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We study a simplified version of the Sachdev-Ye-Kitaev (SYK) model with real interactions by exact diagonalization. Instead of satisfying a continuous Gaussian distribution, the interaction strengths are assumed to be chosen from discrete values with a finite separation. A quantum phase transition from a chaotic state to an integrable state is observed by increasing the discrete separation. Below the critical value, the discrete model can well reproduce various physical quantities of the original SYK model, including the volume law of the ground-state entanglement, level distribution, thermodynamic entropy, and out-of-time-order correlation (OTOC) functions. For systems of size up to N = 20, we find that the transition point increases with system size, indicating that a relatively weak randomness of interaction can stabilize the chaotic phase. Our findings significantly relax the stringent conditions for the realization of SYK model, and can reduce the complexity of various experimental proposals down to realistic ranges.

Performing quantum measurements produces not only the expectation value of a physical observable O but also the probability distribution of all possible outcomes. The full counting statistics (FCS) Z(φ, O), a Fourier transform of this distribution, contains the complete information of the measurement. In this work, we study the FCS of Q A , the charge operator in subsystem A, for 1D systems described by non-Hermitian SYK models, which are solvable in the large-N limit. In both the volume-law entangled phase for interacting systems and the critical phase for non-interacting systems, the conformal symmetry emerges, which gives F(φ, Q A ) ≡ log Z(φ, Q A ) ∼ φ 2 log |A|. In short-range entangled phases, the FCS shows area-law behavior F(φ, Q A ) ∼ (1 − cos φ)|∂A|, regardless of the presence of interactions. Our results suggest the FCS is a universal probe of entanglement phase transitions in non-Hermitian systems with conserved charges, which does not require the introduction of multiple replicas. We also discuss the consequence of discrete symmetry, long-range hopping, and generalizations to higher dimensions.