Complex Energy Plane Research Articles

Scale-tailored localization and its observation in non-Hermitian electrical circuits

Anderson localization and non-Hermitian skin effect are two paradigmatic wave localization phenomena, resulting from wave interference and the intrinsic non-Hermitian point gap, respectively. In this study, we unveil a novel localization phenomenon associated with long-range asymmetric coupling, termed scale-tailored localization, where the number of induced localized modes and their localization lengths scale exclusively with the coupling range. We show that the long-range coupling fundamentally reshapes the energy spectra and eigenstates by creating multiple connected paths on the lattice. Furthermore, we present experimental observations of scale-tailored localization in non-Hermitian electrical circuits utilizing adjustable voltage followers and switches. The circuit admittance spectra possess separate point-shaped and loop-shaped components in the complex energy plane, corresponding respectively to skin modes and scale-tailored localized states. Our findings not only expand and deepen the understanding of peculiar effects induced by non-Hermiticity but also offer a feasible experimental platform for exploring and controlling wave localizations.

The possible KK¯* and DD¯* bound and resonance states by solving the Schrodinger equation

The Schrodinger equation with a Yukawa type of potential is solved analytically. When different boundary conditions are taken into account, a series of solutions are indicated as a Bessel function, the first kind of Hankel function and the second kind of Hankel function, respectively. Subsequently, the scattering processes of KK¯* and DD¯* are investigated. In the KK¯* sector, the f 1(1285) particle is treated as a KK¯* bound state, therefore, the coupling constant in the KK¯* Yukawa potential can be fixed according to the binding energy of the f 1(1285) particle. Consequently, a KK¯* resonance state is generated by solving the Schrodinger equation with the outgoing wave condition, which lies at 1417 − i18 MeV on the complex energy plane. It is reasonable to assume that the KK¯* resonance state at 1417 − i18 MeV might correspond to the f 1(1420) particle in the review of the Particle Data Group. In the DD¯* sector, since the X(3872) particle is almost located at the DD¯* threshold, its binding energy is approximately equal to zero. Therefore, the coupling constant in the DD¯* Yukawa potential is determined, which is related to the first zero point of the zero-order Bessel function. Similarly to the KK¯* case, four resonance states are produced as solutions of the Schrodinger equation with the outgoing wave condition. It is assumed that the resonance states at 3885 − i1 MeV, 4029 − i108 MeV, 4328 − i191 MeV and 4772 − i267 MeV might be associated with the Zc(3900), the X(3940), the χ c1(4274) and χ c1(4685) particles, respectively. It is noted that all solutions are isospin degenerate.

Assessment of ϒ -states above BB¯ -threshold using a constituent-quark-model based meson-meson coupled-channels framework

The ϒ(10753) state has been recently observed by the Belle and Belle II collaborations with enough global significance to motivate an assessment of the high-energy spectrum usually predicted by any reasonable quark model. In the framework of a constituent quark model which satisfactorily describes a wide range of properties of conventional hadrons containing heavy quarks, the quark-antiquark and meson-meson degrees of freedom have been incorporated with the goal of elucidating the influence of open-bottom meson-meson thresholds into the ϒ states whose masses are within the energy range of the ϒ(10753)’s mass. It is well known that such effects could be relevant enough as to dynamically generate new states and thus provide a plausible explanation of the nature of the ϒ(10753) state. In particular, we have performed a coupled-channels calculation in which the bare states ϒ(4S), ϒ(3D), ϒ(5S), and ϒ(4D) are considered together with the threshold channels BB¯, BB¯*, B*B¯*, BsB¯s, BsB¯s*, and Bs*B¯s*. Among the results we have described, the following are of particular interest: (i) a richer complex spectrum is gained when thresholds are present and bare bound states are sufficiently nonrelativistic; (ii) those poles obtained in the complex energy plane do not have to appear as simple peaks in the relevant cross sections; and (iii) the ϒ(10753) candidate is interpreted as a dressed hadronic resonance whose structure is an equally mixture of a conventional bb¯ state and B*B¯* molecule. Published by the American Physical Society 2024

