Real-to-complex spectral transitions mark the onset of amplification in non-Hermitian systems, but their thresholds are often treated as model-specific quantities. Here we develop a general, non-perturbative activated-mode principle that governs the real-to-complex threshold across broad classes of non-Hermitian systems. A central insight is that only a small Hilbert subspace is ``activated" at the transition onset, which can be variationally determined through the competition between spectral detuning and mode-level projected non-Hermitian couplings. The result is a closed-form exceptional-activation condition for arbitrarily large ``disturbances", rather than a perturbative estimate. We apply our framework to three contrasting illustrative problems, establishing (i) a closed-form threshold for critical non-Hermitian skin amplification at \emph{all} system sizes; (ii) a new link between impurity tunneling threshold and exceptional point switching; and (iii) activation channel switching without underlying topological phase transition. Overall, our findings recast real-to-complex transitions as generic mode-selection problems independent of any specific symmetry.
Real-to-complex spectral transitions mark the onset of amplification in non-Hermitian systems, but their thresholds are often treated as model-specific quantities. Here we develop a general, non-perturbative activated-mode principle that governs the real-to-complex threshold across broad classes of non-Hermitian systems. A central insight is that only a small Hilbert subspace is activated" at the transition onset, which can be variationally determined through the competition between spectral detuning and mode-level projected non-Hermitian couplings. The result is a closed-form exceptional-activation condition for arbitrarily large disturbances", rather than a perturbative estimate. We apply our framework to three contrasting illustrative problems, establishing (i) a closed-form threshold for critical non-Hermitian skin amplification at \emph{all} system sizes; (ii) a new link between impurity tunneling threshold and exceptional point switching; and (iii) activation channel switching without underlying topological phase transition. Overall, our findings recast real-to-complex transitions as generic mode-selection problems independent of any specific symmetry.
The conventional time-dependent perturbation theory in quantum mechanics is established within the framework of Hermitian Hamiltonians, applicable for describing quantum transitions and associated energy level responses in such systems. However, this theory has fundamental limitations when applied to non-Hermitian systems. Consequently, researchers have systematically extended time-dependent perturbation theory to non-Hermitian systems, establishing a corresponding mature framework. Building on this foundation, this study extends the theory to investigate the transition dynamics induced by non-Hermitian interactions in non-Hermitian Hamiltonian systems. We employ a biorthogonal basis representation for a three-level non-Hermitian system. This work investigates a system comprising an unperturbed static non-Hermitian Hamiltonian and a periodically driven time-dependent perturbation Hamiltonian. Taking the three-level system as a concrete example, we combine analytical methods with numerical simulations to solve and analyze its dynamical evolution equations. These complementary approaches reveal that when system parameters complete a full cycle around an exceptional point, the transitional behavior exhibits specific evolutionary patterns. In this system, quantum transition probabilities exhibit significant asymmetry and non-conservation that depend on the initial and final states, revealing inherent directional characteristics in the dynamical process. Furthermore, for a three-level, periodically driven non-Hermitian system with time-dependent perturbations, this asymmetry is even more pronounced, manifesting as a distinct disparity between forward and reverse transition probabilities. The periodic driving actively amplifies the asymmetry in the transition process. By designing the perturbation spectrum, selective manipulation of specific quantum states can be achieved. Moreover, transition probabilities can be significantly enhanced under resonance conditions, while non-Hermiticity further breaks the system's inherent symmetry, leading to substantial amplification of transitions in a single direction. By precisely tuning the drive frequency, interactions between specific coupling channels can be selectively enhanced or suppressed. The amplification of channel asymmetry by non-Hermitian properties provides a novel mechanism for directional control of quantum states and opens new pathways for realizing related quantum technologies.
Non-Hermitian systems exhibit unique spectral properties, including the non-Hermitian skin effect and exceptional points, often influenced by boundary conditions. The modulation of these phenomena by generalized boundary conditions remains unexplored and not understood. Here, we analyze the Hatano-Nelson model with generalized boundary conditions induced by complex hopping amplitudes at the boundary. Using similarity transformations, we determine the conditions yielding real energy spectra and skin effect, and identify the emergence of exceptional points where spectra transition from real to complex. We demonstrate that tuning the boundary hopping amplitudes precisely controls the non-Hermitian skin effect, i.e., the localization of eigenmodes at the lattice edges. These findings reveal the sensitivity of spectral and localization properties to boundary conditions, providing a framework for engineering quantum lattice models with tailored spectral and localization features, with potential applications in quantum devices.
