Sachdev–Ye–Kitaev model

In condensed matter physics and black hole physics, the Sachdev–Ye–Kitaev (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. 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 are amenable to digital quantum simulation, with pioneering experiments implemented in nuclear magnetic resonance.

Model
Let $$n$$ be an integer and $$m$$ an even integer such that $$2\leq m\leq n$$, and consider a set of Majorana fermions $$\psi_1,\dotsc,\psi_n$$ which are fermion operators satisfying conditions:
 * 1) Hermitian $$\psi_i^{\dagger}=\psi_i$$;
 * 2) Clifford relation $$\{\psi_i,\psi_j\}=2\delta_{ij}$$.

Let $$J_{i_1 i_2 \cdots i_m}$$ be random variables whose expectations satisfy:


 * 1) $$\mathbf{E}(J_{i_1i_2\cdots i_m})=0$$;
 * 2) $$\mathbf{E}(J_{i_1i_2\cdots i_m}^2)=1$$.

Then the SYK model is defined as


 * $$H_{\rm SYK}=i^{m/2}\sum_{1 \leq i_1 < \cdots < i_m \leq n}J_{i_1i_2\cdots i_m}\psi_{i_1}\psi_{i_2}\cdots\psi_{i_m}$$.

Note that sometimes an extra normalization factor is included.

The most famous model is when $$m=4$$:


 * $$H_{\rm SYK}=-\frac{1}{4!}\sum_{i_1, \dotsc, i_4 = 1}^n J_{i_1i_2i_3 i_4}\psi_{i_1}\psi_{i_2}\psi_{i_3}\psi_{i_4}$$,

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