Commuting SYK: a pseudo-holographic model

Abstract

In this work, we study a type of commuting SYK model in which all terms in the Hamiltonian are commutative to each other. Because of the commutativity, this model has a large number of conserved charges and is integrable. After the ensemble average of random couplings, we can solve this model exactly in any $N$. Though this integral model is not holographic, we do find that it has some holography-like features, especially the near-perfect size winding in high temperatures. Therefore, we would like to call it pseudo-holographic. We also find that the size winding of this model has a narrowly peaked size distribution, which is different from the ordinary SYK model. We apply the traversable wormhole teleportation protocol in the commuting SYK model and find that the teleportation has a few features similar to the semiclassical traversable wormhole but in different parameter regimes. We show that the underlying physics is not entirely determined by the size-winding mechanism but involves the peaked-size mechanism and thermalization. Lastly, we comment on the recent simulation of the dynamics of traversable wormholes on Google's quantum processor.

Figures (8)

• Figure 1: The comparison between the simplified saddle approximation and exact results with $N=100$ and for high temperature $\beta=1/3$ (a,c,e) and intermediate temperature $\beta=0.95<\beta_c$ (b,d,f). In the $P_{2n+1},|Q_{2n+1}|$ plots, the blue dots are exact values of $P_{2n+1}$ and the red dots are exact values of $|Q_{2n+1}|$; the yellow joint lines are saddle approximation of $P_{2n+1}$ and the green joint lines are saddle approximation of $|Q_{2n+1}|$. In the $\arg Q_{2n+1}$ plots, the blue dots are exact values of $\arg Q_{2n+1}$ and the yellow joint lines are saddle approximation of $\arg Q_{2n+1}$. In (b) where the simplified saddle approximation does not work very well, we plot the saddle approximation using (\ref{['eq:x0']}). The purple joint line is $P_{2n+1},$the gray joint line is $|Q_{2n+1}|$ in the left picture, and the green joint line is $\arg Q_{2n+1}$ in the right picture. For $n$ not too big, the saddle approximation is improved, but for large $n$ it loses accuracy and we need to improve the saddle location $x_{0}$ further in (\ref{['eq:x0']}).
• Figure 2: The exact results with $N=8$ and for high temperature $\beta=0.1$ (a,c,e) and low temperature $\beta=1$ (b,d,f). In the $P_{2n+1},|Q_{2n+1}|$ plots, the joint blue dots are exact values of $P_{2n+1}$ and the joint yellow dots are exact values of $|Q_{2n+1}|$. These two series of joint dots are plotted with different thicknesses for clear comparison. In the $\arg Q_{2n+1}$ plots, the joint blue dots are exact values of $\arg Q_{2n+1}$.
• Figure 3: The traversable wormhole teleportation protocol (copied from Gao:2019nyj)
• Figure 4: The optimized $\max|\Im H_{i\mu}|-\max|\Im H_{-i\mu}|$ for the scale $t_{l}\sim-t_{r}\sim O(1)$. The blue curves are $-\Im H_{i\mu}$ and the yellow curves are $\Im H_{-i\mu}$. (a) Lower temperature $\beta=0.95$, the optimal $\mu=0.0468\pi$ and the injection time is at $t_{r}=-1.791$. (d) High temperature $\beta=0.1$, the optimal $\mu=0.0509\pi$ and the injection time is at $t_{r}=-1.507$.
• Figure 5: The optimized $\max|\Im H_{i\mu}|-\max|\Im H_{-i\mu}|$ for $N=8$. The blue curves are $-\Im H_{i\mu}$ and the yellow curves are $\Im H_{-i\mu}$. The dashed lines are extended plots for $t_{l}<0$, which is an unphysical regime. (a) Even-lower temperature $\beta=4$, the optimal $\mu=0.221\pi$ and the injection time is at $t_{r}=-0.474$. (b) Low temperature $\beta=2.5$, the optimal $\mu=0.195\pi$ and the injection time is at $t_{r}=-0.614$. (c) Intermediate temperature $\beta=1$, the optimal $\mu=0.139\pi$ and the injection time is at $t_{r}=-0.720$. (d) High temperature $\beta=0.1$, the optimal $\mu=0.131\pi$ and the injection time is at $t_{r}=-0.742$.