Observing S-Matrix Pole Flow in Resonance Interplay

We provide an overview of experiments exploring resonances in the collision of ultracold clouds of atoms. Using a laser-based accelerator that capitalises on the energy resolution provided by the ultracold atomic setting, we unveil resonance phenomena such as Feshbach and shape resonances in their quintessential form by literally photographing the halo of outgoing scattered atoms. We exploit the tunability of magnetic Feshbach resonances to instigate an interplay between scattering resonances. By experimentally recording the scattering in a parameter space spanned by collision energy and magnetic field, we capture the imprint of the S-matrix pole flow in the complex energy plane. After revisiting experiments that place a Feshbach resonance in the proximity of a shape resonance and an anti-bound state, respectively, we discuss the possibility of using S-matrix pole interplay between two Feshbach resonances to create a bound-state-in-the-continuum.

Toward the quark mass dependence of Tcc+ from lattice QCD

The DD* scattering phase shifts in the Tcc+=ccu¯d¯ channel are extracted from lattice QCD for five different charm quark masses and a fixed light-quark mass corresponding to mπ≃280 MeV. The phase shifts are analyzed employing two approaches: effective range expansion and Lippmann-Schwinger equation derived in the effective field theory. In the latter case, the results imply an attraction at short range parametrized by contact terms and a slight repulsion at long range mediated by one-pion exchange with mπ>mD*−mD. The poles in the amplitude across the complex energy plane are extracted and their trajectories are discussed as the charm quark mass is varied. Two complex conjugate poles corresponding to a resonance below threshold are found for mc close to the physical value. They turn into a pair of virtual states at the largest mc studied. With further increasing mc, one virtual pole representing Tcc+ is expected to move towards the two-body threshold and turn into a bound state. The light-quark mass dependence of the Tcc+ pole is briefly discussed using the data on DD* scattering from other lattice collaborations. Published by the American Physical Society 2024

Occupation-Dependent Particle Separation in One-Dimensional Non-Hermitian Lattices.

We unveil an exotic phenomenon arising from the intricate interplay between non-Hermiticity and many-body physics, namely, an occupation-dependent particle separation for hardcore bosons in a one-dimensional lattice driven by unidirectional non-Hermitian pumping. Taking hardcore bosons as an example, we find that a pair of particles occupying the same unit cell exhibit an opposite non-Hermitian pumping direction to that of unpaired ones occupying different unit cells. By turning on an intracell interaction, many-body eigenstates split in their real energies, forming separable clusters in the complex energy plane with either left-, right-, or bipolar-types of non-Hermitian skin effect (NHSE). The dependency of skin accumulating directions on particle occupation is further justified with local sublattice correlation and entanglement entropy of many-body eigenstates. Dynamically, this occupation-dependent NHSE manifests as uni- or bidirectional pumping for many-body initial states, allowing for spatially separating paired and unpaired particles. Our results unveil the possibility of designing and exploring novel non-Hermitian phases originated from particle nonconservation in subsystems (e.g., orbitals, sublattices, or spin species) and their spatial configurations.

Lattice QCD study of πΣ−K¯N scattering and the Λ(1405) resonance

A lattice QCD computation of the coupled channel πΣ–K¯N scattering amplitudes in the Λ(1405) region is detailed. Results are obtained using a single ensemble of gauge field configurations with Nf=2+1 dynamical quark flavors and mπ≈200 MeV and mK≈487 MeV. Hermitian correlation matrices using both single baryon and meson-baryon interpolating operators for a variety of different total momenta and irreducible representations are used. Several parametrizations of the two-channel scattering K-matrix are utilized to obtain the scattering amplitudes from the finite-volume spectrum. The amplitudes, continued to the complex energy plane, exhibit a virtual bound state below the πΣ threshold and a resonance pole just below the K¯N threshold. Published by the American Physical Society 2024

