Repulsive Case Research Articles

Positive and sign-changing solutions for the nonlinear Schrödinger systems with synchronization and separation

In this paper, we consider the following nonlinear Schrödinger system: { − Δ u + P ( x ) u = μ 1 u 3 + β u v 2 , x ∈ R 3 , − Δv + Q ( x ) v = μ 2 v 3 + β u 2 v , x ∈ R 3 , where P ( x ) , Q ( x ) are positive radial potentials, μ 1 , μ 2 > 0 , β ∈ R is a coupling constant. We constructed a new type of solutions which are different from the ones obtained by Peng and Wang [Arch Rational Mech Anal. 2013;208:305–339]. This new family of solutions to system have a more complex concentration structure and are centered at the points lying on the top and the bottom circles of a cylinder with height h. Meanwhile, we examine the effect of nonlinear coupling on the solution structure. In the repulsive case, we construct an unbounded sequence of non-radial positive vector solutions of segregated type. In the attractive case, we construct an unbounded sequence of non-radial positive vector solutions of synchronized type. Moreover, we prove that there exist infinitely many sign-changing solutions whose energy can be arbitrarily large.

Stretched Central Double Bonds in Dialumene and Disilene by Amino Substituents: A Case of Lone Pair Repulsion.

There have been remarkable advances in the syntheses and applications of groups 13 and 14 homonuclear ethene analogues. However, successes are largely limited to aryl- and/or silyl-substituted species. Analogues bearing two or more heteroatoms are still scarce. In this work, we employed the block-localized wavefunction (BLW) method at the density functional theory (DFT) level to study dialumene and disilene bearing two amino substituents whose optimal geometries exhibit significantly stretched central M=M (M = Al or Si) double bonds compared with aryl- and/or silyl-substituted species. Computational analyses showed that the repulsion between the lone electron pairs of amino substituents and M=M π bond plays a critical role in the elongation of the M=M bonds. Evidently, replacing the substituent groups -NH2 with -BH2 can enhance the planarity and shorten the central double bonds due to the absence of lone pair electrons in BH2.

Gaussian variational method to Fermi-Hubbard model in one and two dimensions

The study of ground-state properties of the Fermi-Hubbard model is a long-lasting task in the research of strongly correlated systems, and is crucial for the understanding of notable quantum phenomena including superconductivity and magnetism. Owing to the exponentially growing complexity of the system, a quantitative analysis usually demands high computational cost and is restricted to small samples, especially in two or higher dimensions. Here, we introduce a variational method in the frame of fermionic Gaussian states, and obtain the ground states of one- and two-dimensional attractive Hubbard models via imaginary-time evolution. We calculate the total energy and benchmark the results in a wide range of interaction strength and filling factor with those obtained via exact two-body results, the density matrix renormalization group based on matrix product states (MPS), and projector Quantum Monte Carlo (QMC) method. For both 1D and 2D cases, the Gaussian variational method presents accurate results for total energy with a maximum systematic error in the intermediate interaction region. The accuracy of these results has negligible dependence on the system size. We further calculate the double occupancy and find excellent agreement with MPS and QMC, as well as the experimental results of cold quantum gases in optical lattices. The results suggest that the Gaussian pairing state is a good approximation to the ground states of attractive Hubbard model, in particular in the strong and weak coupling limits. Moreover, we generalize the method to the attractive Hubbard model with a finite spin-polarization, which can be mapped to the repulsive interaction case via particle-hole transformation, and obtain accurate results of ground state energy and double occupancy. Our work demonstrates the ability of the Gaussian variational method to extract ground state properties of strongly correlated many-body systems with negligible computational cost, especially of large size and in higher dimensions.

