Sachdev-Ye-Kitaev model

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

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

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

The more famous model is when ${\displaystyle m=4}$, then the model becomes

${\displaystyle H_{SYK}=-{\frac {1}{4!}}\sum _{i_{1}\ldots i_{4}\in \{1,N\}}J_{i_{1}i_{2}i_{3}i_{4}}\psi _{i_{1}}\psi _{i_{2}}\psi _{i_{3}}\psi _{i_{4}}}$,

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