Reconciling experimental and lattice data of Zc(3900) in a J/ψπ−DD¯* coupled-channel analysis

We study the J/ψπ and DD¯* coupled-channel system within a covariant framework. The J/ψπ and DD¯* invariant-mass distributions measured at 4.23 and 4.26 GeV by BESIII and the finite-volume energy levels from recent lattice QCD simulations are simultaneously fitted. Phase shifts and inelasticities of the J/ψπ and DD¯* scattering are predicted using the resulting amplitudes. Poles corresponding to the Zc(3900) state are found in the complex energy plane and their couplings with J/ψπ and DD¯* are determined. Our results indicate that the current lattice data do not preclude the existence of the Zc(3900) state. Published by the American Physical Society 2024

Nucleon Resonance Masses from QCD Sum Rules

We calculate the masses of the nucleon resonances [N+(1440), N-(1535), N- (1650), N+(1710), N+(1880), N-(1895), N+(2100)] using an adaptation of the method of finite energy QCD sum rules (FESR) introduced recently in the determination of the masses of the vector meson resonances. This variation of finite energy sum rules involves as usual integration over a closed contour consisting of a circle of a large radius R and a cut in the complex energy (squared) plane. A specific integration kernel is used, however, which minimizes the integral over the cut. We obtain remarkably stable results over a wide range R, where R is the radius of the integration contour. The sum rule predictions agree well with the experimental values. We make use of the nucleonic correlator discussed in our previous FESR calculation of the nucleon mass.

Spectral Structure and Doublon Dissociation in the Two‐Particle Non‐Hermitian Hubbard Model

AbstractStrongly‐correlated systems in non‐Hermitian models are an emergent area of research. Herein, a non‐Hermitian Hubbard model is considered, where the single‐particle hopping amplitudes on the lattice are not reciprocal, and provide exact analytical results of the spectral structure in the two‐particle sector of Hilbert space under different boundary conditions. The analysis unveils some interesting spectral and dynamical effects of purely non‐Hermitian nature and that deviate from the usual scenario found in the single‐particle regime. Specifically, a spectral phase transition of the Mott‐Hubbard band on the infinite lattice is predicted as the interaction energy is increased above a critical value, from an open to a closed loop in complex energy plane, and the dynamical dissociation of doublons, i.e., instability of two‐particle bound states, in the bulk of the lattice, with a sudden revival of the doublon state when the two particles reach the lattice edge. Particle dissociation observed in the bulk of the lattice is a clear manifestation of non‐Hermitian dynamics arising from the different lifetimes of single‐particle and two‐particle states, whereas the sudden revival of the doublon state at the boundaries is a striking burst edge dynamical effect peculiar to non‐Hermitian systems with boundary‐dependent energy spectra, here predicted for the first time for correlated particles.

Molecular states in systems* *Supported by the Doctoral Program of Tian Chi Foundation of Xinjiang Uyghur Autonomous Region of China (51052300506), the NSFC and the Deutsche Forschungsgemeinschaftn(DFG, German Research Foundation) through the funds provided to the Sino-German Collaborative Research Center TRR110 “Symmetries and the Emergence of Structure in QCD” (NSFC 12070131001, DFG Project-ID 196253076 - TRR 110), the NSFC (11835015,

The possible hadronic molecules in systems with , and are investigated with interactions described by light meson exchanges. By varying the cutoff in a phenomenologically reasonable range of GeV, we find ten near-threshold (bound or virtual) states in the single-channel case. After introducing the coupled-channel dynamics of ----- systems, these states, except those below the lowest channels in each sector, move into the complex energy plane and become resonances in the mass range GeV. Their spin-parities and nearby thresholds are , , , , , , , , , and . The impact of the -term in the one-boson-exchange model on these states is presented. Setting GeV as an illustrative value, it is found that is a stable bound state (becoming unstable if the coupling to lower channels is turned on), and are physical resonances in cases where the -term is included or excluded, and the other seven states are physical resonances or "virtual-state-like" poles near thresholds, depending on whether the -term is included. In addition, the partial decay widths of the physical resonances are provided. These double-charm hidden-strangeness pentaquark states, as the partners of the experimentally observed and states, can be searched for in the final states in the future.

