  It has been recently proposed by Maldacena and Qi that an eternal traversable
wormhole in a two dimensional Anti de Sitter space (${\rm AdS}_2$) is the
gravity dual of the low temperature limit of two Sachdev-Ye-Kitaev (SYK) models
coupled by a relevant interaction (which we will refer to as spin operator). In
this paper, we study spectral and eigenstate properties of this coupled SYK
model. We have found that level statistics in the tail of the spectrum, and for
a sufficiently weak coupling, shows substantial deviations from random matrix
theory which suggests that traversable wormholes are not quantum chaotic. By
contrast, for sufficiently strong coupling, corresponding to the black hole
phase, level statistics are well described by random matrix theory. This
transition in level statistics coincides approximately with a previously
reported Hawking-Page transition for weak coupling. We have shown explicitly
that this thermodynamic transition turns into a sharp crossover as the coupling
increases. Likewise, this critical coupling also corresponds with the one at
which the overlap between the ground state and the thermofield double state
(TFD) is smallest. In the range of sizes we can reach by exact diagonalization,
the ground state is well approximated by the TFD state only in the strong
coupling limit. This is due to the fact that the ground state is close to the
eigenstate of the spin operator corresponding to the lowest eigenvalue which is
an exact TFD state at infinite temperature. In this region, the spectral
density is separated into blobs centered around the eigenvalues of the spin
operator. For weaker couplings, the exponential decay of coefficients in a
tensor product basis, typical of the TFD, becomes power law. Finally, we have
also found that the total Hamiltonian has an additional discrete symmetry which
has not been reported previously.
