Dominance of Replica Off-Diagonal Configurations and Phase Transitions in a PT Symmetric Sachdev-Ye-Kitaev Model

We show that, after ensemble averaging, the low temperature phase of a conjugate pair of uncoupled, quantum chaotic, non-Hermitian systems such as the Sachdev-Ye-Kitaev (SYK) model or the Ginibre ensemble of random matrices is dominated by saddle points that couple replicas and conjugate replicas. This results in a nearly flat free energy that terminates in a first-order phase transition. In the case of the SYK model, we show explicitly that the spectrum of the effective replica theory has a gap. These features are strikingly similar to those induced by wormholes in the gravity path integral which suggests a close relation between both configurations. For a nonchaotic SYK, the results are qualitatively different: the spectrum is gapless in the low temperature phase and there is an infinite number of second order phase transitions unrelated to the restoration of replica symmetry.

Automated summary

We demonstrate that the low temperature phase of uncoupled, quantum chaotic, non-Hermitian systems such as the SYK model or the Ginibre ensemble of random matrices is governed by saddle points that couple replicas and conjugate replicas after ensemble averaging. This leads to an almost flat free energy that terminates in a first-order phase transition. In the case of the SYK model, we explicitly show that the spectrum of the effective replica theory has a gap. These features resemble those induced by wormholes in the gravity path integral, suggesting a strong connection between these configurations. For a nonchaotic SYK model, the results differ qualitatively with a gapless spectrum in the low temperature phase and an infinite number of second order phase transitions unrelated to replica symmetry restoration.