The Analytic Wavefunction

The wavefunction in quantum field theory is an invaluable tool for tackling a variety of problems, including probing the interior of Minkowski spacetime and modelling boundary observables in de Sitter spacetime. Here we study the analytic structure of wavefunction coefficients in Minkowski as a function of their kinematics. We introduce an off-shell wavefunction in terms of amputated time-ordered correlation functions and show that it is analytic in the complex energy plane except for possible singularities on the negative real axis. These singularities are determined to all loop orders by a simple energy-conservation condition. We confirm this picture by developing a Landau analysis of wavefunction loop integrals and corroborate our findings with several explicit calculations in scalar field theories. This analytic structure allows us to derive new UV/IR sum rules for the wavefunction that fix the coefficients in its low-energy expansion in terms of integrals of discontinuities in the corresponding UV-completion. In contrast to the analogous sum rules for scattering amplitudes, the wavefunction sum rules can also constrain total-derivative interactions. We explicitly verify these new relations at one-loop order in simple UV models of a light and a heavy scalar. Our results, which apply to both Lorentz invariant and boost-breaking theories, pave the way towards deriving wavefunction positivity bounds in flat and cosmological spacetimes.

Magnetic flux response of non-Hermitian topological phases

We derive the response of non-Hermitian topological phases with intrinsic point gap topology to localized magnetic flux insertions. In two spatial dimensions, we identify the necessary and sufficient conditions for a flux skin effect that localizes an extensive number of in-gap modes at a flux core. In three dimensions, we furthermore establish the existence of: a flux spectral jump, where flux tube insertion fills up the entire point gap only at a single parallel crystal momentum; a higher-order flux skin effect, which occurs at the ends of flux tubes in presence of pseudo-inversion symmetry; and a flux Majorana mode that represents a spectrally isolated mid-gap state in the complex energy plane. We uniquely associate each non-Hermitian symmetry class with intrinsic point gap topology with one of these cases or a trivial flux response, and discuss possible experimental realizations.

Anatomy of open-boundary bulk in multiband non-Hermitian systems

Although the non-Bloch band theory is a milestone in elaborating bulk energy bands of non-Hermitian systems under the open-boundary condition (OBC), vital issues related to multivalued functions of non-Hermitian energy bands remain unsolved. In this paper, we anatomize the bulk properties of one-dimensional multiband non-Hermitian systems under OBC. We put forward the energy-band branches (EBBs) to settle the multivalued functions of non-Hermitian energy bands, which become gapped or gapless corresponding to disconnected or connected EBBs in the complex energy plane, where the branch points and branch cuts play a crucial role. We clarify the precise significance of the non-Hermitian skin effect, which illustrates the asymptotic behavior of EBB eigenstates (bulk eigenstates) in the deep bulk and compensates previous non-Bloch band theory. We also obtain a general expression of open-boundary Green's functions based on such EBBs and generalized Brillouin zones, useful for studies on non-Hermitian dynamical evolution.

Continuum of Bound States in a Non-Hermitian Model.

In a Hermitian system, bound states must have quantized energies, whereas free states can form a continuum. We demonstrate how this principle fails for non-Hermitian systems, by analyzing non-Hermitian continuous Hamiltonians with an imaginary momentum and Landau-type vector potential. The eigenstates, which we call "continuum Landau modes" (CLMs), have Gaussian spatial envelopes and form a continuum filling the complex energy plane. We present experimentally realizable 1D and 2D lattice models that host CLMs; the lattice eigenstates are localized and have other features matching the continuous model. One of these lattices can serve as a rainbow trap, whereby the response to an excitation is concentrated at a position proportional to the frequency. Another lattice can act a wave funnel, concentrating an input excitation onto a boundary over a wide frequency bandwidth. Unlike recent funneling schemes based on the non-Hermitian skin effect, this requires a simple lattice design with reciprocal couplings.

