Nonlocal Potential Research Articles

Elastic properties and thermodynamic anomalies of supersolids

We study a supersolid in the context of a Gross-Pitaevskii theory with a nonlocal effective potential. We employ a homogenization technique which allows us to calculate the elastic moduli, supersolid fraction, and other state variables of the system. Our methodology is verified against numerical simulations of elastic deformations. We can also verify that the long-wavelength Goldstone modes that emerge from this technique agree with Bogoliubov theory. We find a thermodynamic anomaly that the supersolid does not obey the thermodynamic relation ∂P/∂V|N=−n(∂P/∂N|V), which we claim is a feature unique to supersolids. Published by the American Physical Society 2024

Emergent pair symmetries in systems with poor man's Majorana modes

Few-site Kitaev chains are promising for realizing Majorana zero modes without topological protection but fully nonlocal, which are known as poor man's Majorana modes. While several signatures have already been reported both theoretically and experimentally, it still remains unknown what is the nature of superconducting correlations in the presence of poor man's Majorana modes. In this paper, we study few-site Kitaev chains and demonstrate that they host pair correlations with distinct symmetries, entirely determined by the underlying quantum numbers. In particular, we find that a two-site Kitaev chain hosts local (odd-frequency) and nonlocal (odd- and even-frequency) pair correlations, both spin polarized and highly tunable by the system parameters. Interestingly, the odd-frequency pair correlations exhibit a divergent behavior around zero frequency when the nonlocal p-wave pair potential and electron tunneling are of the same order, an effect that can be controlled by the on-site energies. Since a divergent odd-frequency pairing is directly connected to the intrinsic spatial nonlocality of Majorana zero modes in topological superconductors, the divergent odd-frequency pairing here reflects the intrinsic Majorana nonlocality of poor man's Majorana modes but without any relation to topology. Our findings could help understanding the emergent pair correlations in few-site Kitaev chains. Published by the American Physical Society 2024

An exact treatment of localization of electromagnetic plus separable potential

Equivalent local potential with energy-momentum dependence is developed for the combined interaction of Hulthén modified Yamaguchi potential by developing its exact Jost solution. The generated local potentials are applied to compute scattering phase parameters for (p-p) and (p-d) systems through the phase function method (PFM). The same are also calculated for the nonlocal potential from the expression of the Fredholm determinant. Our obtained data for both the energy-momentum dependent local and the pure nonlocal interactions are in reasonable agreement with the standard data. Reasonable correspondence between the results for the phase equivalent and the nonlocal potentials indicate that our equivalent local analysis is in proper order.

Multidimensional Hyperbolic Equations with Nonlocal Potentials: Families of Global Solutions

In spaces of arbitrary dimensions, hyperbolic differential–difference equations with potentials undergoing translations along arbitrary spatial directions are investigated. The fundamental novelty of this study is that, unlike previous investigations, no restrictions are imposed on the symbol of the differential–difference operator contained in the equation. For the investigated equation, we succeed to explicitly construct a multiparametric family of classical global solutions.

Comparison of energy-dependent and independent interactions-a case study

In the present text we construct velocity-dependent or equivalently energy-dependent potential (EDP) to an energy-independent nonlocal potential (EIP) of rank-1 by Taylor series method. The phase shifts for the nucleon-nucleon and alpha-nucleon systems are computed for these two potentials by exploiting the variable phase method and the Fredholm determinant, respectively. The velocity-dependent potential is found to be superior to central nonlocal interaction in generating the scattering phase shifts up to high energy region.

