Physics: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 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. The model is amenable to digital quantum simulation, with pioneering experiments implemented in a NMR setting.

SYK model

Let [math]\displaystyle{ n }[/math] be an integer and [math]\displaystyle{ m }[/math] and even integer such that [math]\displaystyle{ 2\leq m\leq n }[/math], consider a set of Majorana Fermions [math]\displaystyle{ \psi_{i_1},\cdots,\psi_{i_n} }[/math]which are Fermion operators satisfy conditions: (1) Hermitian [math]\displaystyle{ \psi_i^{\dagger}=\psi_i }[/math]; (2) Clifford relation [math]\displaystyle{ \{\psi_i,\psi_j\}=2\delta_{ij} }[/math]. Choosing a random variable [math]\displaystyle{ J_{i_1i_2\cdots i_m} }[/math] such that the expectation satisfy: (1) [math]\displaystyle{ \mathbf{E}(J_{i_1i_2\cdots i_m})=0 }[/math]; and (2) [math]\displaystyle{ \mathbf{E}(J_{i_1i_2\cdots i_m}^2)=1 }[/math], Then the SYK model is defined as

[math]\displaystyle{ H_{SYK}=i^{m/2}\sum_{1\leq i_1\lt \cdots\lt i_m\leq n}J_{i_1i_2\cdots i_m}\psi_{i_1}\psi_{i_2}\cdots\psi_{i_n} }[/math].

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

The more famous model is when [math]\displaystyle{ m=4 }[/math], then the model becomes

[math]\displaystyle{ H_{SYK}=-\frac{1}{4!}\sum_{i_1 \ldots i_4\in \{1,N\} }J_{i_1i_2i_3 i_4}\psi_{i_1}\psi_{i_2}\psi_{i_3}\psi_{i_4} }[/math],

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