Zero-energy Resonance Research Articles

Zero energy bound states on nano atomic line defect in iron-based high temperature superconductors

Motivated by recent scanning tunneling microscopy experiments on Fe atomic line defect in iron-based high temperature superconductors, we explore the origin of the zero energy bound states near the endpoints of the line defect by employing the two-orbit four-band tight binding model. With increasing the strength of the Rashba spin–orbit coupling along the line defect, the zero energy resonance peaks move simultaneously forward to negative energy for s+− pairing symmetry, but split for s++ pairing symmetry. The superconducting order parameter correction due to As(Te, Se) atoms missing does not shift the zero energy resonance peaks. Such the zero energy bound states are induced by the weak magnetic order rather than the strong Rashba spin–orbit coupling on Fe atomic line defect.

Negative ion of hydrogen in dense semi-classical plasmas: Stability and zero-energy resonances

The effects of dense semi-classical plasma (DSCP) on the ground state of the negative ion of hydrogen (H−) and on the dynamics of electron–hydrogen scattering have been investigated. DSCP is described by an effective potential which takes care of the collective effects of the plasma at large distances as well as the quantum mechanical effects of diffraction at small distances. An elaborative wave function is employed in the Rayleigh–Ritz variational method to compute the ground state energy of H− for various values of the plasma parameters. In particular, critical values of the plasma parameters are calculated accurately to make a detailed study on the stability of the ion embedded in DSCP. Furthermore, parameters related to the ground state of H− and H are used in the effective range theory to study the effects of DSCP on the dynamics of low-energy e − H(1s) scattering. Special emphasis is given to investigate the phenomenon of zero-energy resonances by computing the singlet scattering length near the critical values of the plasma parameters.

Dynamical formulation of low-energy scattering in one dimension

The transfer matrix M of a short-range potential may be expressed in terms of the time-evolution operator for an effective two-level quantum system with a time-dependent non-Hermitian Hamiltonian. This leads to a dynamical formulation of stationary scattering. We explore the utility of this formulation in the study of the low-energy behavior of the scattering data. In particular, for the exponentially decaying potentials, we devise a simple iterative scheme for computing terms of arbitrary order in the series expansion of M in powers of the wavenumber. The coefficients of this series are determined in terms of a pair of solutions of the zero-energy stationary Schrödinger equation. We introduce a transfer matrix for the latter equation, express it in terms of the time-evolution operator for an effective two-level quantum system, and use it to obtain a perturbative series expansion for the solutions of the zero-energy stationary Schrödinger equation. Our approach allows for identifying the zero-energy resonances for scattering potentials in both full line and half-line with zeros of the entries of the zero-energy transfer matrix of the potential or its trivial extension to the full line.

Scattering of slow electron from hydrogen atom in non-ideal classical plasmas: Zero-energy resonances

The scattering of slow electrons from the hydrogen atom embedded in non-ideal classical plasma has been investigated by using the effective range theory. A pseudopotential, which has been derived from a solution of Bogolyubov's hierarchy equations, is used to represent the interactions among the charged particles in plasma. A detailed study is made on the effects of non-ideality of plasma on the scattering dynamics for a wide range of the non-ideality parameter. In particular, the zero-energy resonance phenomenon has been studied in detail by evaluating the singlet scattering length. It is found that the scattering length suffers significant changes when the non-ideality parameter approaches its critical value.

Stability and collision dynamics of electron–proton in dense semi-classical hydrogen plasma

Fully quantum mechanical calculations have been carried out to investigate the properties of bound states and collision dynamics of an electron–proton system embedded in dense semi-classical plasmas. In particular, the investigation includes the stability of the hydrogen atom and the dynamics of electron scattering from a proton for a wide range of plasma parameters. The interaction between the electron and proton is modeled by a pseudopotential which takes care of the quantum mechanical effects of diffraction at short distances as well as the collective effect at large distances. A large basis set is employed in the Rayleigh–Ritz variational principle to study the stability of the hydrogen atom. On the other hand, the Schwinger variational method in the momentum space is applied to perform a detailed study on the scattering process. Particular emphasis is made to investigate the scattering dynamics at low energies. Scattering length has been calculated quite accurately to explore the phenomenon of zero-energy resonance. Furthermore, effects of diffraction and effects of screening on the scattering length, full cross section, and transport cross section have been investigated in detail.

