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 Jinwu Ye, and later modified by Alexei Kitaev to the present commonly used form. The model is believed to 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. Many condensed matter systems, such as quantum dot coupled to topological superconducting wires, graphene flake with irregular boundary, and kagome optical lattice with impurities, are proposed to be modeled by it. Some variants of the model is amenable to digital quantum simulation, with pioneering experiments implemented in a NMR setting.

Model

Let $n$ be an integer and $m$ an 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 satisfying 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

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

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

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

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

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

See also

• Non-Fermi liquid