Full counting statistics across the entanglement phase transition of non-Hermitian Hamiltonians with charge conservation

Performing quantum measurements produces not only the expectation value of a physical observable O but also the probability distribution P (o ) of all possible outcomes o . The full counting statistics (FCS) Z (ϕ ,O ) ≡∑_oe^{i ϕ o}P (o ) , a Fourier transform of this distribution, contains the complete information of the measurement outcome. In this work, we study the FCS of Q_A, the charge operator in subsystem A , for one-dimensional systems described by non-Hermitian Sachdev-Ye-Kitaev-like models, which are solvable in the large-N limit. In both the volume-law entangled phase for interacting systems and the critical phase for noninteracting systems, the conformal symmetry emerges, which gives F (ϕ ,Q_A) ≡lnZ (ϕ ,Q_A) ∼ϕ²ln|A | . In short-range entangled phases, the FCS shows area-law behavior which can be approximated as F (ϕ ,Q_A) ∼(1 −cosϕ ) |∂ A | for ζ ≫J , 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 consequences of discrete symmetry, long-range hopping, and generalizations to higher dimensions.