A non-Hermitian complex symmetric 2 × 2-matrix toy model is used to study projective Hilbert space structures in the vicinity of exceptional points (EPs). The bi-orthogonal eigenvectors of a diagonalizable matrix are Puiseux-expanded in terms of the root vectors at the EP. It is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behaviour in the EP-limit can be resolved by projectively extending the original Hilbert space. The complementary normalization conditions correspond then to two different affine charts of this enlarged projective Hilbert space. Geometric phase and phase-jump behaviour are analysed, and the usefulness of the phase rigidity as measure for the distance to EP configurations is demonstrated. Finally, EP-related aspects of PT-symmetrically extended quantum mechanics are discussed and a conjecture concerning the quantum brachistochrone problem is formulated. PACS numbers: 03.65.Fd, 03.65.Vf, 03.65.Ca, 02.40.Xx 1.A non-Hermitian complex symmetric 2 × 2-matrix toy model is used to study projective Hilbert space structures in the vicinity of exceptional points (EPs). The bi-orthogonal eigenvectors of a diagonalizable matrix are Puiseux-expanded in terms of the root vectors at the EP. It is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behaviour in the EP-limit can be resolved by projectively extending the original Hilbert space. The complementary normalization conditions correspond then to two different affine charts of this enlarged projective Hilbert space. Geometric phase and phase-jump behaviour are analysed, and the usefulness of the phase rigidity as measure for the distance to EP configurations is demonstrated. Finally, EP-related aspects of PT-symmetrically extended quantum mechanics are discussed and a conjecture concerning the quantum brachistochrone problem is formulated. PACS numbers: 03.65.Fd, 03.65.Vf, 03.65.Ca, 02.40.Xx 1.

Jan 8, 201598 citations

Extended exceptional points in projected non-Hermitian systems

Xiao-Ran Wang, Fei Yang, Xianqi Tong +3 more

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.

New Journal of PhysicsMar 1, 20242 citations

Geometric phase from encircling an exceptional point of a quantum resonance in the complex-scaling method

Okuto Morikawa, Shoya Ogawa, Soma Onoda

Non-Hermitian operators are now routinely used to describe few-mode systems such as optical resonators and superconducting qubits, and exceptional points (EPs) are defective spectral singularities of such non-Hermitian operators. In contrast, the scattering-theoretic formulation of EP physics for unbounded Hamiltonians remains less settled. In this work, we formulate the geometric phase associated with encircling an EP when the underlying eigenstates are quantum resonances within a one-dimensional scattering model. To do this, we employ the complex-scaling method, where resonance poles of the S matrix are realized as discrete eigenvalues of the non-Hermitian dilated Hamiltonian, to construct situations in which resonant and scattering states coalesce into an EP in the complex energy plane, that is, the resonance pole is embedded into the continuum spectrum. We analyze the self-orthogonality in the vicinity of an EP, the Berry phase, and the Chern characteristic. Our results clarify how EP branch structure and geometric holonomy arise directly from resonance poles in scattering theory, thereby connecting non-Hermitian spectral topology with the traditional theory of quantum resonances.Non-Hermitian operators are now routinely used to describe few-mode systems such as optical resonators and superconducting qubits, and exceptional points (EPs) are defective spectral singularities of such non-Hermitian operators. In contrast, the scattering-theoretic formulation of EP physics for unbounded Hamiltonians remains less settled. In this work, we formulate the geometric phase associated with encircling an EP when the underlying eigenstates are quantum resonances within a one-dimensional scattering model. To do this, we employ the complex-scaling method, where resonance poles of the S matrix are realized as discrete eigenvalues of the non-Hermitian dilated Hamiltonian, to construct situations in which resonant and scattering states coalesce into an EP in the complex energy plane, that is, the resonance pole is embedded into the continuum spectrum. We analyze the self-orthogonality in the vicinity of an EP, the Berry phase, and the Chern characteristic. Our results clarify how EP branch structure and geometric holonomy arise directly from resonance poles in scattering theory, thereby connecting non-Hermitian spectral topology with the traditional theory of quantum resonances.

