Sachdev-Ye-Kitaev model



Algebraic Quantum Field Theory

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



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

The Lagrangian density definign 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}_{ijkl} \chi_i \chi_j \chi_k \chi_l



Review includes

The model is due to

with further discussion in

Further developments in

  • Biao Lian, S. L. Sondhi, Zhenbin Yang, The chiral SYK model (arXiv:1906.03308)

See also

In AdS/CFT and string theory

Discussion in view of AdS-CFT duality, specifically AdS-CFT in condensed matter physics, includes:

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

Last revised on July 23, 2019 at 15:25:33. See the history of this page for a list of all contributions to it.