Resonant scattering between dark matter and baryons: Revised direct detection and CMB limits

Traditional dark matter models, e.g., weakly interacting massive particles (WIMPs), assume dark matter (DM) is weakly coupled to the standard model so that elastic scattering between dark matter and baryons can be described perturbatively by the Born approximation; most direct detection experiments are analyzed according to that assumption. We show that when the fundamental DM-baryon interaction is attractive, dark matter-nucleus scattering is nonperturbative in much of the relevant parameter range. The cross section exhibits rich resonant behavior with a highly nontrivial dependence on atomic mass; furthermore, the extended rather than pointlike nature of nuclei significantly impacts the cross sections and must therefore be properly taken into account. The repulsive case also shows significant departures from perturbative predictions and also requires full numerical calculation. These nonperturbative effects change the boundaries of exclusion regions from existing direct detection, astrophysical and CMB constraints. Near a resonance value of the parameters the typical velocity-independent Yukawa behavior, $\ensuremath{\sigma}\ensuremath{\sim}{v}^{0}$, does not apply. We take the nontrivial velocity dependence into account in our analysis, however it turns out that this more accurate treatment has little impact on limits given current constraints. Correctly treating the extended size of the nucleus and doing an exact integration of the Schr\"odinger equation does have a major impact relative to past analyses based on the Born approximation and naive form factors, so those improvements are essential for interpreting observational constraints. We report the corrected exclusion regions superseding previous limits from XQC, CRESST Surface Run, CMB power spectrum and extensions with Lyman-$\ensuremath{\alpha}$ and Milky Way satellites, and Milky Way gas clouds. Some limits become weaker, by an order of magnitude or more, than previous bounds in the literature which were based on perturbation theory and pointlike sources, while others become stronger. Gaps which open by correct treatment of some particular constraint can sometimes be closed using a different constraint. We also discuss the dependence on mediator mass and give approximate expressions for the velocity dependence near a resonance. Sexaquark ($uuddss$) DM with mass around 2 GeV, which exchanges QCD mesons with baryons, remains unconstrained for most of the parameter space of interest. A statement in the literature that a DM-nucleus cross section larger than ${10}^{\ensuremath{-}25}\text{ }\text{ }{\mathrm{cm}}^{2}$ implies dark matter is composite, is corrected.

NLS ground states on the half-line with point interactions

We investigate the existence and the uniqueness of NLS ground states of fixed mass on the half-line in the presence of a point interaction at the origin. The nonlinearity is of power type, and the regime is either L^{2}-subcritical or L^{2}-critical, while the point interaction is either attractive or repulsive. In the L^{2}-subcritical case, we prove that ground states exist for every mass value if the interaction is attractive, while ground states exist only for sufficiently large masses if the interaction is repulsive. In the latter case, if the power is less or equal to four, ground states coincide with the only bound state. If instead, the power is greater than four, then there are values of the mass for which two bound states exist, and neither of the two is a ground state, and values of the mass for which two bound states exist, and one of them is a ground state. In the L^{2}-critical case, we prove that ground states exist for masses strictly below a critical mass value in the attractive case, while ground states never exist in the repulsive case.

Solution of master equations by fermionic-duality: Time-dependent charge and heat currents through an interacting quantum dot proximized by a superconductor

We analyze the time-dependent solution of master equations by exploiting fermionic duality, a dissipative symmetry applicable to a large class of open systems describing quantum transport. Whereas previous studies mostly exploited duality relations after partially solving the evolution equations, we here systematically exploit the invariance under the fermionic duality mapping from the very beginning when setting up these equations. Moreover, we extend the resulting simplifications -so far applied to the local state evolution- to non-local observables such as transport currents. We showcase the exploitation of fermionic duality for a quantum dot with strong interaction -covering both the repulsive and attractive case- proximized by contact with a large-gap superconductor which is weakly probed by charge and heat currents into a wide-band normal-metal electrode. We derive the complete time-dependent analytical solution of this problem involving non-equilibrium Cooper pair transport, Andreev bound states and strong interaction. Additionally exploiting detailed balance we show that even for this relatively complex problem the evolution towards the stationary state can be understood analytically in terms of the stationary state of the system itself via its relation to the stationary state of a dual system with inverted Coulomb interaction, superconducting pairing and applied voltages.

