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 ${\displaystyle n}$ be an integer and ${\displaystyle m}$ and even integer such that ${\displaystyle 2\leq m\leq n}$, consider a set of Majorana Fermions ${\displaystyle \psi _{i_{1}},\cdots ,\psi _{i_{n}}}$which are Fermion operators satisfy conditions: (1) Hermitian ${\displaystyle \psi _{i}^{\dagger }=\psi _{i}}$; (2) Clifford relation ${\displaystyle \{\psi _{i},\psi _{j}\}=2\delta _{ij}}$. Choosing a random variable ${\displaystyle J_{i_{1}i_{2}\cdots i_{m}}}$ such that the expectation satisfy: (1) ${\displaystyle \mathbf {E} (J_{i_{1}i_{2}\cdots i_{m}})=0}$; and (2) ${\displaystyle \mathbf {E} (J_{i_{1}i_{2}\cdots i_{m}}^{2})=1}$, Then the SYK model is defined as

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

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

The more famous model is when ${\displaystyle m=4}$, then the model becomes

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

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

References

1. Sachdev, Subir; Ye, Jinwu (1993-05-24). "Gapless spin-fluid ground state in a random quantum Heisenberg magnet". Physical Review Letters. 70 (21): 3339–3342. arXiv:cond-mat/9212030. Bibcode:1993PhRvL..70.3339S. doi:10.1103/PhysRevLett.70.3339. PMID 10053843.
2. "Alexei Kitaev, Caltech & KITP, A simple model of quantum holography (part 1)". online.kitp.ucsb.edu. Retrieved 2019-11-02.
3. "Alexei Kitaev, Caltech, A simple model of quantum holography (part 2)". online.kitp.ucsb.edu. Retrieved 2019-11-02.
4. García-Álvarez, L.; Egusquiza, I.L.; Lamata, L.; del Campo, A.; Sonner, J.; Solano, E. (2017). "Digital Quantum Simulation of Minimal AdS/CFT". Physical Review Letters. 119: 040501. arXiv:1607.08560. Bibcode:2017PhRvL.119d0501G. doi:10.1103/PhysRevLett.119.040501.
5. Luo, Z.; You, Y.-Z.; Li, J.; Jian, C.-M.; Lu, D.; Xu, C.; Zeng, B.; Laflamme, R. (2019). "Quantum simulation of the non-fermi-liquid state of Sachdev-Ye-Kitaev model". npj Quantum Information. 5: 53. arXiv:1712.06458. Bibcode:2019npjQI...5...53L. doi:10.1038/s41534-019-0166-7.