Entanglement Phase Transition in Chaotic non-Hermitian Systems

TL;DR

This work examines entanglement phase transitions in chaotic non-Hermitian spin chains where the non-Hermitian term commutes with the spin-spin coupling, focusing on a non-Hermitian transverse-field Ising model and a non-Hermitian XX model under a transverse field. Using post-selected quantum trajectories and a Faber polynomial-based time-evolution scheme, it reveals a gapless-to-gapped transition at a model-dependent $\gamma_c$, with the complex gap displaying nonmonotonic, oscillatory behavior due to level crossings of the maximal imaginary level. The steady-state entanglement transitions from volume-law to area-law across the transition, while level-crossing events induce abrupt jumps in entanglement, highlighting an unconventional entanglement landscape in commuting non-Hermitian systems. These findings connect to Yang-Lee-type phenomena yet show distinct spectral behavior in the presence of level crossings, providing insight into entanglement dynamics in non-Hermitian many-body physics.

Abstract

We have studied entanglement phase transition in a class of chaotic non-Hermitian spin chain in which its spin-spin coupling term commutes with the non-Hermitian term. Two models are investigated: transverse field Ising model with a complex longitudinal field and non-Hermitian XX model with a transverse field. Through calculating their complex spectra, we find these models are subject to a gapless-gapped phase transition with dissipation rate if the transverse field is larger than a model-dependent value. Interestingly, the variation of the complex gap with the dissipation rate is not monotonous, instead it manifest oscillations before entering the gapped phase. By simulating their non-unitary evolution, we show that the entanglement entropy of the steady state would transition from a volume-law to an area-law scaling with the increase of the dissipation rate. Meanwhile, some unexpected results about the entanglement entropy appear in the volume-law phase. These unusual features of the complex gap and the steady-state entanglement can be attributed to level crossings between the maximal imagine level and other levels. Our work reveals a novel entanglement transition in chaotic non-Hermitian systems.

Figures (5)

• Figure 1: Complex gap and spectra of NHTFI model with respect to $\gamma$. (a) The gap for different $\Omega$ with the chain length N=12. (b) The gap for different N with $\Omega=2$. The imaginary part (c) and real part (d) of eigenenergies of several levels with N=12 and $\Omega=2$.
• Figure 2: Entanglement dynamics of the NHTFI model and entanglement entropy of the steady state in N, with $\Omega=0.9$ in (a)-(b), and $\Omega=2$ in (c)-(d).
• Figure 3: Complex spectral gap with respect to $\gamma$ in NHXX model. The chain length N=10 in (a)-(b), and N=12 in (c)-(d). The values of $\Omega$ are listed in the legend of each figure.
• Figure 4: The imaginary part (a) and real part (b) of the eigenenergies with respect to $\gamma$ for chain length N=10. The parameter $\Omega=2$.
• Figure 5: Complex gap with respect to $\gamma$ for different chain length and the entanglement entropy of the steady state as a function of N. In (a)-(b) the parameter $\Omega=0.9$ while $\Omega=2$ in (c)-(d).