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^[1] and later modified by Alexei Kitaev to the present commonly used form^[2][3]. 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^[4], with pioneering experiments implemented in a NMR setting^[5].

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.

References

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