Full Counting Statistics across the Entanglement Phase Transition of Non-Hermitian Hamiltonians with Charge Conservations

Abstract

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.