Electric-field-induced bound states and resonances in heteronuclear atomic collisions

The variations of the bound states of heteronuclear collision complexes in a static electric field are investigated. Using a prototype model, we engineer the interatomic interaction of the collision complex to have an $l$-wave bound state or quasibound state (virtual state for $s$ wave) close to the threshold. The variation of such a state with electric field is calculated to find out whether it crosses the threshold and produces a zero-energy resonance. We identify the cases in which an electric-field-induced resonance can occur at low field. The prototype model is used to interpret the calculated results with realistic potentials for heteronuclear alkali-metal collision complexes. We also calculate the probability density of the low-energy scattering state and find that the short-range probability density can still be enhanced in the absence of electric-field-induced resonance.

On absence of threshold resonances for Schrödinger and Dirac operators

Using a unified approach employing a homogeneous Lippmann-Schwinger-type equation satisfied by resonance functions and basic facts on Riesz potentials, we discuss the absence of threshold resonances for Dirac and Schrodinger operators with sufficiently short-range interactions in general space dimensions. More specifically, assuming a sufficient power law decay of potentials, we derive the absence of zero-energy resonances for massless Dirac operators in space dimensions $n \geq 3$, the absence of resonances at $\pm m$ for massive Dirac operators (with mass $m > 0$) in dimensions $n \geq 5$, and recall the well-known case of absence of zero-energy resonances for Schr\"odinger operators in dimension $n \geq 5$.

Designer fermion models in functionalized graphene bilayers

We propose a method to realize a broad class of tunable fermionic Hamiltonians in graphene bilayer. For that matter, we consider graphene bilayer functionalized with sp$^3$ defects that induce zero energy resonances hosting an individual electron each. The application of an off-plane electric field opens up a gap, so that the zero energy resonance becomes an in-gap bound state whose confinement scales inversely with the gap. Controlling both the distance among the defects and the applied electric field, we can define fermionic models, even lattices, whose hoppings and Coulomb interactions can be tuned. We consider in detail the case of triangular and honeycomb artificial lattices and we show how, for a given arrangement of the sp$^3$ centers, these lattices can undergo an electrically controlled transition between the weak and strong coupling regimes.

Scattering in non-ideal classical plasmas: Scattering length and zero-energy resonances

A fully quantum mechanical calculation has been performed to investigate the scattering of charged particles in nonideal classical plasmas (NICPs). Interactions among the charged particles in NICP have been depicted by a pseudopotential, derived from a sequential solution of Bogolyubov's chain equations. The Schwinger variational method (SVM) has been employed in the momentum space to compute the scattering phase shifts accurately for various plasma parameters. A detailed study has been made on the dynamics of electron-electron (e-e) and electron-proton (e-p) scattering in NICP for a wide range of plasma parameters. Special attention is paid to explore the scattering dynamics at low incident energies. In particular, scattering length and zero-energy resonances have been investigated in detail. Furthermore, effects of nonideality of plasma on the total cross section and transport cross section have also been studied.

1D Schrödinger operators with Coulomb-like potentials

We study the convergence of 1D Schrödinger operators Hε with the potentials which are regularizations of a class of pseudopotentials having, in particular, the form αδ′(x) + βδ(x) + γ/|x| or αδ′(x) + βδ(x) + γ/x. The limit behavior of Hε in the norm resolvent topology, as ε → 0, essentially depends on a way of regularization of the Coulomb potential and the existence of zero-energy resonances for δ′-like potential. All possible limits are described in terms of point interactions at the origin. As a consequence of the convergence results, different kinds of L∞(R)-approximations to the even and odd Coulomb potentials, both penetrable and impenetrable in the limit, are constructed.

Modeling atom–atom interactions at low energy by Jost–Kohn potentials

More than 65 years ago, Jost and Kohn (1952 Phys. Rev. 87, 977) derived an explicit expression for a class of short-range model potentials from a given effective range expansion with the s-wave scattering length as being negative. For as > 0, they calculated another class of short-range model potentials (Jost and Kohn 1953 Dan. Mat. Fys. Medd 27, no. 9) using a method based on an adaptation from Gel’fand−Levitan theory (Gel’fand and Levitan 1951 Dokl. Akad. Nauk. USSR 77, 557-60) of inverse scattering. We here revisit the methods of Jost and Kohn in order to explore the possibility of modeling resonant finite-range interactions at low energy. We show that the Jost−Kohn potentials can account for zero-energy resonances. The s-wave phase shift for positive scattering length is expressed in an analytical form as a function of the binding energy of a bound state. We show that, for small binding energy, both the scattering length and the effective range are strongly influenced by the binding energy; below a critical binding energy the effective range becomes negative provided the scattering length is large. As a consistency check we carry out some simple calculations to show that Jost−Kohn potentials can reproduce the standard results of contact interaction in the limit of the effective range going to zero.

