In this work, we investigate many-body phase transitions in a one-dimensional anisotropic XY model subject to a complex-valued transverse field. Within the biorthogonal framework, we calculate the ground-state correlation functions and entanglement entropy, confirming that their scaling behavior remains identical to that in the Hermitian XY model. The preservation of Hermitian phase transition features in the non-Hermitian setting is rooted in the persistence and emergence of symmetries and their breaking. Specifically, the ferromagnetic (FM) phase arises from the breaking of a $Z_2$ symmetry, while the Luttinger liquid (LL) phase is enabled by the emergence of a $U(1)$ symmetry together with the degeneracy of the real part of the energy spectrum. The nontrivial topology of the LL phase are characterized by the winding number around the exceptional point (EP). Given that non-Hermitian systems are inherently open, this research opens a new avenue for exploring conventional quantum phase transitions that are typically vulnerable to decoherence and environmental disruption in open quantum systems.
In this work, we investigate many-body phase transitions in a one-dimensional anisotropic XY model subject to a complex-valued transverse field. Within the biorthogonal framework, we calculate the ground-state correlation functions and entanglement entropy, confirming that their scaling behavior remains identical to that in the Hermitian XY model. The preservation of Hermitian phase transition features in the non-Hermitian setting is rooted in the persistence and emergence of symmetries and their breaking. Specifically, the ferromagnetic (FM) phase arises from the breaking of a Z2 symmetry, while the Luttinger liquid (LL) phase is enabled by the emergence of a U(1) symmetry together with the degeneracy of the real part of the energy spectrum. The nontrivial topology of the LL phase are characterized by the winding number around the exceptional point (EP). Given that non-Hermitian systems are inherently open, this research opens a new avenue for exploring conventional quantum phase transitions that are typically vulnerable to decoherence and environmental disruption in open quantum systems.
We study many-body phase transitions in a one-dimensional ferromagnetic transversed field Ising model with an imaginary field, and show that the system exhibits three phase transitions: one second-order phase transition and two PT phase transitions. The second-order phase transition occurring in the ground state is investigated via biorthogonal and self-normal entanglement entropy, for which we develop an approach to perform finite-size scaling theory to extract the central charge for small systems. Compared with the second-order phase transition, the first PT transition is characterized by the appearance of an exceptional point in the full energy spectrum, while the second PT transition only occurs in specific excited states. Furthermore, we interestingly show that both exceptional points are second-order in terms of scalings of imaginary parts of the energy. This work provides an exact solution for many-body phase transitions in non-Hermitian systems.
The study of quantum criticality has historically been confined to Hermitian systems, where closed-system approximations neglect environmental interactions. However, a significant portion of quantum phenomena occurs in open systems, necessitating a departure from this conventional framework. This paper explores non-Hermitian quantum criticality in open many-body systems, delving into the unique phases and phase transitions that emerge due to dissipation, gain, and other non-conservative interactions. We establish a comprehensive framework that integrates concepts from biorthogonal quantum mechanics, Liouvillian dynamics, and tensor network methods to characterize critical phenomena, including dynamical quantum phase transitions and the emergence of non-Hermitian quantum many-body scars. Through an extensive literature review, we highlight how the interplay of non-Hermiticity with many-body correlations leads to novel critical behaviors, distinct from their Hermitian counterparts, and in some cases, surprisingly resilient or even enhanced. We discuss the role of exceptional points as hallmarks of non-Hermitian systems, their influence on energy spectra, and their implications for criticality and topological invariants. Furthermore, we examine the methodologies for defining observables and statistical ensembles in these non-Hermitian settings, emphasizing the distinctions and occasional controversies that arise. The findings provide critical insights into the fundamental principles governing quantum criticality in dissipative environments and pave new avenues for engineering and understanding open quantum technologies, from quantum sensing to robust quantum information processing.
