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,[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. Many condensed matter systems, such as quantum dot coupled to topological superconducting wires,[4] graphene flake with irregular boundary,[5] and kagome optical lattice with impurities,[6] are proposed to be modeled by it. Some variants of the model are amenable to digital quantum simulation,[7] with pioneering experiments implemented in nuclear magnetic resonance.[8]

Model

Let ${\displaystyle n}$ be an integer and ${\displaystyle m}$ an even integer such that ${\displaystyle 2\leq m\leq n}$, and consider a set of Majorana fermions ${\displaystyle \psi _{1},\dotsc ,\psi _{n}}$ which are fermion operators satisfying conditions:

1. Hermitian ${\displaystyle \psi _{i}^{\dagger }=\psi _{i}}$;
2. Clifford relation ${\displaystyle \{\psi _{i},\psi _{j}\}=2\delta _{ij}}$.

Let ${\displaystyle J_{i_{1}i_{2}\cdots i_{m}}}$ be random variables whose expectations satisfy:

1. ${\displaystyle \mathbf {E} (J_{i_{1}i_{2}\cdots i_{m}})=0}$;
2. ${\displaystyle \mathbf {E} (J_{i_{1}i_{2}\cdots i_{m}}^{2})=1}$.

Then the SYK model is defined as

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

Note that sometimes an extra normalization factor is included.

The most famous model is when ${\displaystyle m=4}$:

${\displaystyle H_{\rm {SYK}}=-{\frac {1}{4!}}\sum _{i_{1},\dotsc ,i_{4}=1}^{n}J_{i_{1}i_{2}i_{3}i_{4}}\psi _{i_{1}}\psi _{i_{2}}\psi _{i_{3}}\psi _{i_{4}}}$,

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

See also

• Non-Fermi liquid

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. S2CID 1103248.
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. Chew, Aaron; Essin, Andrew; Alicea, Jason (2017-09-29). "Approximating the Sachdev-Ye-Kitaev model with Majorana wires". Phys. Rev. B. 96 (12): 121119. arXiv:1703.06890. Bibcode:2017PhRvB..96l1119C. doi:10.1103/PhysRevB.96.121119. S2CID 119222270.
5. Chen, Anffany; Ilan, R.; Juan, F.; Pikulin, D.I.; Franz, M. (2018-06-18). "Quantum Holography in a Graphene Flake with an Irregular Boundary". Phys. Rev. Lett. 121 (3): 036403. arXiv:1802.00802. Bibcode:2018PhRvL.121c6403C. doi:10.1103/PhysRevLett.121.036403. PMID 30085787. S2CID 51940526.
6. Wei, Chenan; Sedrakyan, Tigran (2021-01-29). "Optical lattice platform for the Sachdev-Ye-Kitaev model". Phys. Rev. A. 103 (1): 013323. arXiv:2005.07640. Bibcode:2021PhRvA.103a3323W. doi:10.1103/PhysRevA.103.013323. S2CID 234363891.
7. 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 (4): 040501. arXiv:1607.08560. Bibcode:2017PhRvL.119d0501G. doi:10.1103/PhysRevLett.119.040501. PMID 29341740. S2CID 5144368.
8. 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. S2CID 195344916.