Two-local modifications of SYK model with quantum chaos

Abstract

The Sachdev--Ye--Kitaev (SYK) model may provide us with a good starting point for the experimental study of quantum chaos and holography in the laboratory. Still, the four-local interaction of fermions makes quantum simulation challenging, and it would be good to search for simpler models that keep the essence. In this paper, we argue that the four-local interaction may not be important by introducing a few models that have two-local interactions. The first model is a generalization of the spin-SYK model, which is obtained by replacing the spin variables with SU($d$) generators. Simulations of this class of models might be straightforward on qudit-based quantum devices. We study the case of $d=3, 4, 5, 6$ numerically and observe quantum chaos already for two-local interactions in a wide energy range. We also introduce modifications of spin-SYK and SYK models that have similar structures to the SU($d$) model (e.g., $H=\sum_{p,q}J_{pq}χ_pχ_{p+1}χ_qχ_{q+1}$ instead of the original SYK Hamiltonian $H=\sum_{p,q,r,s}J_{pqrs}χ_pχ_qχ_rχ_{s}$), which shows strongly chaotic features although the interaction is essentially two-local. These models may be a good starting point for the quantum simulation of the original SYK model.

Figures (8)

• Figure 1: Plots of the density of states $\rho(E)$ for $d>2$ qudit SYK model \ref{['Hamiltonian:Qudit_SYK']} with the coupling constant normalization \ref{['eqn:SUqSpin-normalization']}. The $q=2$ model shows soft edges, while $q=3$ shows hard edges. The number of samples is $2^5\times 3^{12-L}$ for $d=3$ (for both $q=2$ and $q=3$), $5\times 4^{11-L}$ for $d=4$, $2^8\times 5^{7-L}$ for $d=5$, $2\times 6^{9-L}$ for $d=6$, so that the number of eigenvalues is at least $1.7\times10^7$.
• Figure 2: Nearest-Neighbor Level Spacings for qudit SYK model \ref{['Hamiltonian:Qudit_SYK']} for $d=3, 4, 5, 6$ and $q=2$, and for $d=3$ and $q=3$. Multiple values of sites $L$ are plotted in the same figure. The distribution converges to that of GUE random matrix as $L$ increases. The GUE result is from Dietz:1990.
• Figure 3: Neighboring gap ratio for the fixed-$i$-unfolded spectrum for the first $30$ neighboring gap pairs for $d=3$ qudit SYK model \ref{['Hamiltonian:Qudit_SYK']} (left) and the distribution of the gap ratio over the entire energy spectrum (right). [Top] $q=2$. [Bottom] $q=3$.
• Figure 4: $|Z(t)|^2/|Z(0)|^2$ for various $L$ for $d=3$ qudit SYK model \ref{['Hamiltonian:Qudit_SYK']}. Left: $q=2$. Right: $q=3$. The curves are in consecutive order with regards to $L$.
• Figure 5: Density of states in $q=2$, $M=2$ overlapping clusters SYK model \ref{['Hamiltonian:q=2_M=2_overlapping_clusters_SYK']}. For $6\leq N\leq 34$, $2^{24-N/2}$ samples are used. For $N=36$, $11$ samples are used.