Multi-Peak Solutions for Coupled Nonlinear Schrödinger Systems in Low Dimensions

In this paper, we construct the solutions to the following nonlinear Schrödinger system -ϵ2Δu+P(x)u=μ1up+βup-12vp+12inRN,-ϵ2Δv+Q(x)v=μ2vp+βup+12vp-12inRN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} -\\epsilon ^{2}\\Delta u+P(x)u= \\mu _{1} u^{p}+\\beta u^{\\frac{p-1}{2}}v^{\\frac{p+1}{2}} \\ \\ \\ \ ext {in} \\ \\ \\mathbb {R}^{N},\\\\ -\\epsilon ^{2}\\Delta v+Q(x)v= \\mu _{2} v^{p}+\\beta u^{\\frac{p+1}{2}}v^{\\frac{p-1}{2}} \\ \\ \\ \ ext {in} \\ \\ \\mathbb {R}^{N}, \\end{array}\\right. } \\end{aligned}$$\\end{document}where 3< p<+infty , Nin {1,2}, epsilon >0 is a small parameter, the potentials P, Q satisfy 0<P_{0} le P(x)le P_{1} and Q(x) satisfies 0<Q_{0} le Q(x)le Q_{1}. We construct the solution for attractive and repulsive cases. When x_{0} is a local maximum point of the potentials P and Q and P(x_{0})=Q(x_{0}), we construct k spikes concentrating near the local maximum point x_{0}. When x_{0} is a local maximum point of P and overline{x}_{0} is a local maximum point of Q, we construct k spikes of u concentrating at the local maximum point x_{0} and m spikes of v concentrating at the local maximum point overline{x}_{0} when x_{0}ne overline{x}_{0}. This paper extends the main results established by Peng and Wang (Arch Ration Mech Anal 208:305–339, 2013) and Peng and Pi (Discrete Contin Dyn Syst 36:2205–2227, 2016), where the authors considered the case N=3, p=3.

Tuning attraction and repulsion between active particles through persistence

We consider the interplay between persistent motion, which is a generic property of active particles, and a recoil interaction which causes particles to jump apart on contact. The recoil interaction exemplifies an active contact interaction between particles, which is inelastic and is generated by the active nature of the constituents. It is inspired by the “shock” dynamics of certain microorganisms, such as Pyramimonas octopus, and always generates an effective repulsion between a pair of passive particles. Highly persistent particles can be attractive or repulsive, according to the shape of the recoil distribution. We show that the repulsive case admits an unexpected transition to attraction at intermediate persistence lengths, that originates in the advective effects of persistence. This allows active particles to fundamentally change the collective effect of active interactions amongst them, by varying their persistence length.

Normalized solutions for a fourth-order Schrödinger equation with a positive second-order dispersion coefficient

In this paper, we study normalized solutions to a fourth-order Schrödinger equation with a positive second-order dispersion coefficient in the mass supercritical regime. Unlike the well-studied case where the second-order term is zero or negative, the geometrical structure of the corresponding energy functional changes dramatically and this makes the solution set richer. Under suitable control of the second-order dispersion coefficient and mass, we find at least two radial normalized solutions, a ground state and an excited state, together with some asymptotic properties. It is worth pointing out that in the considered repulsive case, the compactness analysis of the related Palais-Smale sequences becomes more challenging. This forces the implementation of refined estimates of the Lagrange multiplier and the energy level to obtain normalized solutions.

Functional renormalization of spinless triangular-lattice fermions: N-patch vs. truncated-unity scheme

We study competing orders of spinless fermions in the triangular-lattice Hubbard model with nearest-neighbor interaction. We calculate the effective, momentum-resolved two-particle vertex in an unbiased way in terms of the functional renormalization group method and compare two different schemes for the momentum discretization, one based on dividing the Fermi surface into patches and one based on a channel decomposition. We study attractive and repulsive nearest-neighbor interaction and find a competition of pairing and charge instabilities. In the attractive case, a Pomeranchuk instability occurs at Van Hove filling and f-wave and p-wave pairing emerge when the filling is reduced. In the repulsive case, we obtain a charge density wave at Van Hove filling and extended p-wave pairing with reduced filling. The p-wave pairing solution is doubly degenerate and can realize chiral p+ip superconductivity with different Chern numbers in the ground state. We discuss implications for strongly correlated spin-orbit coupled hexagonal electron systems such as moiré heterostructures.Graphic

Vibronic recovering of functionality of quantum cellular automata based on bi-dimeric square cells with violated condition of strong Coulomb repulsion.

