Euclidean-to-Lorentzian wormhole transition and gravitational symmetry breaking in the Sachdev-Ye-Kitaev model

We study a two-site Sachdev-Ye-Kitaev model with complex couplings and a weak inter-site interaction. At low temperatures, the system is dual to a Euclidean wormhole in Jackiw-Teitelboim gravity plus matter. Interestingly, the energy spectrum becomes real for sufficiently strong inter-site coupling despite the Hamiltonian being non-Hermitian. In gravity, this complex-to-real transition corresponds to a Euclidean-to-Lorentzian transition: a dynamical restoration of the gravitational SL(2,R) symmetry of the Lorentzian wormhole, broken to U(1) in the Euclidean wormhole. We show this by identifying an order parameter for the symmetry breaking and by matching the oscillating patterns of the Green's functions. Above the transition, the system can be continued to Lorentzian signature and is dual to an eternal traversable wormhole. Additionally, we observe a thermal phase transition from the wormhole to two black holes and provide a detailed matching of the associated physical quantities. The analysis of level statistics reveals that in a broad range of parameters the dynamics is quantum chaotic in the universality class of systems with time reversal invariance.