Point-Like Perturbed Fractional Laplacians Through Shrinking Potentials of Finite Range

We construct the rank-one, singular (point-like) perturbations of the d-dimensional fractional Laplacian in the physically meaningful norm-resolvent limit of fractional Schrodinger operators with regular potentials centred around the perturbation point and shrinking to a delta-like shape. We analyse both possible regimes, the resonance-driven and the resonance-independent limit, depending on the power of the fractional Laplacian and the spatial dimension. To this aim, we also qualify the notion of zero-energy resonance for Schrodinger operators formed by a fractional Laplacian and a regular potential.

Some Remarks on 1D Schrödinger Operators With Localized Magnetic and Electric Potentials

One-dimensional Schr\"odinger operators with singular perturbed magnetic and electric potentials are considered. We study the strong resolvent convergence of two families of the operators with potentials shrinking to a point. Localized $\delta$-like magnetic fields are combined with $\delta\,'$-like perturbations of the electric potentials as well as localized rank-two perturbations. The limit results obtained heavily depend on zero-energy resonances of the electric potentials. In particular, the approximation for a wide class of point interactions in one dimension is obtained.

Scattering of charged particles in quantum plasmas: Zero energy resonances

The scattering of charged particles in quantum plasmas (QPs) has been investigated by employing a fully quantum mechanical treatment within the framework of the Schwinger variational principle in the momentum space. The effective potential in QP has been described by a modified Debye-Huckel potential. Scattering phase shifts for various plasma screening strengths have been obtained accurately by a convergent scheme of the Schwinger variational method. The accuracy of the results has been corroborated by solving the corresponding Schrodinger equation with accurate numerical techniques. The nature of scattering for a wide range of plasma screening has been studied. The role of quantum mechanical effects in plasma is examined by comparing the results in QP with the corresponding results in classical weakly coupled plasma for which effective potential has been described by Debye-Huckel potential. Special emphasis has been made to study the dynamics at low energies. In particular, a detailed investigation has been made on the zero-energy resonance phenomenon.

Vison-Majorana complex zero-energy resonance in the Kitaev spin liquid

We study the effect of site dilution in Kitaev's model. We derive an analytical solution of the dynamical spin correlation functions for arbitrary configurations of $Z_2$ fluxes. By incorporating this solution into classical Monte Carlo scheme, we address how a site vacancy affects the experimental observables, such as the static spin susceptibility and the spin lattice relaxation rate, $1/T_1$. As a result, we found an enhancement of dynamical magnetic response in the vicinity of vacancy, which leads to Friedel-like oscillation in local $1/T_1$, in contrast to limited influences on the static susceptibility. Furthermore, we found a sharp zero-energy peak in the magnetic excitation spectrum, which is attributed to the Vison & Majorana zero mode trapped near the site vacancy. This zero mode can be interpreted as fractionalized spin hole into an Ising triplet with differentiated magnetic axes, which leads to the characteristic temperature and field-orientational dependence of $1/T_1$.

Entropy Signatures of Topological Phase Transitions

We review the behavior of the entropy per particle in various two-dimensional electronic systems. The entropy per particle is an important characteristic of any many body system that tells how the entropy of the ensemble of electrons changes if one adds one more electron. Recently, it has been demonstrated how the entropy per particle of a two-dimensional electron gas can be extracted from the recharging current dynamics in a planar capacitor geometry. These experiments pave the way to the systematic studies of entropy in various crystal systems including novel two-dimensional crystals such as gapped graphene, germanene and silicene. Theoretically, the entropy per particle is linked to the temperature derivative of the chemical potential of the electron gas by the Maxwell relation. Using this relation, we calculate the entropy per particle in the vicinity of topological transitions in various two-dimensional electronic systems. We show that the entropy experiences quantized steps at the points of Lifshitz transitions in a two-dimensional electronic gas with a parabolic energy spectrum. In contrast, in doubled-gapped Dirac materials, the entropy per particles demonstrates characteristic spikes once the chemical potential passes through the band edges. The transition from a topological to trivial insulator phase in germanene is manifested by the disappearance of a strong zero-energy resonance in the entropy per particle dependence on the chemical potential. We conclude that studies of the entropy per particle shed light on multiple otherwise hidden peculiarities of the electronic band structure of novel two-dimensional crystals.

