nLab Sachdev-Ye-Kitaev model

Idea

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. $\mathcal{N} = 4$ 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 $N$ 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

Definition

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

The Lagrangian density defining the SYK model is this:

$$L = \frac{1}{2} \sum_{i=1}^N \chi_i \partial_t \chi_i - \frac{1}{4!} \sum_{i,j,k,l=1}^N \mathcal{J}_{i j k l} \chi_i \chi_j \chi_k \chi_l$$

Related concepts

• strange metal

• Jackiw-Teitelboim gravity

• AdS-CFT in condensed matter physics

• AdS-CFT correspondence

• BFSS matrix model

References

General

The model is due to

• Subir Sachdev, Jinwu Ye, Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet, Phys. Rev. Lett. 70:3339, 1993 (arXiv:cond-mat/9212030)

• Alexei Kitaev, A simple model of quantum holography, Talks at KITP, April 7, 2015 and May 27, 2015. (part I, part II)

Textbook accounts:

• Sean Hartnoll, Andrew Lucas, Subir Sachdev, Section 5.11 of: Holographic quantum matter, MIT Press 2018 (arXiv:1612.07324, publisher)

Further review:

• Subir Sachdev, The SYK model, talk at Aspen Center for Physics, 2018 (pdf)

• Vladimir Rosenhaus, An introduction to the SYK model, Journal of Physics A: Mathematical and Theoretical, Volume 52, Number 32 (arXiv:1807.03334, doi:10.1088/1751-8121/ab2ce1)

• Matteo Laudonio, Romain Pascalie, Adrian Tanasa, Combinatorial aspects of the Sachdev-Ye-Kitaev model (arXiv:2001.11849)

• Subir Sachdev, Quantum glasses, reparameterization invariance, Sachdev-Ye-Kitaev models [arXiv:2402.17824]

See also

• Wikipedia, SYK model of non-Fermi liquids and black holes

Further developments:

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

• Alexandre Streicher, SYK Correlators for All Energies (arxiv:1911.10171)

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:

• Subir Sachdev, Statistical mechanics of strange metals and black holes (arXiv:2205.02285)

SYK-model in $AdS_2/CFT_1$

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:

• Juan Maldacena, Douglas Stanford, Comments on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94, 106002 (2016)(arXiv:1604.07818)

• Subir Sachdev, Holographic metals and the fractionalized Fermi liquid, Phys. Rev. Lett. 105:151602, 2010 (arXiv:1006.3794)

Review:

• Gábor Sárosi, $AdS_2$ holography and the SYK model, Proceedings of Science 323 (arXiv:1711.08482, doi:10.22323/1.323.0001)

• Juan Maldacena, Toy models for black holes II, talk at PiTP 2018 From QBits to spacetime (recording)

• Dmitrii A. Trunin, Pedagogical introduction to SYK model and 2D Dilaton Gravity (arXiv:2002.12187)

Relation to black holes in terms of Majorana dimer states:

• Ioanna Kourkoulou, Juan Maldacena, Pure states in the SYK model and nearly-$AdS_2$ gravity (arXiv:1707.02325)

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

• Jordan S. Cotler, Guy Gur-Ari, Masanori Hanada, Joseph Polchinski, Phil Saad, Stephen Shenker, Douglas Stanford, Alexandre Streicher, Masaki Tezuka, Black Holes and Random Matrices, JHEP 1705:118, 2017 (arXiv:1611.04650)

• Tomoki Nosaka, Tokiro Numasawa, Quantum Chaos, Thermodynamics and Black Hole Microstates in the mass deformed SYK model (arXiv:1912.12302)

On non-perturbative effects and resurgence:

• Paolo Gregori, Ricardo Schiappa, From Minimal Strings towards Jackiw-Teitelboim Gravity: On their Resurgence, Resonance, and Black Holes (arXiv:2108.11409)

See also

• Yuri D. Lensky, Xiao-Liang Qi, Pengfei Zhang, Size of bulk fermions in the SYK model (arXiv:2002.01961)

• Xiao-Liang Qi, Pengfei Zhang, The Coupled SYK model at Finite Temperature (arXiv:2003.03916)

• Akash Goel, Herman Verlinde, Towards a String Dual of SYK (arXiv:2103.03187)

• Tarek Anous, Felix M. Haehl, The quantum $p$-spin glass model: A user manual for holographers (arXiv:2106.03838)

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 $1/N^2$, JHEP 04 (2018) 146 (arXiv:1801.02696)

• Yiyang Jia, Jacobus J. M. Verbaarschot, Section 4 of: Large $N$ 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)

following:

• 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 $N$ double-scaled SYK model, JHEP 03 (2019) 079 (arxiv:1811.02584)

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

and specifically in relation to Jackiw-Teitelboim gravity:

• Andreas Blommaert, Thomas Mertens, Henri Verschelde, The Schwarzian Theory - A Wilson Line Perspective, JHEP 1812 (2018) 022 (arXiv:1806.07765)

• Andreas Blommaert, Thomas Mertens, Henri Verschelde, Fine Structure of Jackiw-Teitelboim Quantum Gravity, JHEP 1909 (2019) 066 (arXiv:1812.00918)