Entanglement complexification transition driven by a single non-Hermitian impurity

TL;DR

The paper analyzes a Hermitian gapless chain decorated by a single non-Hermitian impurity and develops a fully analytical framework to track its entanglement structure. By solving the right/left eigenproblems and constructing biorthogonal states, the authors derive closed forms for the equal-time correlation function $C(l,m)$ and decompose it into a translational piece and an impurity-induced contribution, uncovering an entanglement complexification transition where the logarithmic EE remains, but the effective central charge becomes complex in a new nonunitary defect-CFT regime. An analytic continuation from unitary defect CFT explains the real-$c_{ ext{eff}}$ regime, while a breakdown of this continuation defines the complex regime, for which a dedicated analytical formula matches numerics and is further enriched by bound-state contributions when $|t_{R}t_{L}|>1$. The results establish a rare, exactly solvable boundary-non-Hermitian platform that connects impurity physics, defect-CFT, and timelike entanglement, with potential experimental relevance in open photonic or phononic systems.

Abstract

While non-Hermitian bulk systems and their sensitivity to boundary conditions have been extensively studied, how a non-Hermitian boundary affects the entanglement structure of Hermitian critical systems remains largely unexplored. Here we present a fully analytical framework by exactly solving a Hermitian gapless chain with a single non-Hermitian impurity acting as a non-Hermitian boundary. When the entanglement cut is placed at the impurity, we uncover a sharp \emph{entanglement complexification transition}: the logarithmic entanglement entropy retains its scaling form, but the effective central charge evolves from real to complex values, accompanied by a spectral collapse of the correlation matrix. We demonstrate that the real regime follows analytic continuation from a unitary defect conformal field theory (CFT), whereas the complex regime lies entirely beyond this framework. For the latter, we derive an analytical formula in perfect agreement with numerics. Our results reveal that a single non-Hermitian impurity can drive a Hermitian critical system into a nonunitary defect-CFT phase, establishing a rare analytically solvable platform for boundary non-Hermiticity.

Figures (4)

• Figure 1: (a) Effective central charge $c_{\text{eff}}$ versus impurity parameters $t_{R}$ and $t_{L}$ for Partition-I. In the first quadrant (Q1: $t_{R},t_{L}>0$), $c_{\text{eff}}$ is real; in the second (Q2: $t_{R}<0$, $t_{L}>0$), it becomes complex and we plot $\mathrm{Re}(c_{\text{eff}})$. Along $t_{R}t_{L}=t_{\text{eff}}^{2}$, $c_{\text{eff}}$ stays constant; the dashed line denotes the Hermitian limit with symmetric hopping $t_{\text{eff}}$. (b) Norm of correlation matrix $|C^{A}|$ and real/imaginary parts of EE $S_{A}$ for subsystem $A$ ($L_{A}=100$) as functions of $t_{R}$ at fixed $t_{L}=0.5$ using Partition-I. (c) Model Hamiltonian $\hat{H}_{d}$: the red bond marks the impurity ($t_{R},t_{L}$), black bonds denote uniform Hermitian hopping $t$. Partition-I and Partition-II cut as indicated; $L_{0}$ is the impurity–subsystem distance and $L_{A}$ the subsystem size.
• Figure 2: Scaling behaviors of EE induced by the impurity in the parameter region Q1. (a) and (b) show the logarithmic scaling of the EE with the varying parameter $t_{R}$, using Partition-II and Partition-I, respectively. (c) presents $c_{\text{eff}} - 1/2$ as a function of $t_{R}$ for Partition-I, where the blue points and red line are numerical results and fit function in Eq. \ref{['eq_central_formu']}, respectively. (d) displays the scaling of the EE for two sets of dual parameters, $(t_{R}, t_{L})$ and $\left( 1/t_{R}, 1/t_{L} \right)$, using Partition-I. Here, $t_{L} = 0.2$, $L_{0} = 500$ for Partition-II, and the total size of the system $N=5000$.
• Figure 3: Scaling behaviors of EE induced by the impurity in the parameter region Q2. Both the real and imaginary parts of the EE, $S^{I}_{A}$ in (a) and (b), exhibit logarithmic scaling with Partition-I. As $t_{R}$ decreases, the real part of $c_{\text{eff}}$ decreases, while the imaginary part increases. (c) and (d) show $\Re(c_{\text{eff}})-1/2$ and $\Im(c_{\text{eff}})$ of the phase in Q2 as functions of $t_{R}$ with $L_{A}=2048$ for Partition-I, where the blue points and red line are numerical results and fit function in Eq. \ref{['eq_anoma_central']}, respectively. Here, $t_{L} = 0.5$, and the total size of the system $N=5000$.
• Figure S1: The black line is the integral contour for $l+m$ being odd.