Physics: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 [math]\displaystyle{ n }[/math] be an integer and [math]\displaystyle{ m }[/math] an even integer such that [math]\displaystyle{ 2\leq m\leq n }[/math], and consider a set of Majorana fermions [math]\displaystyle{ \psi_1,\dotsc,\psi_n }[/math] which are fermion operators satisfying conditions:
- Hermitian [math]\displaystyle{ \psi_i^{\dagger}=\psi_i }[/math];
- Clifford relation [math]\displaystyle{ \{\psi_i,\psi_j\}=2\delta_{ij} }[/math].
Let [math]\displaystyle{ J_{i_1 i_2 \cdots i_m} }[/math] be random variables whose expectations satisfy:
- [math]\displaystyle{ \mathbf{E}(J_{i_1i_2\cdots i_m})=0 }[/math];
- [math]\displaystyle{ \mathbf{E}(J_{i_1i_2\cdots i_m}^2)=1 }[/math].
Then the SYK model is defined as
- [math]\displaystyle{ H_{\rm 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_m} }[/math].
Note that sometimes an extra normalization factor is included.
The most famous model is when [math]\displaystyle{ m=4 }[/math]:
- [math]\displaystyle{ 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} }[/math],
where the factor [math]\displaystyle{ 1/4! }[/math] is included to coincide with the most popular form.
See also
References
- ↑ 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. doi:10.1103/PhysRevLett.70.3339. PMID 10053843. Bibcode: 1993PhRvL..70.3339S.
- ↑ "Alexei Kitaev, Caltech & KITP, A simple model of quantum holography (part 1)". http://online.kitp.ucsb.edu/online/entangled15/kitaev/.
- ↑ "Alexei Kitaev, Caltech, A simple model of quantum holography (part 2)". http://online.kitp.ucsb.edu/online/entangled15/kitaev2/.
- ↑ Chew, Aaron; Essin, Andrew; Alicea, Jason (2017-09-29). "Approximating the Sachdev-Ye-Kitaev model with Majorana wires". Phys. Rev. B 96 (12): 121119. doi:10.1103/PhysRevB.96.121119. Bibcode: 2017PhRvB..96l1119C.
- ↑ 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. doi:10.1103/PhysRevLett.121.036403. PMID 30085787. Bibcode: 2018PhRvL.121c6403C.
- ↑ Wei, Chenan; Sedrakyan, Tigran (2021-01-29). "Optical lattice platform for the Sachdev-Ye-Kitaev model". Phys. Rev. A 103 (1): 013323. doi:10.1103/PhysRevA.103.013323. Bibcode: 2021PhRvA.103a3323W.
- ↑ 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. doi:10.1103/PhysRevLett.119.040501. PMID 29341740. Bibcode: 2017PhRvL.119d0501G.
- ↑ 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. doi:10.1038/s41534-019-0166-7. Bibcode: 2019npjQI...5...53L.
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