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, Ji1⋯iqL=Ji1⋯iqR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {J}_{i_1\cdots {i}_q}^{(L)}={J}_{i_1\cdots {i}_q}^{(R)} $$\end{document}, 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 Ji1⋯iqL=Ji1⋯iqR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\langle {J}_{i_1\cdots {i}_q}^{(L)}={J}_{i_1\cdots {i}_q}^{(R)}\right\rangle $$\end{document} 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.