quant-pharXiv:2512.24528Jan 1, 2026

Multiblock exceptional points in open quantum systems

Aysel Shiralieva, Grigory A. Starkov, Björn Trauzettel

Open quantum systems can be approximately described by non-Hermitian Hamiltonians (NHHs) and Liouvillian superoperators. The two approaches differ by quantum jump terms corresponding to a measurement of the system by its environment. We analyze the emergence of exceptional points (EPs) in NHHs and Liouvillian superoperators. In particular, we show how EPs in NHHs relate to a novel type of EPs -- multi-block EPs -- in the no-jump Liouvillian, i.e. the Liouvillian superoperator in absence of quantum jump terms. We further analyze how quantum jump terms modify the multi-block structure. To illustrate our general findings, we present two prime examples: qubits and qutrits coupled to additional ground state levels that serve as sinks of the population. In those examples, we can navigate through the EP block structure by a variation of physical parameters. We analyze how the dynamics of the population of the states is affected by the order of the EPs. Additionally, we demonstrate that the quantum geometric tensor serves as a sensitive indicator of EPs of different kinds.Open quantum systems can be approximately described by non-Hermitian Hamiltonians (NHHs) and Liouvillian superoperators. The two approaches differ by quantum jump terms corresponding to a measurement of the system by its environment. We analyze the emergence of exceptional points (EPs) in NHHs and Liouvillian superoperators. In particular, we show how EPs in NHHs relate to a novel type of EPs -- multi-block EPs -- in the no-jump Liouvillian, i.e. the Liouvillian superoperator in absence of quantum jump terms. We further analyze how quantum jump terms modify the multi-block structure. To illustrate our general findings, we present two prime examples: qubits and qutrits coupled to additional ground state levels that serve as sinks of the population. In those examples, we can navigate through the EP block structure by a variation of physical parameters. We analyze how the dynamics of the population of the states is affected by the order of the EPs. Additionally, we demonstrate that the quantum geometric tensor serves as a sensitive indicator of EPs of different kinds.

arXiv:2509.11856Physical Review AMar 16, 20261 citation

Puiseux series about exceptional singularities dictated by symmetry-allowed Hessenberg forms of perturbation matrices

Ipsita Mandal

We develop a systematic framework for determining the nature of exceptional points of $n^{\rm th}$ order (EP $_n$ s) in non-Hermitian (NH) systems, represented by complex square matrices. By expressing symmetry-preserving perturbations ( $\propto ε$ ) in the Jordan-normal basis of the defective matrix at an EP $_n$ , we show that the upper- $\ell$ Hessenberg structure of the perturbation matrix directly dictates the leading-order eigenvalue- and eigenvector-splitting to be $\propto ε^{1/\ell}$ , when expanded in a Puiseux series. Applying this to three-band NH models invariant under parity (P), charge-conjugation (C), or parity-time-reversal (PT), we find that EP $_3$ s in P- and C-symmetric systems are restricted to at most $\sim ε^{1/2}$ branch points, while PT-symmetric systems generically support EP $_3$ s with the strongest possible singularities (viz. $\sim ε^{1/3}$ ). We illustrate these results with concrete three-dimensional models in which exceptional curves and surfaces emerge. We further show that fine-tuned perturbations can suppress the leading-order branch point to a less-singular splitting, which have implications for designing direction-dependent EP-based sensors. The appendix extends the analysis to four-band P- and C-symmetric models, establishing the existence of EP $_4$ s with $\sim ε^{1/4}$ and $\sim ε^{1/2}$ singularities, respectively.We develop a systematic framework for determining the nature of exceptional points of $n^{\rm th}$ order (EP$_n$s) in non-Hermitian (NH) systems, represented by complex square matrices. By expressing symmetry-preserving perturbations ($\propto ε$) in the Jordan-normal basis of the defective matrix at an EP$_n$, we show that the upper-$\ell$ Hessenberg structure of the perturbation matrix directly dictates the leading-order eigenvalue- and eigenvector-splitting to be $\propto ε^{1/\ell}$, when expanded in a Puiseux series. Applying this to three-band NH models invariant under parity (P), charge-conjugation (C), or parity-time-reversal (PT), we find that EP$_3$s in P- and C-symmetric systems are restricted to at most $\sim ε^{1/2}$ branch points, while PT-symmetric systems generically support EP$_3$s with the strongest possible singularities (viz. $\sim ε^{1/3}$). We illustrate these results with concrete three-dimensional models in which exceptional curves and surfaces emerge. We further show that fine-tuned perturbations can suppress the leading-order branch point to a less-singular splitting, which have implications for designing direction-dependent EP-based sensors. The appendix extends the analysis to four-band P- and C-symmetric models, establishing the existence of EP$_4$s with $\sim ε^{1/4}$ and $\sim ε^{1/2}$ singularities, respectively.

