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We show that the low temperature phase of a conjugate pair of uncoupled,quantum chaotic, nonhermitian systems such as the Sachdev-Ye-Kitaev (SYK) modelor the Ginibre ensemble of random matrices are dominated by replica symmetrybreaking configurations with a nearly flat free energy that terminates in afirst order phase transition. In the case of the SYK model, we show explicitlythat the spectrum of the effective replica theory has a gap. These features arestrikingly similar to those induced by wormholes in the gravity path integralwhich suggests a close relation between both configurations. For a non-chaoticSYK, the results are qualitatively different: the spectrum is gapless in thelow temperature phase and there is an infinite number of second order phasetransitions unrelated to the restoration of replica symmetry.

Replica Symmetry Breaking and Phase Transitions in a PT Symmetric Sachdev-Ye-Kitaev Model