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,{{Cite journal|last1=Sachdev|first1=Subir|last2=Ye|first2=Jinwu|date=1993-05-24|title=Gapless spin-fluid ground state in a random quantum Heisenberg magnet|journal=Physical Review Letters|volume=70|issue=21|pages=3339–3342|doi=10.1103/PhysRevLett.70.3339|pmid=10053843|arxiv=cond-mat/9212030|bibcode=1993PhRvL..70.3339S|s2cid=1103248 }} and later modified by Alexei Kitaev to the present commonly used form.{{Cite web|url=http://online.kitp.ucsb.edu/online/entangled15/kitaev/|title=Alexei Kitaev, Caltech & KITP, A simple model of quantum holography (part 1)|website=online.kitp.ucsb.edu|access-date=2019-11-02}}{{Cite web|url=http://online.kitp.ucsb.edu/online/entangled15/kitaev2/|title=Alexei Kitaev, Caltech, A simple model of quantum holography (part 2)|website=online.kitp.ucsb.edu|access-date=2019-11-02}} 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,{{Cite journal|last1=Chew|first1=Aaron|last2=Essin|first2=Andrew|last3=Alicea|first3=Jason|date=2017-09-29|title=Approximating the Sachdev-Ye-Kitaev model with Majorana wires|journal=Phys. Rev. B|volume=96|issue=12|pages=121119|doi=10.1103/PhysRevB.96.121119|arxiv=1703.06890 |bibcode=2017PhRvB..96l1119C |s2cid=119222270 }} graphene flake with irregular boundary,{{Cite journal|last1=Chen|first1=Anffany|last2=Ilan|first2=R.|last3=Juan|first3=F.|last4=Pikulin|first4=D.I.|last5=Franz|first5=M.|date=2018-06-18|title=Quantum Holography in a Graphene Flake with an Irregular Boundary|journal=Phys. Rev. Lett.|volume=121|issue=3|pages=036403|doi=10.1103/PhysRevLett.121.036403|pmid=30085787 |arxiv=1802.00802|bibcode=2018PhRvL.121c6403C |s2cid=51940526 }} and kagome optical lattice with impurities,{{Cite journal|last1=Wei|first1=Chenan|last2=Sedrakyan|first2=Tigran|date=2021-01-29|title=Optical lattice platform for the Sachdev-Ye-Kitaev model|journal=Phys. Rev. A|volume=103|issue=1|pages=013323|doi=10.1103/PhysRevA.103.013323|arxiv=2005.07640|bibcode=2021PhRvA.103a3323W|s2cid=234363891 }} are proposed to be modeled by it. Some variants of the model are amenable to digital quantum simulation,{{Cite journal|last1=García-Álvarez|first1=L.|last2=Egusquiza|first2=I.L.|last3=Lamata|first3=L.|last4=del Campo|first4=A.|last5=Sonner|first5=J.|last6=Solano|first6=E.|date=2017|title=Digital Quantum Simulation of Minimal AdS/CFT|journal=Physical Review Letters|volume=119|issue=4 |pages=040501|doi= 10.1103/PhysRevLett.119.040501|pmid=29341740 |arxiv=1607.08560|bibcode=2017PhRvL.119d0501G |s2cid=5144368 }} with pioneering experiments implemented in nuclear magnetic resonance.{{Cite journal|last1=Luo|first1=Z.|last2=You|first2=Y.-Z.|last3=Li|first3=J.|last4=Jian|first4=C.-M.|last5=Lu|first5=D.|last6=Xu|first6=C.|last7=Zeng|first7=B.|last8=Laflamme |first8=R.|date=2019|title=Quantum simulation of the non-fermi-liquid state of Sachdev-Ye-Kitaev model|journal=npj Quantum Information |volume=5|pages=53|doi= 10.1038/s41534-019-0166-7|arxiv=1712.06458|bibcode=2019npjQI...5...53L|s2cid=195344916 }}

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 $\psi_1,\dotsc,\psi_n$ which are fermion operators satisfying conditions:

Hermitian $\psi_i^{\dagger}=\psi_i$ ;
Clifford relation $\{\psi_i,\psi_j\}=2\delta_{ij}$ .

Let $J_{i_1 i_2 \cdots i_m}$ be random variables whose expectations satisfy:

$\mathbf{E}(J_{i_1i_2\cdots i_m})=0$ ;
$\mathbf{E}(J_{i_1i_2\cdots i_m}^2)=1$ .

Then the SYK model is defined as

: $H_{\rm SYK}=i^{m/2}\sum_{1 \leq i_1 < \cdots < i_m \leq n}J_{i_1i_2\cdots i_m}\psi_{i_1}\psi_{i_2}\cdots\psi_{i_m}$ .

Note that sometimes an extra normalization factor is included.

The most famous model is when $m=4$ :

: $H_{\rm SYK}=-\frac{1}{4!}\sum_{i_1, \dotsc, i_4 = 1}^n J_{i_1i_2i_3 i_4}\psi_{i_1}\psi_{i_2}\psi_{i_3}\psi_{i_4}$ ,

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

References

Category:Lattice models

Sachdev–Ye–Kitaev model

Model

See also

References