Lee-Yang theory offers a unifying framework for understanding classical phase transitions and dynamical quantum phase transitions through the analysis of partition functions and Loschmidt echoes. Recently, this framework is extended to characterize quantum phase transitions of quantum Ising models by introducing the concepts of non-Hermitian parity-symmetry breaking and fidelity zeros. Here, we generalize the theory by studying a broad class of quantum models, including the XY, the XXZ, the XYZ, and the Zp clock models in one dimension, subject to a complex magnetic field. For the XY, XXZ and XYZ models, we find that the complex field breaks parity symmetry and induces oscillations of the ground state between the two parity sectors, giving rise to fidelity zeros within the ordered phases. For the Z3 clock model, the complex field splits the real part of the ground-state energy between the neutral sector (q=0) and the charged sectors (q=1,2), while preserving the degeneracy within the charged sector. Fidelity zeros arise only after projecting out one of the charged sectors. For the Z4 and Z5 clock models, the ground states are instead projected to oscillate between the neutral sector (q=0) and the charged sectors q=2 and q=1, respectively, giving rise to fidelity zeros. Finite-size scaling of these zeros yields critical exponents in full agreement with analytical predictions, demonstrating that this approach is applicable not only to the Ising model with Z2 symmetry, but also to more general Heisenberg-type models and systems with higher discrete symmetries.
Non-Hermitian spin systems have attracted extensive interest due to their unconventional magnetic properties, rich entanglement resources, and unusual quantum criticality phenomena. In this paper, based on the exact solution of a one-dimensional non-Hermitian spin-1/2 XY model with rotation-time-reversal (RT) symmetry [X. Z. Zhang et al., Phys. Rev. A 87, 012114 (2013)], we study the ground-state energy density, magnetization, correlation functions, quantum entanglement, and correlation in RT-symmetric and RT-broken phases as well as their characteristics at the exceptional point. We find that the energy density rises faster with the non-Hermitian parameter $|\ensuremath{\gamma}|$ in the symmetric region than in the broken one and is elevated rapidly at the exceptional point, and the decay of magnetization has similar results. We analyze the effect of $\ensuremath{\gamma}$ on the system and find that the energy density decreases linearly with the external magnetic field h for $\ensuremath{\gamma}>0,$ while it is bifurcated when $\ensuremath{\gamma}=0$. In addition, the behaviors of the energy density indicate that the phase transition caused by symmetry breaking is second order, which is further demonstrated by the magnetization, quantum entanglement, etc. The numerical results of the correlation functions surprisingly indicate that the RT-broken phase has quasi-long-range order, which is quite different from the Hermitian XY model. Especially, the crossover behavior of the ground-state entanglement shows that it increases with $\ensuremath{\gamma}$ in the symmetric region, which is opposite to the case of the broken one, and its maximum always appears at the exceptional point. The above behaviors at the phase boundary are actually ascribed to the fierce competition between $\ensuremath{\gamma}$ and h, which results in a rapid decline of magnetization and the appearance of the maximum of entanglement.
model with a complex transverse field, where the imaginary transverse field can be generated by three-level atoms with spontaneous decay. Many-body spectrum can be obtained analytically. The phase diagram in thermodynamical limit indicates that the imaginary transverse field induces non-Hermitian degeneracy and shrinks the ferromagnetic phase for anisotropic interaction. We also analyze the effect of imaginary field on quantum criticality through ground state geometry phase. It causes a shift of transition point but does not change the order of phase transition in anisotropic interaction region. However, the phase transition is related to the original Hermitian degeneracy in anisotropic interaction region, which indicates that the imaginary transverse field cannot change phase boundary and critical behavior.
In this manuscript, we study the non-Hermitian spin-1/2 XY model in the presence of the alternating, imaginary and transverse magnetic fields. For the two-site spin system, we solve exactly the energy spectrum and phase diagram, also calculate the ground-state and thermal entanglements by using the concept of the concurrence. It is found that the two-site concurrence in the eigenstate which only depends on the imaginary magnetic field η is always equal to one in the region of PT symmetry, while it decreases with η in the PT-symmetric broken region. Especially, the concurrence shows the non-analytic behavior at the exceptional point, and the same is true in the case of the biorthogonal basis, which indicates that the concurrence can characterize the phase transition in this non-Hermitian system. The interesting thing is that η weakens the thermal entanglement when the system is isotropic and enhances the entanglement when the system becomes the Ising model. For the one-dimensional spin chain, the magnetization and entanglement are further studied by using the two-spin cluster mean-field approximation. The results show that their variations have opposite trends with the magnetic fields. Moreover, the system exists the first-order quantum phase transitions for some anisotropic parameters in the PT-symmetry region, and the entanglement changes suddenly at the quantum phase transition point.