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We study a two-site Sachdev-Ye-Kitaev model with complex couplings and a weakinter-site interaction. At low temperatures, the system is dual to a Euclideanwormhole in Jackiw-Teitelboim gravity plus matter. Interestingly, the energyspectrum becomes real for sufficiently strong inter-site coupling despite theHamiltonian being non-Hermitian. In gravity, this complex-to-real transitioncorresponds to a Euclidean-to-Lorentzian transition: a dynamical restoration ofthe 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 forthe symmetry breaking and by matching the oscillating patterns of the Green'sfunctions. Above the transition, the system can be continued to Lorentziansignature and is dual to an eternal traversable wormhole. Additionally, weobserve a thermal phase transition from the wormhole to two black holes andprovide a detailed matching of the associated physical quantities. The analysisof level statistics reveals that in a broad range of parameters the dynamics isquantum chaotic in the universality class of systems with time reversalinvariance.

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