Three-body resonances in the φ4 theory

We study the properties of three-body resonances using a lattice complex scalar φ4 theory with two scalars, with parameters chosen such that one heavy particle can decay into three light ones. We determine the two- and three-body spectra for several lattice volumes using variational techniques, and then analyze them with two versions of the three-particle finite-volume formalism: the Relativistic Field Theory approach and the Finite-Volume Unitarity approach. We find that both methods provide an equivalent description of the energy levels, and we are able to fit the spectra using simple parametrizations of the scattering quantities. By solving the integral equations of the corresponding three-particle formalisms, we determine the pole position of the resonance in the complex energy plane and thereby its mass and width. We find very good agreement between the two methods at different values of the coupling of the theory.

Complex eigenenergy of the giant dipole resonance forO16by the Jost function within the random-phase approximation framework

The Jost function method is extended within the framework of the random phase approximation (RPA) theory to find poles on the complex energy plane that exhibit complex RPA eigenenergies. Poles corresponding to the RPA excited states such as the giant resonance of $^{16}\mathrm{O}$ electric dipole excitations were successfully found on the complex energy plane. Although the giant resonance has been known as a single resonance with large strength and width, it is found that, at least within the RPA framework, the $^{16}\mathrm{O}$ electric dipole giant resonance is formed by multiple poles, each of which is an independent pole with different widths, origins, response properties to residual interactions, and components structures of the density fluctuation.

Isovector meson masses from QCD sum rules

We present a calculation of the masses of the isovector mesons (vector, scalar and pseudoscalar including the established recurrences) using a new method of finite energy QCD sum rules. The method is based on the idea of choosing a suitable integration kernel which minimizes the occurring integral over the cut in the complex energy (squared) plane. We obtain remarkably stable results in a wide range [Formula: see text], where [Formula: see text] is the radius of the integration contour. The sum rule predictions agree with the experimental values within the expected accuracy showing that QCD describes single resonances.

Contour prescriptions in string-inspired nonlocal field theories

In quantum field theory, a consistent prescription to define and deform integration contours in the complex energy plane is needed to evaluate loop integrals and compute scattering amplitudes. In some nonlocal field theories, including string field theory, interaction vertices contain transcendental functions of momenta that can diverge along certain complex directions, thus making it impossible to use standard techniques, such as Wick rotation, to perform loop integrals. The aim of this paper is to investigate the viability of several contour prescriptions in the presence of nonlocal vertices. We consider three ``different'' prescriptions, and establish their (in)equivalence in local and nonlocal theories. In particular, we prove that all these prescriptions turn out to be equivalent in standard local theories, while this is not the case for nonlocal theories where amplitudes must be defined first in Euclidean space, and then analytically continued to Minkowski. We work at one-loop level and focus on the bubble diagram. In addition to proving general results for a large class of nonlocal theories, we show explicit calculations in a string inspired nonlocal scalar model.

Dynamical signatures of point-gap Weyl semimetal

We demonstrate a few unique dynamical properties of point-gap Weyl semimetal, an intrinsic non-Hermitian topological phase in three dimensions. We consider a concrete model where a pair of Weyl points reside on the imaginary axis of the complex energy plane, opening up a point gap characterized by a topological invariant, the three-winding number $W_3$. This gives rise to surface spectra and dynamical responses that differ fundamentally from those in Hermitian Weyl semimetals. First, we predict a time-dependent current flow along the magnetic field in the absence of an electric field, in sharp contrast to the current driven by the chiral anomaly, which requires both electric and magnetic fields. Second, we reveal a novel type of boundary-skin mode in the wire geometry which becomes localized at two corners of the wire cross section. We explain its origin and show its experimental signatures in wave-packet dynamics.