cond-mat.mes-hallCross-listedarXiv:2603.25603Mar 27, 2026

Quantum and Classical Exceptional Points at the Nanoscale: Properties and Applications

Yuwei Lu, Wei Li, Xuehua Wang

Exceptional points (EPs) are the spectral singularities and one of the central concepts of non-Hermitian physics, originating from the inevitable energy exchange with the surrounding environment. EPs exist in diverse physical systems and give rise to many counterintuitive effects, offering rich opportunities to control the dynamics and alter the properties of optical, electronic, acoustic, and mechanical states. The last two decades have witnessed the flourishing of non-Hermitian physics and associated applications related to coalesced eigenstates at EPs in a plethora of classical systems. While stemming from the quantum mechanism, the implementation of EPs in real quantum systems still faces challenges of tuning and stabilizing the systems at EPs, as well as the additional noises that hinder the observation of relevant phenomena. This review mainly focuses on summarizing the current efforts and opportunities offered by quantum EPs that result from or produce observable quantum effects. We introduce the concepts of Hamiltonian and Liouvillian EPs in the quantum regime and focus on their different properties in connection with quantum jumps and decoherence. We then provide a comprehensive discussion covering the theoretical and experimental advances in accessing EPs in diverse quantum systems and platforms. Special attention is paid to EP-based quantum-optics applications with state-of-art technologies. Finally, we present a discussion on the existing challenges of constructing quantum EPs at the nanoscale and an outlook on the fundamental science and applied technologies of quantum EPs, aiming to provide valuable insights for future research and building quantum devices with high performance and advanced functionalities.Exceptional points (EPs) are the spectral singularities and one of the central concepts of non-Hermitian physics, originating from the inevitable energy exchange with the surrounding environment. EPs exist in diverse physical systems and give rise to many counterintuitive effects, offering rich opportunities to control the dynamics and alter the properties of optical, electronic, acoustic, and mechanical states. The last two decades have witnessed the flourishing of non-Hermitian physics and associated applications related to coalesced eigenstates at EPs in a plethora of classical systems. While stemming from the quantum mechanism, the implementation of EPs in real quantum systems still faces challenges of tuning and stabilizing the systems at EPs, as well as the additional noises that hinder the observation of relevant phenomena. This review mainly focuses on summarizing the current efforts and opportunities offered by quantum EPs that result from or produce observable quantum effects. We introduce the concepts of Hamiltonian and Liouvillian EPs in the quantum regime and focus on their different properties in connection with quantum jumps and decoherence. We then provide a comprehensive discussion covering the theoretical and experimental advances in accessing EPs in diverse quantum systems and platforms. Special attention is paid to EP-based quantum-optics applications with state-of-art technologies. Finally, we present a discussion on the existing challenges of constructing quantum EPs at the nanoscale and an outlook on the fundamental science and applied technologies of quantum EPs, aiming to provide valuable insights for future research and building quantum devices with high performance and advanced functionalities.

ACS NanoMay 6, 202522 citations

We develop a systematic framework for determining the nature of exceptional points of

n^{\rm th}

order (EP

_n

s) in non-Hermitian (NH) systems, represented by complex square matrices. By expressing symmetry-preserving perturbations (

\propto ε

) in the Jordan-normal basis of the defective matrix at an EP

_n

, we show that the upper-

\ell

Hessenberg structure of the perturbation matrix directly dictates the leading-order eigenvalue- and eigenvector-splitting to be

\propto ε^{1/\ell}

, when expanded in a Puiseux series. Applying this to three-band NH models invariant under parity (P), charge-conjugation (C), or parity-time-reversal (PT), we find that EP

_3

s in P- and C-symmetric systems are restricted to at most

\sim ε^{1/2}

branch points, while PT-symmetric systems generically support EP

_3

s with the strongest possible singularities (viz.

\sim ε^{1/3}

). We illustrate these results with concrete three-dimensional models in which exceptional curves and surfaces emerge. We further show that fine-tuned perturbations can suppress the leading-order branch point to a less-singular splitting, which have implications for designing direction-dependent EP-based sensors. The appendix extends the analysis to four-band P- and C-symmetric models, establishing the existence of EP

_4

s with

\sim ε^{1/4}

and

\sim ε^{1/2}

singularities, respectively.