Sachdev–Ye–Kitaev model

In condensed matter physics and black hole physics, the Sachdev-Ye-Kitaev (SYK) model commonly known as SYK model is an exactly solvable model initially proposed by Subir Sachdev and his graduate student Jinwu Ye and later modified by Alexei Kitaev to the present commonly used form. The model has be believed to o bring insights into the understanding of strongly correlated materials and it also has a close relation with the discrete model of AdS/CFT, and Fermionic code.

SYK model

Let $n$ be an integer and $m$ and even integer such that $2\leq m\leq n$, consider a set of Majorana Fermions $\psi _{i_{1}},\cdots ,\psi _{i_{n}}$which are Fermion operators satisfy conditions: (1) Hermitian $\psi _{i}^{\dagger }=\psi _{i}$; (2) Clifford relation $\{\psi _{i},\psi _{j}\}=2\delta _{ij}$. Choosing a random variable $J_{i_{1}i_{2}\cdots i_{m}}$ such that the expectation satisfy: (1) $\mathbf {E} (J_{i_{1}i_{2}\cdots i_{m}})=0$; and (2) $\mathbf {E} (J_{i_{1}i_{2}\cdots i_{m}}^{2})=1$, Then the SYK model is defined as

$H_{SYK}=i^{m/2}\sum _{1\leq i_{1}<\cdots <i_{m}\leq n}J_{i_{1}i_{2}\cdots i_{m}}\psi _{i_{1}}\psi _{i_{2}}\cdots \psi _{i_{n}}$.

Note that sometimes, an extra normalization factor will be added.

The more famous model is when $m=4$, then the model becomes

$H_{SYK}=-{\frac {1}{4!}}\sum _{1\leq i_{1}<\cdots <i_{4}\leq n}J_{i_{1}i_{2}i_{3}i_{4}}\psi _{i_{1}}\psi _{i_{2}}\psi _{i_{3}}\psi _{i_{4}}$,

notice that here the factor $1/4!$ is added for coincidence with the usually used form.