On SYK traversable wormhole with imperfectly correlated disorders

In this paper, we study the phase structure of two Sachdev-Ye-Kitaev models (L-system and R-system) coupled by a simple interaction, with imperfectly correlated disorder. When the disorder of the two systems is perfectly correlated, J( i1)((L))...(iq) = J((R)) (i1)..(.iq), this model is known to exhibit a phase transition at a finite temperature between the two-black hole phase at high temperature and the traversable wormhole phase at low temperature. We find that, as the correlation (SIC)J((L)) (i1)...(iq) J(i1)((R))...(iq)) is decreased, the critical temperature becomes lower. At the same time, the transmission between the L-system and R-system in the low-temperature phase becomes more suppressed, while the chaos exponent of the whole system becomes larger. Interestingly we also observe that when the correlation is smaller than some q-dependent critical value the phase transition completely disappears in the entire parameter space. At zero temperature, the energy gap becomes larger as we decrease the correlation. We also use a generalized thermofield double state as a variational state. Interestingly, this state coincides with the ground state in the large q limit.