21/N^2, JHEP 04 (2018) 146 (arXiv:1801.02696)

  • Yiyang Jia, Jacobus J. M. Verbaarschot, Section 4 of: Large NN expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs, JHEP 11 (2018) 031 (arXiv:1806.03271)

  • Micha Berkooz, Prithvi Narayan, Joan Simón, Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction, JHEP 08 (2018) 192 (arxiv:1806.04380)


  • László Erdős, Dominik Schröder, Phase Transition in the Density of States of Quantum Spin Glasses, D. Math Phys Anal Geom (2014) 17: 9164 (arXiv:1407.1552)

which in turn follows

  • Philippe Flajolet, Marc Noy, Analytic Combinatorics of Chord Diagrams, pages 191–201 in Daniel Krob, Alexander A. Mikhalev,and Alexander V. Mikhalev, (eds.), Formal Power Series and Algebraic Combinatorics, Springer 2000 (doi:10.1007/978-3-662-04166-6_17)

With emphasis on holographic content:

  • Micha Berkooz, Mikhail Isachenkov, Vladimir Narovlansky, Genis Torrents, Section 5 of: Towards a full solution of the large NN double-scaled SYK model, JHEP 03 (2019) 079 (arxiv:1811.02584)

  • Vladimir Narovlansky, Slide 23 (of 28) of: Towards a Solution of Large NN Double-Scaled SYK, 2019 (pdf)

and specifically in relation to Jackiw-Teitelboim gravity:

Last revised on February 29, 2024 at 04:00:08. See the history of this page for a list of all contributions to it.