Sachdev–Ye–Kitaev model

<h2 id="model">Model</h2>
<p>Let 
  
    
      
        n
      
    
    {\displaystyle n}
  
 be an integer and 
  
    
      
        m
      
    
    {\displaystyle m}
  
 an even integer such that 
  
    
      
        2
        ≤
        m
        ≤
        n
      
    
    {\displaystyle 2\leq m\leq n}
  
, and consider a set of <a href="/facts/Majorana_fermion/39gzL0Nv">Majorana fermions</a> 
  
    
      
        
          ψ
          
            1
          
        
        ,
        …
        ,
        
          ψ
          
            n
          
        
      
    
    {\displaystyle \psi _{1},\dotsc ,\psi _{n}}
  
 which are fermion operators satisfying conditions:
</p>
<ol><li>Hermitian 
  
    
      
        
          ψ
          
            i
          
          
            †
          
        
        =
        
          ψ
          
            i
          
        
      
    
    {\displaystyle \psi _{i}^{\dagger }=\psi _{i}}
  
;</li>
<li>Clifford relation 
  
    
      
        {
        
          ψ
          
            i
          
        
        ,
        
          ψ
          
            j
          
        
        }
        =
        2
        
          δ
          
            i
            j
          
        
      
    
    {\displaystyle \{\psi _{i},\psi _{j}\}=2\delta _{ij}}
  
.</li></ol>
<p>Let 
  
    
      
        
          J
          
            
              i
              
                1
              
            
            
              i
              
                2
              
            
            ⋯
            
              i
              
                m
              
            
          
        
      
    
    {\displaystyle J_{i_{1}i_{2}\cdots i_{m}}}
  
 be random variables whose expectations satisfy:
</p>
<ol><li>
  
    
      
        
          E
        
        (
        
          J
          
            
              i
              
                1
              
            
            
              i
              
                2
              
            
            ⋯
            
              i
              
                m
              
            
          
        
        )
        =
        0
      
    
    {\displaystyle \mathbf {E} (J_{i_{1}i_{2}\cdots i_{m}})=0}
  
;</li>
<li>
  
    
      
        
          E
        
        (
        
          J
          
            
              i
              
                1
              
            
            
              i
              
                2
              
            
            ⋯
            
              i
              
                m
              
            
          
          
            2
          
        
        )
        =
        1
      
    
    {\displaystyle \mathbf {E} (J_{i_{1}i_{2}\cdots i_{m}}^{2})=1}
  
.</li></ol>
<p>Then the SYK model is defined as
</p>

H
          
            
              S
              Y
              K
            
          
        
        =
        
          i
          
            m
            
              /
            
            2
          
        
        
          ∑
          
            1
            ≤
            
              i
              
                1
              
            
            <
            ⋯
            <
            
              i
              
                m
              
            
            ≤
            n
          
        
        
          J
          
            
              i
              
                1
              
            
            
              i
              
                2
              
            
            ⋯
            
              i
              
                m
              
            
          
        
        
          ψ
          
            
              i
              
                1
              
            
          
        
        
          ψ
          
            
              i
              
                2
              
            
          
        
        ⋯
        
          ψ
          
            
              i
              
                m
              
            
          
        
      
    
    {\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}}}
  
.
<p>Note that sometimes an extra normalization factor is included.
</p><p>The most famous model is when 
  
    
      
        m
        =
        4
      
    
    {\displaystyle m=4}
  
:
</p>

H
          
            
              S
              Y
              K
            
          
        
        =
        −
        
          
            1
            
              4
              !
            
          
        
        
          ∑
          
            
              i
              
                1
              
            
            ,
            …
            ,
            
              i
              
                4
              
            
            =
            1
          
          
            n
          
        
        
          J
          
            
              i
              
                1
              
            
            
              i
              
                2
              
            
            
              i
              
                3
              
            
            
              i
              
                4
              
            
          
        
        
          ψ
          
            
              i
              
                1
              
            
          
        
        
          ψ
          
            
              i
              
                2
              
            
          
        
        
          ψ
          
            
              i
              
                3
              
            
          
        
        
          ψ
          
            
              i
              
                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}}}
  
,
<p>where the factor 
  
    
      
        1
        
          /
        
        4
        !
      
    
    {\displaystyle 1/4!}
  
 is included to coincide with the most popular form.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Fermi_liquid_theory/BubT8nPx">Non-Fermi liquid</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>"Alexei Kitaev, Caltech & KITP, A simple model of quantum holography (part 1)". online.kitp.ucsb.edu. Retrieved 2019-11-02. <a href="http://online.kitp.ucsb.edu/online/entangled15/kitaev/" target="_blank">http://online.kitp.ucsb.edu/online/entangled15/kitaev/</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>"Alexei Kitaev, Caltech, A simple model of quantum holography (part 2)". online.kitp.ucsb.edu. Retrieved 2019-11-02. <a href="http://online.kitp.ucsb.edu/online/entangled15/kitaev2/" target="_blank">http://online.kitp.ucsb.edu/online/entangled15/kitaev2/</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
</ol>

Sachdev–Ye–Kitaev model open-in-new

Sachdev–Ye–Kitaev model