Abstract The behavior of quantum states at exceptional points (EPs) and at critical points associated with quantum phase transitions (QPTs) is intriguing yet puzzling. In this study, we present an alternative method for obtaining the Berry potentials using the evolution generator along the parameter induced dimension and demonstrate that they are singular at these critical points. Although these singularities may appear to indicate a breakdown in quantum state evolution, we show that the information carried by quantum states evolving across these critical points is not destroyed. Specifically, when the evolution generator of the full Hilbert space bundle is taken into account, the quantum states remain insensitive to the critical points. In physical terms, it is similar to the classical image of an object smoothly passing through a black hole’s event horizon. Further similarities between EPs and QPTs are explored in this work.
We study the impact of quantum phase transitions (QPTs) on the distribution of exceptional points (EPs) of the Hamiltonian in the complex-extended parameter domain. Analyzing first- and second-order QPTs in the Lipkin-Meshkov-Glick model we find an exponentially and polynomially close approach of EPs to the respective critical point with increasing size of the system. If the critical Hamiltonian is subject to random perturbations of various kinds, the averaged distribution of EPs close to the critical point still carries decisive information on the QPT type. We therefore claim that properties of the EP distribution represent a parametrization-independent signature of criticality in quantum systems.
We investigate the exceptional points (EPs) and their pseudospectra in black hole perturbation theory. By considering a Gaussian bump modification to the Regge-Wheeler potential with variable amplitude, position, and width parameters,
(ε,d,σ0), a continuous line of EPs (exceptional line, EL) in this three-dimensional parameter space is revealed. Notably, the EL exhibits an anisotropic spectral response: parameters migrating along the EL direction leaves the coalesced QNM spectra nearly unchanged, while moving parameters away from the EL induces the characteristic
ε1/2 scaling, highlighting the directional nature of spectral instability in exceptional structures. We find that the vorticity
ν=±1/2 and the Berry phase
γ=π for loops encircling the EL, while
ν=0 and
γ=0 for those do not encircle the EL. In the neighborhood of an eigenvalue, through matrix perturbation theory, we prove that the
ε-pseudospectrum contour size scales as
ε1/q at an EP , where
q is the order of the largest Jordan block of the Hamiltonian-like operator associated with that eigenvalue, contrasting with the linear
ε scaling at non-EPs. Numerical implements confirm this observation, demonstrating enhanced spectral instability at EPs for non-Hermitian systems including black holes.
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 $\ensuremath{\alpha}$.
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.
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.
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.