Sachdev–Ye–Kitaev model

Let ${\displaystyle n}$ be an integer and ${\displaystyle m}$ an even integer such that ${\displaystyle 2\leq m\leq n}$, and consider a set of Majorana fermions ${\displaystyle \psi _{1},\dotsc ,\psi _{n}}$ which are fermion operators satisfying conditions:

1. Hermitian ${\displaystyle \psi _{i}^{\dagger }=\psi _{i}}$;
2. Clifford relation ${\displaystyle \{\psi _{i},\psi _{j}\}=2\delta _{ij}}$.

Let ${\displaystyle J_{i_{1}i_{2}\cdots i_{m}}}$ be random variables whose expectations satisfy:

1. ${\displaystyle \mathbf {E} (J_{i_{1}i_{2}\cdots i_{m}})=0}$;
2. ${\displaystyle \mathbf {E} (J_{i_{1}i_{2}\cdots i_{m}}^{2})=1}$.

Then the SYK model is defined as

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

Note that sometimes an extra normalization factor is included.

The most famous model is when ${\displaystyle m=4}$:

${\displaystyle H_{\rm {SYK}}=-{\frac {1}{4!}}\sum _{i_{1},\dotsc ,i_{4}=1}^{n}J_{i_{1}i_{2}i_{3}i_{4}}\psi _{i_{1}}\psi _{i_{2}}\psi _{i_{3}}\psi _{i_{4}}}$,

where the factor ${\displaystyle 1/4!}$ is included to coincide with the most popular form.

• Non-Fermi liquid