General Theory of Constructing Potential with Bound States in the Continuum

Abstract We present a general theory of potentials that support bound states at positive energies (bound states in the continuum). On the theoretical side, we prove that, for systems described by nonlocal potentials of the form $V(r,r^{\prime })$, bound states at positive energies are as common as those at negative energies. At the same time, we show that a local potential of the form $V(r)$ rarely supports a positive-energy bound state. On the practical side, we show how to construct a (naturally nonlocal) potential that supports an arbitrary normalizable state at an arbitrary positive energy. We demonstrate our theory with numerical examples both in momentum and coordinate spaces with emphasis on the important role played by nonlocal potentials. Finally, we discuss how to observe bound states at positive energies, and where to search for nonlocal potentials that may support them.

On Global Solutions of Two-Dimensional Hyperbolic Equations with General-Kind Nonlocal Potentials

In the case of one spatial independent variable, we study hyperbolic differential-difference equations with potentials represented as linear combinations of translations of the desired function along the spatial variable. The qualitative novelty of this investigation is that, unlike previous research, it is not assumed that the real part of the symbol of the differential-difference operator contained in the equation has a constant sign. Previously, it was possible to remove that substantial restriction (i.e., the specified sign constancy) only for the case where the nonlocal term (i.e., the translated potential) is unique. In the present paper, we consider the case of the general-kind one-variable nonlocal potential, i.e., equations with an arbitrary amount of translated terms. No commensurability assumptions are imposed on the translation lengths. The following results are presented: We find a condition relating the coefficients at the nonlocal terms of the investigated equation and the length of the translations, providing the global solvability of the investigated equation. Under this condition, we explicitly construct a three-parametric family of smooth global solutions of the investigated equation.

Taming Negative Ion Resonances Using Nonlocal Exchange-Correlation Functionals.

The characterization of negative ion resonances poses a fundamental challenge to density functional methods due to the unbound nature of resonances. To overcome this challenge, we propose one-particle nonlocal exchange-correlation (xc) potentials combining the exact-exchange (EXX) and the random phase approximation (RPA) correlation potentials. The negative ion resonances are identified by perturbing the real Hermitian nonlocal xc potentials using complex absorbing local potentials. Our studies show that the nonlocal EXX+RPA potential significantly enhances the description of positions and widths of negative ion resonance states compared to potentials that exclude dynamic polarization in RPA or include only EXX. The use of low-scaling algorithms simplifies the computation of the RPA potential, thereby providing a practical solution for resonance-state characterization within the density functional framework. A theoretical framework and the underlying assumptions required for combining real Hermitian nonlocal xc potentials with complex local potentials are discussed.

Possible Σc*Σ¯ molecular states

We investigate the possibility of deuteronlike Σc*Σ¯ bound states within the one-boson-exchange model and systematically analyze the effects of the contact range δ3(r→) potential, the tensor term from the vector-meson exchange, and nonlocal potentials due to the dependence on the sum of the initial and final state center-of-mass momenta. We find that the pion-exchange potential including the δ3(r→) term and the tensor term of the ρ-exchange potential exhibit comparable magnitudes but opposite signs for any S-wave baryon-antibaryon systems. For the Σc*Σ¯ system, it is most likely to form bound states with mass around 3.7 GeV in the I(JP)=0(2−) and 1(2−) channels. Published by the American Physical Society 2024

Density and energy dependent semi-microscopic optical potential of nucleon–nucleus elastic scattering

A nonlocal velocity-dependent optical potential was used to study the angular distribution of the differential cross section of nucleon–nucleus elastic scattering. The potential parameters were calculated by fitting elastic angular distributions over a wide range of energy (10-65 MeV) and mass (12-208). The real part of the local optical potential was calculated using the CDM3Y6-Paris interaction, whereas the imaginary part was a Woods-Saxon form factor and its derivative.

