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) $Z(\phi, O)\equiv \sum_o e^{i\phi o}P(o)$, 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 1D systems described by non-Hermitian SYK-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 non-interacting systems, the conformal symmetry emerges, which gives $F(\phi, Q_A)\equiv \log Z(\phi, Q_A)\sim \phi^2\log |A|$. In short-range entangled phases, the FCS shows area-law behavior which can be approximated as $F(\phi, Q_A)\sim (1-\cos\phi) |\partial A|$ 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 consequence of discrete symmetry, long-range hopping, and generalizations to higher dimensions.