Abstract Exceptional points are interesting physical phenomena in non-Hermitian physics at which the eigenvalues are degenerate and the eigenvectors coalesce. In this paper, we find that in projected non-Hermitian two-level systems (sub-systems under projecting partial Hilbert space) the singularities of exceptional points (EPs) is due to basis defectiveness rather than energy degeneracy or state coalescence. This leads to the discovery of extended exceptional points (EEPs). For EEPs, more subtle structures (e.g. the so-called Bloch peach), additional classification, and ‘hidden’ quantum phase transitions are explored. By using the topologically protected sub-space from two edge states in the non-Hermitian Su–Schrieffer–Heeger model as an example, we illustrate the physical properties of different types of EEPs.
Abstract Exceptional points are interesting physical phenomena in non-Hermitian physics at which the eigenvalues are degenerate and the eigenvectors coalesce. In this paper, we find that in projected non-Hermitian two-level systems (sub-systems under projecting partial Hilbert space) the singularities of exceptional points (EPs) is due to basis defectiveness rather than energy degeneracy or state coalescence. This leads to the discovery of extended exceptional points (EEPs). For EEPs, more subtle structures (e.g. the so-called Bloch peach), additional classification, and ‘hidden’ quantum phase transitions are explored. By using the topologically protected sub-space from two edge states in the non-Hermitian Su–Schrieffer–Heeger model as an example, we illustrate the physical properties of different types of EEPs.
An exceptional point (EP) is exclusive for non-Hermitian systems and distinct from that at a degeneracy point (DP), supporting intriguing dynamics which can be utilized to probe quantum phase transitions and prepare eigenstates in a Hermitian many-body system. In this paper, we investigate the transition from a DP for a Hermitian system to an EP driven by non-Hermitian terms. We present a theorem on the existence of a transition between a DP and EP for a general system. Specifically, one of twofold degenerate eigenstates of a Hermitian system becomes a coalescing state when a selected non-Hermitian term is added. The obtained EP is robust to the strength of non-Hermitian terms. We illustrate the theorem by an exactly solvable quasi-one-dimensional model, which allows for the existence of a transition between fully degenerate and exceptional spectra driven by non-Hermitian tunnelings in real and k spaces, respectively. We also study the EP dynamics for generating coalescing edge modes in Su-Schrieffer-Heeger-like models. This finding reveals the ubiquitous connection between DPs and EPs.
The appearance of topological singularities, namely exceptional points (EPs) is an intriguing feature of parameter-dependent open quantum or wave systems. EPs are the special type of nonHermitian degeneracies where two (or more) eigenstates of the underlying system coalesce. In this paper, we present a three level non-Hermitian Hamiltonian which hosts three interacting eigenstates. The matrix elements are optimized in such a way that the intermediate eigenstate interacts with both the other states and the underlying system hosts at least two different second order EPs. The impact of quasi-static parameter variation along a cyclic contour around the embedded EPs on the dynamics of interacting eigenvalues is well investigated in the context of cascaded state conversion. Such dynamics of the eigenvalues shows a clear signature of the third order EP with a combined effect of both the second order EPs. Moreover, we examine the accumulation of phases around the identified EPs and study the hallmark of phase exchange during cascaded state conversions accompanied by the parametric encirclement of the third order EP.
Open photonic systems provide a versatile platform for non-Hermitian physics, enabling control over complex spectra, transport, and light-matter interactions. Exceptional points (EPs), at which eigenvalues and eigenvectors coalesce and the governing operator becomes defective, play a central role because they combine branch-point spectral topology, nonanalytic perturbative response, and controllable eigenstate conversion. This Review provides a unified framework for EP photonics by systematically distinguishing exceptional degeneracies according to the underlying operator, spectral variable, boundary conditions, and experimentally accessible observables. We discuss Hamiltonian EPs, absorbing EPs associated with scattering zeros, real-frequency scattering-matrix and Jones-matrix EPs, Bloch and Floquet EPs, and Liouvillian EPs in open quantum systems. We review their spectral topology, static and dynamical encircling, higher-order exceptional structures, and coexistence with bound states in the continuum, together with applications in sensing, lasing, coherent absorption, directional scattering, polarization and wavefront control, nonlinear optics, optical storage, nonreciprocal photonics, and quantum photonics. We also critically assess the current limitations, practical challenges, and future perspectives of EP-based photonic technologies, with particular attention to robustness, noise, scalability, and experimentally measurable performance.
Abstract Exceptional points (EPs) occur in non-Hermitian systems where eigenvalues and eigenvectors coalesce. In open quantum systems, EPs are typically studied via non-Hermitian effective Hamiltonians or Liouvillians, but these omit the full generality of quantum dynamics, which requires description using quantum channels. Here, we introduce a general strategy for generating quantum-channel EPs in the single-qubit setting. We show that quantum channels naturally fall into two distinct phases, purely real or complex-conjugate eigenvalues, without the need of enforcing additional symmetries, and interpolating between different phases induces a phase transition where EPs can emerge. We experimentally simulate such channels on a nuclear magnetic resonance quantum computer, observing second-order quantum-channel EPs with 93% fidelity, and extend the approach to three channels to uncover exceptional lines and a third-order quantum-channel EP. Our results promise potential applications from enhanced quantum sensing to quantum control, such as Jordan-chain mediated asymmetric conversion.
Exceptional points are degeneracies characteristic of non-Hermitian operators, where eigenvalues and eigenvectors coalesce, rendering the Hamiltonian defective. We investigate the exceptional-point structure and topological properties of a generalized non-Hermitian Rice-Mele model with balanced gain and loss, as well as next-nearest-neighbor hopping. The system hosts only second-order exceptional points under both periodic and open boundary conditions. Under periodic boundary conditions, the exceptional points in parameter space lie on lines and ellipses that are independent of the next-nearest-neighbor hopping, since the latter enters the bulk Hamiltonian only as an identity contribution. Under open boundary conditions, this independence is broken: the next-nearest-neighbor hopping not only shifts the energy of existing exceptional points but also generates new ones, with a specific condition signaling a topological gap closing observed only in the open-boundary spectrum. At special parameter points, multiple simultaneous second-order exceptional points yield degenerate configurations whose degeneracy grows with system size. Exceptional point locations are identified numerically via the condition number of the eigenvector matrix and confirmed by Jordan decomposition. The topological phase diagram, computed via a winding number framework for non-Hermitian systems without symmetry protection, reveals sectors with zero, one, and two edge states; the bulk-boundary correspondence is confirmed, and the non-Hermitian skin effect is absent.
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 $\ensuremath{\alpha}$.