Performing quantum measurements produces not only the expectation value of a physical observable $O$ but also the probability distribution $P\left(o\right)$ of all possible outcomes $o$. The full counting statistics (FCS) $Z(\varphi ,O)\equiv {\sum }_o{e}^{i\varphi o}P\left(o\right)$, 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 $F(\varphi ,Q_A)\equiv lnZ(\varphi ,Q_A)\sim {\varphi }^{2}ln\left|A\right|$. In short-range entangled phases, the FCS shows area-law behavior which can be approximated as $F(\varphi ,Q_A)\sim (1-cos\varphi )\left|\partial A\right|$ for $\zeta \gg J$, 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.

• Received 1 March 2023
• Accepted 12 September 2023

DOI:https://doi.org/10.1103/PhysRevB.108.094308