Partially deorbitalized meta-GGA

Mejia-Rodriguez and Trickey recently proposed a procedure for removing the explicit dependence of meta-GGA exchange-correlation energy functionals Exc on the kinetic energy density τ. We present a simple modification to this approach in which the exact Kohn-Sham τ is used as input for Exc but the functional derivative of τ with respect to the density ρ, required to calculate the potential term ∫d3r′δExc∕δτ(r′)∣ρ⋅δτ(r′)∕δρ(r), is evaluated using an approximate kinetic energy density functional. This ‘half-way’ strategy ensures that the Kohn-Sham potential is a local multiplicative function (as opposed to the non-local potential of a generalized Kohn-Sham approach) while preserving the inherent non-locality of the functional itself. Electronic structure codes can be easily modified to use the new method. We validate it by quantifying the accuracy of the predicted lattice parameters, bulk moduli, magnetic moments and cohesive energies of a large set of periodic solids. An unanticipated benefit of this method is to gauge the quality of approximate kinetic energy functionals by checking if the self-consistent solution is indeed at the variational minimum.

Ring-like partially nonlocal extreme wave of a (3+1)-dimensional NLS system with partially nonlocal nonlinearity and external potential

This paper aims to study extreme waves localized in (3+1)-dimensional space based on a (3+1)-dimensional nonautonomous partially nonlocal NLS system under the linear and parabolic potentials, which is simplified into a (2+1)-dimensional autonomous equation via a converting formula. Based on solutions of the autonomous NLS system, an approximate solution of ring-like extreme wave is found. Characteristics and evolution of ring-like extreme wave are explored in an exponential system. The influence of the magnitude and exponential parameters of diffraction on ring-like extreme wave is studied. Moreover, the impact of the Hermite polynomial parameter q, the radius parameter R and the thickness parameter w on ring-like extreme wave is also discussed.

Cauchy Problem with Summable Initial-Value Functions for Parabolic Equations with Translated Potentials

We study the Cauchy problem for differential–difference parabolic equations with potentials undergoing translations with respect to the spatial-independent variable. Such equations are used for the modeling of various phenomena not covered by the classical theory of differential equations (such as nonlinear optics, nonclassical diffusion, multilayer plates and envelopes, and others). From the viewpoint of the pure theory, they are important due to crucially new effects not arising in the case of differential equations and due to the fact that a number of classical methods, tools, and approaches turn out to be inapplicable in the nonlocal theory. The qualitative novelty of our investigation is that the initial-value function is assumed to be summable. Earlier, only the case of bounded (essentially bounded) initial-value functions was investigated. For the prototype problem (the spatial variable is single and the nonlocal term of the equation is single), we construct the integral representation of a solution and show its smoothness in the open half-plane. Further, we find a condition binding the coefficient at the nonlocal potential and the length of its translation such that this condition guarantees the uniform decay (weighted decay) of the constructed solution under the unbounded growth of time. The rate of this decay (weighted decay) is estimated as well.

On Global Solutions of Hyperbolic Equations with Positive Coefficients at Nonlocal Potentials

We study hyperbolic equations with positive coefficients at potentials undergoing translations with respect to the spatial independent variable. The qualitative novelty of the investigation is that the real part of the symbol of the differential-difference operator contained in the equation is allowed to change its sign. Earlier, only the case where the said sign is constant was investigated. We find a condition relating the coefficient at the nonlocal term of the investigated equation and the length of the translation, guaranteeing the global solvability of the investigated equation. Under this condition, we explicitly construct a three-parametric family of smooth global solutions of the investigated equation.

Microscopic bell-shape nonlocality: The case of proton scattering off 40Ca at 200 MeV

This work is part of an ongoing effort to build microscopicallydriven nonlocal optical potentials easily tracktable in scattering codes. Based on the separable ‘JνH’ structure proposed recently [1], where the potential can be cast as the product of a radial and nonlocality form factors, we investigate its angular dependence in momentum space. We find that scattering observables have a weak angular dependence between the momentum transfer q = k − k′, and K = (k + k′)/2. The study is focussed on proton elastic scattering off 40Ca at 200 MeV, where the JνH structure is found to be inadequate. We conclude that any improvement of the JνH structure of the potential can be made to the lowest order in multipole expansions in Kq−representation of the potential.