Strong Coulomb repulsion between the two charges in a square planar mixed-valence cell in quantum cellular automata (QCA) allows us to encode the binary information in the two energetically beneficial diagonal distributions of the electronic density. In this article, we pose a question: to what extent is this condition obligatory for the design of the molecular cell? To answer this question, we examine the ability to use a square-planar cell composed of one-electron mixed valence dimers to function in QCA in a general case when the intracell Coulomb interaction U is not supposed to be extremely strong, which means that it is comparable with the characteristic electron transfer energy (violated strong U limit). Using the two-mode vibronic model treated within the semiclassical (adiabatic) and quantum-mechanical approaches, we demonstrate that strong vibronic coupling is able to create a considerable barrier between the two diagonal-type charge configurations, thus ensuring bistability and polarizability of the cells even if the Coulomb barrier is not sufficient. The cases of weak and moderate Coulomb repulsion and strong vibronic coupling are exemplified by consideration of the cation radicals of the two polycyclic derivatives of norbornadiene [C12H12]+ and [C17H16]+ with the terminal C=C chromophores playing the role of redox sites. By using the detailed ab initio data, we reveal the main characteristics of the bi-dimeric cells composed of these molecules and illustrate the pronounced effect of the vibronic recovery clearly manifesting itself in the shape of the cell-cell response function. Revealing such "vibronic recovery" of strong localization when the strong U limit is violated suggests a way to a significant expansion of the class of molecular systems suitable as QCA cells.

Molecular models of hematite, goethite, kaolinite, and quartz: Surface terminations, ionic interactions, nano topography, and water coordination

In light of the modern needs in mineral processing and extractive metallurgy, advanced knowledge about the surface chemistry of minerals is essential, especially their interactions with water. These interactions are intimately related to several processes in mineral technology, such as flotation, wet grinding, dewatering, and leaching. Hematite, goethite, quartz, and kaolinite are the main minerals in iron ore processing circuits. Thus, understanding their differences is fundamental to making lower-grade iron ores processing worthwhile. In this work, we studied the surface physical chemistry of these minerals by molecular modeling and simulation, focusing on surface terminations, protonation degrees, ionic interactions, nano topography, and water coordination. We demonstrate that fundamental differences exist between the terminations of the same mineral, particularly for quartz (100) and hematite (001) surfaces. Iron minerals exhibit strong interactions with water, mainly due to the coordination of surface iron atoms by aquo ligands and their relatively low molecular-scale roughness, in most cases. Such characteristics help explain the behavior of adsorption of reagents and the difficulties in dewatering and drying iron ore. In contrast, the silicate minerals present lower water structuring. We also show how neutral surfaces behave towards ions, cases of both attraction and repulsion were observed and attributed to the surfaces' topmost atomic layering. This work provides novel insights which can connect the fundamentals of surface phenomena with process engineering. Additionally, we provide molecular models to help in related computational studies.

Synchronized and segregated vector solutions for nonlinear fractional Schrödinger systems

We are concerned with the qualitative analysis of synchronized and segregated vector solutions for a class of nonlinear Schrödinger systems driven by the fractional Laplace operator. We investigate both attractive and repulsive cases. In the attractive case, we construct an unbounded sequence of non‐radial positive vector solutions whose components are synchronized, while in the repulsive case, we prove the existence of an unbounded sequence of non‐radial positive vector solutions with segregated components. A key role in the arguments developed in this paper is played by the Lyapunov‐Schmidt reduction method.