Zero-range effective field theory for resonant wino dark matter. Part III. Annihilation effects

Near a critical value of the wino mass where there is a zero-energy S-wave resonance at the neutral-wino-pair threshold, low-energy winos can be described by a zero-range effective field theory (ZREFT) in which the winos interact nonperturbatively through a contact interaction and through Coulomb interactions. The effects of wino-pair annihilation into electroweak gauge bosons are taken into account through the analytic continuation of the real parameters for the contact interaction to complex values. The parameters of ZREFT can be determined by matching wino-wino scattering amplitudes calculated by solving the Schrödinger equation for winos interacting through a real potential due to the exchange of electroweak gauge bosons and an imaginary potential due to wino-pair annihilation into electroweak gauge bosons. ZREFT at leading order gives an accurate analytic description of low-energy wino-wino scattering, inclusive wino-pair annihilation, and a wino-pair bound state. ZREFT can also be applied to partial annihilation rates, such as the Sommerfeld enhancement of the annihilation rate of wino pairs into monochromatic photons.

Zero-range effective field theory for resonant wino dark matter. Part II. Coulomb resummation

Near a critical value of the wino mass where there is a zero-energy S-wave resonance at the neutral-wino-pair threshold, low-energy winos can be described by a zero-range effective field theory (ZREFT) in which the winos interact nonperturbatively through a contact interaction and charged winos also have electromagnetic interactions. At energies near the wino-pair thresholds, the Coulomb interaction from photon exchange between charged winos must also be treated nonperturbatively. The parameters of ZREFT can be determined by matching wino-wino scattering amplitudes calculated by solving the Schrödinger equation for winos interacting through a potential due to the exchange of electroweak gauge bosons. With Coulomb resummation, ZREFT at leading order gives a good description of the low-energy two-body observables for winos.

On energy-critical half-wave maps into $${\mathbb {S}}^2$$ S 2

We consider the energy-critical half-wave maps equation $$\partial_t \mathbf{u} + \mathbf{u} \wedge |\nabla| \mathbf{u} = 0$$ for $\mathbf{u} : [0,T) \times \mathbb{R} \to \mathbb{S}^2$. We give a complete classification of all traveling solitary waves with finite energy. The proof is based on a geometric characterization of these solutions as minimal surfaces with (not necessarily free) boundary on $\mathbb{S}^2$. In particular, we discover an explicit Lorentz boost symmetry, which is implemented by the conformal M\"obius group on the target $\mathbb{S}^2$ applied to half-harmonic maps from $\mathbb{R}$ to $\mathbb{S}^2$. Complementing our classification result, we carry out a detailed analysis of the linearized operator $L$ around half-harmonic maps $\mathbf{Q}$ with arbitrary degree $m \geq 1$. Here we explicitly determine the nullspace including the zero-energy resonances; in particular, we prove the nondegeneracy of $\mathbf{Q}$. Moreover, we give a full description of the spectrum of $L$ by finding all its $L^2$-eigenvalues and proving their simplicity. Furthermore, we prove a coercivity estimate for $L$ and we rule out embedded eigenvalues inside the essential spectrum. Our spectral analysis is based on a reformulation in terms of certain Jacobi operators (tridiagonal infinite matrices) obtained from a conformal transformation of the spectral problem posed on $\mathbb{R}$ to the unit circle $\mathbb{S}$. Finally, we construct a unitary map which can be seen as a gauge transform tailored for a future stability and blowup analysis close to half-harmonic maps. Our spectral results also have potential applications to the half-harmonic map heat flow, which is the parabolic counterpart of the half-wave maps equation.

Zero-range effective field theory for resonant wino dark matter. Part I. Framework

The most dramatic “Sommerfeld enhancements” of neutral-wino-pair annihilation occur when the wino mass is near a critical value where there is a zero-energy S-wave resonance at the neutral-wino-pair threshold. Near such a critical mass, low-energy winos can be described by a zero-range effective field theory in which the winos interact nonperturbatively through a contact interaction. The effective field theory is controlled by a renormalization-group fixed point at which the neutral and charged winos are degenerate in mass and their scattering length is infinite. The parameters of the zero-range effective field theory can be determined by matching wino-wino scattering amplitudes calculated by solving the Schrödinger equation for winos interacting through a potential due to the exchange of weak gauge bosons. If the wino mass is larger than the critical value, the resonance is a wino-pair bound state. The power of the zero-range effective field theory is illustrated by calculating the rate for formation of the bound state in the collision of two neutral winos through the emission of two soft photons.