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_i_1⋯i_q^(L)=J_i_1⋯i_q^(R) , 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 ⟨J_i_1⋯i_q^(L)=J_i_1⋯i_q^(R)⟩ 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.