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

TL;DR

This paper explores a Sachdev-Ye-Kitaev model with complex couplings, revealing a transition from Euclidean to Lorentzian wormholes, symmetry breaking, and a phase change to black holes, with implications for quantum chaos.

Contribution

It demonstrates a gravitational symmetry restoration in the SYK model and connects Euclidean and Lorentzian wormhole phases through a phase transition.

Findings

Energy spectrum becomes real with strong coupling despite non-Hermitian Hamiltonian.

Identifies an order parameter for symmetry breaking and matches Green's function patterns.

Observes a thermal phase transition from wormhole to black holes and quantum chaotic dynamics.

Abstract

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.