A main distinguishing feature of non-Hermitian quantum mechanics is the presence of exceptional points (EPs). They correspond to the coalescence of two energy levels and their respective eigenvectors. Here, we use the Lipkin-Meshkov-Glick (LMG) model as a test bed to explore the strong connection between EPs and the onset of excited state quantum phase transitions (ESQPTs). We show that for finite systems, the exact degeneracies (EPs) obtained with the non-Hermitian LMG Hamiltonian continued into the complex plane are directly linked with the avoided crossings that characterize the ESQPTs for the real (physical) LMG Hamiltonian. The values of the complex control parameter $\ensuremath{\alpha}$ that lead to the EPs approach the real axis as the system size $N\ensuremath{\rightarrow}\ensuremath{\infty}$. This happens for both the EPs that are close to the separatrix that marks the ESQPT and also for those that are far away, although in the latter case, the convergence rate is smaller. With the method of Pad\'e approximants, we can extract the critical value of $\ensuremath{\alpha}$.
Non-Hermitian systems have been at the center of intense research for over a decade, partly due to their nontrivial energy topology formed by intersecting Riemann manifolds with branch points known as exceptional points (EPs). This spectral property can be exploited, e.g., to achieve topologically controlled state permutations that are necessary for implementing a wide class of classical and quantum information protocols. However, the complex-valued spectra of typical non-Hermitian systems lead to instabilities, losses, and breakdown of adiabaticity, which impedes the practical use of EP-induced energy topologies in quantum information protocols based on state permutation symmetries. Indeed, in a given non-Hermitian multiqubit system, the dynamical winding around EPs always results in a predetermined set of attenuated final eigenstates, due to the interplay of decoherence and nonadiabatic transitions, irrespective of the initial conditions. In this work, we address this long-standing problem by introducing a model of interacting qubits governed by an effective non-Hermitian Hamiltonian that hosts a special type of EPs while maintaining a completely real energy spectrum, ensuring the absence of losses in the system’s dynamics. We demonstrate that such non-Hermitian Hamiltonians enable the realization of genuine, in general, non-Abelian permutation groups in the multiqubit system’s eigenspace while dynamically encircling these EPs. Our findings indicate that, contrary to previous beliefs, non-Hermiticity can be utilized to achieve controlled topological state permutations in time-modulated multiqubit systems, thus paving the way for the advancement and development of quantum information protocols in real-world non-Hermitian quantum systems.
Exotic singular objects, known as exceptional points, are ubiquitous in non-Hermitian physics. They might be spectral singularities in energy bands that produce anomalous effects and defectiveness. The quantum entanglement of a generic non-Hermitian model with two different types of spectral exceptional points (SEPs) is systematically investigated in this paper. We discovered a relationship between non-unitary conformal field theories and the k-linear-type SEPs, which is typically associated with PT-symmetry or pesdo-Hermicity spontaneous breaking. The underlying association between k-square-root-type SEPs, which arise concurrently with real (imaginary) gap closing in the complex spectrum, mimicking first-order-phase-transition criticalities, and complex conformal field theories (cCFTs) is addressed through the calculation of complex central charges. From the entanglement spectrum, zero-energy exceptional modes are found to be distinct from normal zero modes or topological boundary modes. Finally, we include a brief discussion of analogous non-Hermitian quantum spin models and endeavor to establish an intuitive understanding of exceptional points through the spin picture in various scenarios.
Quantum matter emerges from the interplay of fluctuations, topology, and entanglement, which - in equilibrium - governs quantized transport, universal criticality, and topological classification. Non-Hermitian systems, widely explored in platforms ranging from electric circuits to photonics, are intrinsically out-of-equilibrium, and display fundamentally new phenomena, including complex spectra, spectral winding, exceptional topology, and non-unitary dynamics. A central challenge is understanding how the complex single-particle spectrum governs universal many-body behavior. We introduce a symmetry-protected dynamical topological index derived directly from the complex spectrum. Through the lens of algebraic topology, more specifically Morse theory, we identify critical points in the spectrum with topological defects, whose curvature and stability are protected under continuous deformations. This links spectral geometry to many-body observables, unifying non-Hermitian band topology, entanglement, and transport. We demonstrate that non-Hermitian quantum criticality in non-interacting systems is controlled by gain-and-loss-selected non-equilibrium steady states, which dynamically generate an emergent imaginary Fermi surface whose Fermi points host scale-invariant gapless modes with logarithmic entanglement scaling and algebraic correlations. Our work establishes a unified framework for non-Hermitian quantum matter, connecting spectral topology to Morse theory, revealing a topological foundation of non-equilibrium quantum criticality.