Sachdev–Ye–Kitaev model explained

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

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,...c,\psi} _{n} which are fermion operators satisfying conditions:

1. Hermitian

\psi ^{\dagger} _{i} = \psi _{i} ;

1. Clifford relation

\{\psi _{i,\psi} _{j\}=2\delta} _{ij} .

Let

J _{i _{1} i _{2} … i _{m}} be random variables whose expectations satisfy:E (J _{i _{1i} _{2 …} i _{m}} ) =0 E (J ^{2 ) =1} _{i _{1i} _{2 …} i _{m}} .

Then the SYK model is defined as

H _{\rm} = i ^{m/2} \sum _{1 \leq i _{1} < … < i _{m} \leq n} J _{i _{1i} _{2 …} i _{m}} \psi _{i _{1}} \psi _{i _{2}} … \psi _{i _{m}} .

Note that sometimes an extra normalization factor is included.

The most famous model is when

m =4 :H _{\rm} =- (1)/(4 !) \sum ^{n} _{i _{1,} ... c, i _{4} =1} 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.

See also

• Non-Fermi liquid

Notes and 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. 10.1103/PhysRevLett.70.3339. 10053843. cond-mat/9212030. 1993PhRvL..70.3339S. 1103248 .
2. Web site: Alexei Kitaev, Caltech & KITP, A simple model of quantum holography (part 1). online.kitp.ucsb.edu. 2019-11-02.
3. Web site: Alexei Kitaev, Caltech, A simple model of quantum holography (part 2). online.kitp.ucsb.edu. 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. 10.1103/PhysRevB.96.121119. 1703.06890 . 2017PhRvB..96l1119C . 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. 10.1103/PhysRevLett.121.036403. 30085787 . 1802.00802. 2018PhRvL.121c6403C . 51940526 .
6. Wei. Chenan. Sedrakyan. Tigran. 2021-01-29. Optical lattice platform for the Sachdev-Ye-Kitaev model. Phys. Rev. A. 103. 1. 013323. 10.1103/PhysRevA.103.013323. 2005.07640. 2021PhRvA.103a3323W. 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. 10.1103/PhysRevLett.119.040501. 29341740 . 1607.08560. 2017PhRvL.119d0501G . 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. 10.1038/s41534-019-0166-7. 1712.06458. 2019npjQI...5...53L. 195344916 .