Counterflow dynamics of two correlated impurities immersed in a bosonic gas

The counterflow dynamics of two correlated impurities in a double well coupled to a one-dimensional bosonic medium is explored. We determine the ground-state phase diagram of the system according to the impurity-medium entanglement and the impurities' two-body correlations. Specifically, bound impurity structures reminiscent of bipolarons for strong attractive couplings as well as configurations with two clustered or separated impurities in the repulsive case are identified. The interval of existence of these phases depends strongly on the impurity-impurity interactions and external confinement of the medium. Accordingly the impurities' dynamical response, triggered by suddenly ramping down the central potential barrier, is affected by the medium's trapping geometry. In particular, for a box-confined medium, repulsive impurity-medium couplings lead, due to attractive induced interactions, to the localization of the impurities around the trap center. In contrast, for a harmonically trapped medium the impurities perform a periodic collision and expansion dynamics further interpreted in terms of a two-body effective model. Our findings elucidate the correlation aspects of the collisional physics of impurities which should be accessible in recent cold-atom experiments.

Numerical and experimental evaluation of the magnetic interaction for frequency up-conversion in piezoelectric vibration energy harvesters

The purpose of this work is to improve the modelling process for the application of permanent magnets in a frequency up-conversion (FuC) mechanism for piezoelectric energy harvesters. More specifically, the aim is to avoid the burdensome finite element analyses (FEA) in the framework of electromechanical devices design. The analytical calculations are compared with experimental tests conducted by an ad-hoc set up and with FEA. After investigations on the interaction, an application of FuC mechanism is proposed on a meso-scale case study in which a low frequency seismic mass (LFM) interacts non-linearly, due to magnetic field, with an high frequency piezoelectric vibration energy harvester (PVEH). Numerical simulations have been carried out in the time domain (step-by-step analysis) under a harmonic low-frequency input acceleration signal. The peculiar behavior, due to non-linear dynamics, is investigated in both the repulsive and the attractive configurations of the magnets. The results confirm the effectiveness of magnetic FuC and show that the repulsive case allows the device to recover a larger amount of energy than the attractive configuration.

Global existence and boundedness of solutions to a parabolic attraction-repulsion chemotaxis system in [formula omitted]: The repulsive dominant case

This paper deals with the initial value problem for a parabolic attraction-repulsion chemotaxis system in R2:{∂tu=Δu−∇⋅(u∇(β1v1−β2v2)),t>0,x∈R2,∂tvj=Δvj−λjvj+u,t>0,x∈R2(j=1,2),u(0,x)=u0(x),vj(0,x)=vj0(x),x∈R2(j=1,2) with positive parameters β1, β2, λ1, λ2 and nonnegative initial data u0, v10, v20. It is well known that the sign of β1−β2 and the mass on u0 play a crucial role in the global boundedness of the system. Specifically, in the attractive dominant case β1>β2, the uniform boundedness of nonnegative solutions has been guaranteed under the condition ∫R2u0dx<4π. Also, in the balance case β1=β2, it has been proven that every nonnegative solution is bounded uniformly in time without any restriction on the mass u0. In this paper, we discuss the global existence and boundedness of nonnegative solutions to the system in the repulsive dominant case β1<β2.

Ranked diffusion, delta Bose gas, and Burgers equation.

We study the diffusion of N particles in one dimension interacting via a drift proportional to their rank. In the attractive case (self-gravitating gas) a mapping to the Lieb-Liniger quantum model allows one to obtain stationary time correlations, return probabilities, and the decay rate to the stationary state. The rank field obeys a Burgers equation, which we analyze. It allows one to obtain the stationary density at large N in an external potential V(x) (in the repulsive case). In the attractive case the decay rate to the steady state is found to depend on the initial condition if its spatial decay is slow enough. Coulomb gas methods allow one to study the final equilibrium at large N.

Pairing and Pair Superfluid Density in One-Dimensional Two-Species Fermionic and Bosonic Hubbard Models.

