  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.