Nonlocal Potentials and Crystalline Order in One and Two Dimensions

We revisit the seminal 1968 proof of the absence of crystalline order in two dimensions and analyze the importance played in the quantum theorem by the assumption of local pair potentials. We relax the assumption of local potentials and consider instead nonlocal pair potentials. We show that the 1/k2 -singularity that occurs in the Bogoliubov inequality, which leads to no crystalline order in two dimensions for local potentials and nonzero temperatures, does not occur for nonlocal potentials. Accordingly, crystalline order in one and two dimensions cannot be ruled out for nonlocal pair potentials at finite temperatures.

A time splitting Chebyshev-Fourier spectral method for the time-dependent rotating nonlocal Schrödinger equation in polar coordinates

In this article, we propose a time splitting Chebyshev-Fourier spectral method to compute the dynamics of rotating nonlocal Schrödinger equation in polar coordinates. Since the rotation term is diagonalizable with Fourier spectral method in azimuthal direction, we split the Hamiltonian into a linear part, i.e., the Laplacian and rotation terms, and the nonlinear part. The linear part is discretized by Chebyshev-Fourier spectral method in space and integrated by Crank-Nicolson or exact matrix exponential. The nonlinear part is solved exactly in physical space, and the nonlocal potential, which is defined as convolution φ:=U⁎|ψ|2 with a singular kernel U, is computed via Kernel Truncation method with spectral accuracy. Then we construct a high order time splitting spectral method. Our scheme is spectrally accurate in space and of second/fourth order in the temporal direction. Extensive numerical results are presented to confirm the accuracy and efficiency for both the nonlocal potential and the wave function, together with one application to a two-component rotating dipolar Bose-Einstein condensates. In addition, we investigate the rotational symmetry preserving performance by a comprehensive comparison with existing scheme.

Direct and Inverse Problems for a Third Order Self-Adjoint Differential Operator with Periodic Boundary Conditions and Nonlocal Potential

Direct and Inverse Problems for a Third Order Self-Adjoint Differential Operator with Periodic Boundary Conditions and Nonlocal Potential

Light bullets in a nonlocal Rydberg medium with PT-symmetric moiré optical lattices

Realizing stable light bullets in high dimensions is a long-standing goal in the study of nonlinear optics. Here, we propose a scheme for the creation of stable light bullets (LBs) in a cold Rydberg atomic gas system with PT symmetry moiré optical lattices. Depending on the synergetic local and nonlocal Kerr nonlinearities, and PT symmetry moiré optical lattices, we obtain an appropriate nonlocal Rydberg long-range interaction and PT symmetry potential which can compensate for the diffraction and dispersion of the probe field. Numerical results show that different spatial and temporal distributions of LBs, including fundamental, multi-pole solitons, and vortical ones are uncovered, and their stabilities are evaluated through linear-stability analysis and direct simulation with perturbation. Furthermore, the solitons exhibit a quasi-elastic collision in the transversal direction during propagation. Our study provides a new route for manipulating LBs via controlled optical nonlinearities and PT symmetry moiré optical lattices in cold Rydberg gases.

Stability and adaptive evolution of higher-order vector vortex solitons in thermally nonlinear media with tunable transverse size

The stability of a higher-order vector vortex (HVV) soliton, comprised of a fundamental-mode beam incoherently coupling with a higher-order vortex beam, is investigated in a thermally nonlocal nonlinear medium. By linear stability analysis, we find the broad and stable interval for HVV solitons and the monotonically increasing relation between the stable interval and the vortical topological charge. The adaptive evolution of this stable HVV soliton in the longitudinally-inhomogeneous medium with tunable transverse size is also discussed. The smooth transition and controllable propagation of HVV solitons, and the oscillating behaviour of HVV breathers under different nonlocal potentials are investigated. In addition, the instability suppression of the HVV soliton by means of gradual change of transverse size is also found.