We use unbiased computational methods to elucidate the onset and properties of pair superfluidity in two-species fermionic and bosonic systems with onsite interspecies attraction loaded in a uniform, i.e., with no confining potential, one-dimensional optical lattice. We compare results from quantum MonteCarlo (QMC) and density matrix renormalization group (DMRG), emphasizing the one-to-one correspondence between the Drude weight tensor, calculated with DMRG, and the various winding numbers extracted from the QMC. Our results show that, for any nonvanishing attractive interaction, pairs form and are the sole contributors to superfluidity; there are no individual contributions due to the separate species. For weak attraction, the pair size diverges exponentially, i.e., Bardeen-Cooper-Schrieffer (BCS) pairing, requiring huge systems to bring out the pair-only nature of the superfluid. This crucial property is largely overlooked in many studies, thereby misinterpreting the origin and nature of the superfluid. We compare and contrast this with the repulsive case and show that the behavior is very different, contradicting previous claims about drag superfluidity and the symmetry of properties for attractive and repulsive interactions. Finally, our results show that the situation is similar for soft-core bosons: superfluidity is due only to pairs, even for the smallest attractive interaction strength compatible with the largest system sizes that we could attain.

Effects of the random single-ion anisotropy on the spin-1 Blume-Emery-Griffiths model

The Blume-Emery-Griffiths (BEG) model is considered on the Bethe lattice (BL) in terms of exact recursion relations (ERR) under the effect of a crystal field (D) which was either turned on or off randomly for a given probability. The repulsive case of biquadratic exchange interaction ((K < 0)) between the nearest-neighbor (NN) spins is assumed and the effects of changing the coordination number are also investigated. The thermal variation of the order-parameters, i.e. dipole and quadrupole moments, is examined to obtain the phase diagrams. Very rich phase diagrams with the second- and first-order phase transition lines, tricritical, bicritical and critical end points and the occurrence of reentrant behavior are observed. Different phase regions were explored which include the ferromagnetic (F), paramagnetic (P), staggered quadrupolar (SQ) and ferrimagnetic (FI) phases.

Core-halo mass relation in scalar field dark matter models and its consequences for the formation of supermassive black holes

Scalar-field dark matter (SFDM) halos exhibit a core-envelope structure with soliton-like cores and CDM-like envelopes. Simulations without self-interaction (free-field case) report a core-halo mass relation $M_c\propto M_{h}^{\beta}$, with either $\beta=1/3$ or $\beta=5/9$, which can be understood if core and halo obey certain energy or velocity scalings. We extend the core-halo mass relations to include SFDM with self-interaction (SI), either repulsive or attractive, and investigate its implications for the gravitational instability and collapse of solitonic cores, leading to supermassive black hole (SMBH) formation. For SFDM parameters that make $\sim$ Kpc-sized cores and CDM-like structure formation on large scales but suppressed on small scales, cores are stable for all galactic halos of interest, from the free-field to the repulsive SI limit. For attractive SI, however, halos masses $M_h\sim (10^{10}-10^{12}) M_\odot$ have cores that collapse to SMBHs with $M_{SMBH}\sim 10^{6}-10^8 M_\odot$, as observations seem to require, while smaller-mass halos have stable cores, for particle masses $m\simeq (2.14\times 10^{-22}-9.9\times 10^{-20})\rm{eV}/c^2$, if the free-field has $\beta=1/3$, or $m = 2.23\times 10^{-21}-1.7\times 10^{-18}\rm{eV}/c^2$, if $\beta=5/9$. For free-field and repulsive cases, however, if previous constraints on particle parameters are relaxed to allow much smaller (sub-galactic scale) cores, then halos can also form SMBHs, for the same range of halo and BH masses, as long as $\beta = 5/9$ is correct for the free-field. In that case, structure formation in SFDM would be largely indistinguishable from Cold Dark Matter (CDM). Such SFDM models might not resolve the small-scale structure problems of CDM, but they would explain the formation of SMBHs quite naturally. Since CDM, itself, has not yet been ruled out, such SFDM models must also be viable (Abbreviated).