Sachdev–Ye–Kitaev model

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Let ${\displaystyle N}$ be an integer and ${\displaystyle q}$ an even integer such that ${\displaystyle 2\leq q\leq N}$, 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. Anticommutation relation ${\displaystyle \{\psi _{i},\psi _{j}\}=\delta _{ij}}$.

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

1. ${\displaystyle \mathbf {E} (J_{i_{1}i_{2}\cdots i_{q}})=0}$;
2. ${\displaystyle \mathbf {E} (J_{i_{1}i_{2}\cdots i_{q}}^{2})={\frac {2^{q-1}}{q}}{\frac {{\mathcal {J}}^{2}(q-1)!}{N^{q-1}}}}$.

Then the ${\displaystyle q}$-body interacting SYK model is defined as^[9]

${\displaystyle H_{\rm {SYK}}=\imath ^{q/2}\sum _{1\leq i_{1}<\cdots <i_{q}\leq N}J_{i_{1}i_{2}\cdots i_{q}}\psi _{i_{1}}\psi _{i_{2}}\cdots \psi _{i_{q}}}$,

where the number of fermions ${\displaystyle N}$ is included in the variance to ensure the energy remains extensive. The ${\displaystyle q}$ dependence in the variance is such that a well-defined large ${\displaystyle q}$ limit may be taken, which yields a controlled ${\displaystyle 1/q}$ expansion.

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

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

• Non-Fermi liquid