# Sachdev-Ye-Kitaev model The SYK model is a 0+1 dimensional field theory, which is, in other words, a quantum mechanical model. It consists of $N$ Majorana fermions, which in 0+1 dimensions just means matrices $\psi_i$ satisfying $\left\{\psi_i, \psi_j\right\}=\delta_{ij}, \quad i, j=1, \ldots, N.$These matrices are $2^{N/2}$ by $2^{N/2}$: $N/2$ qubits needed to build $N$ Majorana fermions. The SYK Hamiltonian is given by$H_{4}=J_{ijkl}\, \psi_i \psi_j \psi_k \psi_l,$where $J_{ijkl}$ are random couplings with an expectation value $\left\langle J_{ijkl}^2\right\rangle=\frac{\mathcal{J}^2}{N^3}$. More generally, it can have $p$ fermions interacting:$H_p=i^{p / 2} \sum_{1 \leq i_1<\cdots<i_p \leq N} J_{i_1 i_2 \cdots i_p} \psi_{i_1} \cdots \psi_{i_p}.$ ## Refs - original - [[1992#Sachdev, Ye]] - Kitaev's talks at KITP (April 7, 2015 and May 27, 2015) - Schwinger-Dyson equation and spectrum - [[2016#Polchinski, Rosenhaus]] - Schwarzian description - [[2016#Maldacena, Stanford]] - [[2017#Kitaev, Suh]] - non-linear soft mode action - [[2024#Berkooz, Frumkin, Mamroud, Seitz]] - [[2024#Bucca, Mezei]] - general reviews - [Wiki](https://en.wikipedia.org/wiki/Sachdev%E2%80%93Ye%E2%80%93Kitaev_model) - talk by Juan [](https://youtu.be/PMhSQo6Tfz0) - Stanford's talk at strings 2017 [](https://youtu.be/m-jwdkIf1gc) - light review [[Sachdev2022Review]][](https://arxiv.org/pdf/2205.02285.pdf) - [[2023#Sachdev (Review)]]: short review, focus on thermodynamics - properties - relation to [[0579 Random matrix theory|random matrix theory]] - [[2016#Cotler, Gur-Ari, Hanada, Polchinski, Saad, Shenker, Stanford, Streicher, Tezuka]] - [[2018#Gharibyan, Hanada, Shenker, Tezuka]] - extensions - lattices of SYK - [[2016#Gu, Qi, Stanford]] - [[2017#Song, Jian, Balents]] - [[2017#Jian, Yao]] - SUSY - [[2016#Fu, Gaiotto, Maldacena, Sachdev]] - higher dimensions - [[2016#Turiaci, Verlinde]] - [[2022#Pasterski, Verlinde (Jan, b)]] - dual of SYK - [[0050 JT gravity]] - as a string theory: [[GoelVerlinde2021]][](https://arxiv.org/abs/2103.03187) - EFT and RG: [[2024#Choun, Kim, Kim]] ## $G$-$\Sigma$ action - introduce bi-local fields $G$ and $\Sigma$ as auxiliary fields - partition function can be written as $Z=\int D G\left(t, t^{\prime}\right) D \Sigma\left(t, t^{\prime}\right) e^{-N I[\Sigma, G]}$ ## Special limits - large $N$ but finite $p$ - Feynman diagrams are dominated by melonic diagrams - large $N$ and $p$ - see [[0503 Double-scaled SYK]]. ## Two-point function - real-time two-point function decays - it is a manifestation of [[0008 Quantum chaos|chaos]] in this model ## Chaos - chaos can be seen by looking at $\left\langle\left\{\psi_a(0), \psi_b(t)\right\}^2\right\rangle$ - SYK saturates the [[0474 Chaos bound|chaos bound]], but see [[2023#Lin, Stanford]] for sub-maximal chaos - an integrable model with chaos-like OTOC: [[2024#Ozaki, Katsura]] ## Related topics - [[0655 JT-Schwarzian correspondence]]