Finite-range Potential Research Articles

The Spectrum of Discrete Schrödinger Operator on a Three Dimensional Triangular Lattice with a Finite-range Potential

The Spectrum of Discrete Schrödinger Operator on a Three Dimensional Triangular Lattice with a Finite-range Potential

Kinetic Theory for the Low-Density Lorentz Gas

The Lorentz gas is one of the simplest and most widely-studied models for particle transport in matter. It describes a cloud of non-interacting gas particles in an infinitely extended array of identical spherical scatterers. The model was introduced by Lorentz in 1905 who, following the pioneering ideas of Maxwell and Boltzmann, postulated that in the limit of low scatterer density, the macroscopic transport properties of the model should be governed by a linear Boltzmann equation. The linear Boltzmann equation has since proved a useful tool in the description of various phenomena, including semiconductor physics and radiative transfer. A rigorous derivation of the linear Boltzmann equation from the underlying particle dynamics was given, for random scatterer configurations, in three seminal papers by Gallavotti, Spohn and Boldrighini-Bunimovich-Sinai. The objective of the present study is to develop an approach for a large class of deterministic scatterer configurations, including various types of quasicrystals. We prove the convergence of the particle dynamics to transport processes that are in general (depending on the scatterer configuration) not described by the linear Boltzmann equation. This was previously understood only in the case of the periodic Lorentz gas through work of Caglioti-Golse and Marklof-Strömbergsson. Our results extend beyond the classical Lorentz gas with hard sphere scatterers, and in particular hold for general classes of spherically symmetric finite-range potentials. We employ a rescaling technique that randomises the point configuration given by the scatterers’ centers. The limiting transport process is then expressed in terms of a point process that arises as the limit of the randomised point configuration under a certain volume-preserving one-parameter linear group action.

Two-particle bound states on a lattice

Two-particle lattice states are important for physics of magnetism, superconducting oxides, and cold quantum gases. The quantum-mechanical lattice problem is exactly solvable for finite-range interaction potentials. A two-body Schrödinder equation can be reduced to a system of linear equations whose numbers scale with the number of interacting sites. For the simplest cases such as on-site or nearest-neighbor attractions, many pair properties can be derived analytically, although final expressions can be quite complicated. In this work, we systematically investigate bound pairs in one-, two-, and three-dimensional lattices. We derive pairing conditions, plot phase diagrams, and compute energies, effective masses and radii. Along the way, we analyze nontrivial physical effects such as light pairs and the dependence of binding thresholds on pair momenta. At the end, we discuss the preformed-pair mechanism of superconductivity and stability of many-pair systems against phase separation. The paper is a combination of original work and pedagogical tutorial.

Universal correlations along the BEC-BCS crossover

Universality of the long-distance behavior across the Bardeen–Cooper–Schrieffer (BEC)-Bose–Einstein condensate (BCS) smooth transition for two-body density correlation functions and the Cooper-pair probability density is demonstrated in a balanced mixture of a two-component Fermi gas at T = 0. It is numerically shown at the mean-field level that these two-body quantities exhibit an exponential decay in terms of the chemical potential and the low-energy behavior of the gap. A general expression is found for the two-body distributions holding for different features of finite-range potentials, such as divergences at the origin, discontinuities at a finite radius, power-law decay, and exponential decay. The correlation length characterizing the long-distance behavior unravels the dependence on the energy needed to break pairs along the BEC-BCS crossover, a quantity meaningful to the stability of the many-body state.

Logarithmic Schrödinger equations in infinite dimensions

We study the logarithmic Schrödinger equation with a finite range potential on RZd. Through a ground-state representation, we associate and construct a global Gibbs measure and show that it satisfies a logarithmic Sobolev inequality. We find estimates on the solutions in arbitrary dimension and prove the existence of weak solutions to the infinite-dimensional Cauchy problem.

Fission barriers of superheavy nuclei for emitted fragment isotopes near proton magic numbers

Fission barriers for supposed doubly magic $^{298}\mathrm{Fl}$ are calculated using a specialized binary macroscopic-microscopic method. The microscopic effects are obtained from the developed binary Strutinsky shell correction method. The main input data are the proton and neutron energy levels calculated with the deformed two-center shell model. One is able in this way to provide the microscopic transition via the level schemes, from the parent to two overlapped and finally separated level schemes. The macroscopic part is acquired by the deformed charged liquid drop model with the double Yukawa-plus-exponential finite range potential. The method is applied to the reactions accompanied by isotopes of Sn and Pb within the fission channels, as being favored by strongly negative shell corrections.

Local Central Limit Theorem for Long-Range Two-Body Potentials at Sufficiently High Temperatures

Dobrushin and Tirozzi [14] showed that, for a Gibbs measure with the finite-range potential, the Local Central Limit Theorem is implied by the Integral Central Limit Theorem. Campanino, Capocaccia, and Tirozzi [7] extended this result for a family of Gibbs measures for long-range pair potentials satisfying certain conditions. We are able to show for a family of Gibbs measures for long-range pair potentials not satisfying the conditions given in [7], that at sufficiently high temperatures, if the Integral Central Limit Theorem holds for a given sequence of Gibbs measures, then the Local Central Limit Theorem also holds for the same sequence. We also extend [7] when the state space is general, provided that it is equipped with a finite measure.

Singular Behavior of the Macroscopic Quantity Near the Boundary for a Lorentz-Gas Model with the Infinite-Range Potential

Possibility of the diverging gradient of the macroscopic quantity near the boundary is investigated by a mono-speed Lorentz-gas model, with a special attention to the regularizing effect of the grazing collision for the infinite-range potential on the velocity distribution function (VDF) and its influence on the macroscopic quantity. By careful numerical analyses of the steady one-dimensional boundary-value problem, it is confirmed that the grazing collision suppresses the occurrence of a jump discontinuity of the VDF on the boundary. However, as the price for that regularization, the collision integral becomes no longer finite in the direction of the molecular velocity parallel to the boundary. Consequently, the gradient of the macroscopic quantity diverges, even stronger than the case of the finite-range potential. A conjecture about the diverging rate in approaching the boundary is made as well for a wide range of the infinite-range potentials, accompanied by the numerical evidence.

Causality and dimensionality in geometric scattering

The scattering matrix which describes low-energy, non-relativistic scattering of spin-1/2 fermions interacting via finite-range potentials can be obtained from a geometric action principle in which space and time do not appear explicitly [1]. In the case of zero-range forces, causality leads to constraints on scattering trajectories in the geometric picture. The effect of spatial dimensionality is also investigated by considering scattering in two and three dimensions. In the geometric formulation it is found that dimensionality is encoded in the phase of the harmonic potential that appears in the geometric action.

Scattering hypervolume of spin-polarized fermions

We analyze the collision of three identical spin-polarized fermions at zero collision energy, assuming arbitrary finite-range potentials, and define the corresponding three-body scattering hypervolume ${D}_{F}$. The scattering hypervolume $D$ was first defined for identical bosons in 2008. It is the three-body analog of the two-body scattering length. We solve the three-body Schr\"odinger equation asymptotically when the three fermions are far apart or one pair and the third fermion are far apart, deriving two asymptotic expansions of the wave function. Unlike the case of bosons for which $D$ has the dimension of length to the fourth power, here the ${D}_{F}$ we define has the dimension of length to the eighth power. We then analyze the interaction energy of three such fermions with momenta $\ensuremath{\hbar}{\mathbf{k}}_{1}$, $\ensuremath{\hbar}{\mathbf{k}}_{2}$, and $\ensuremath{\hbar}{\mathbf{k}}_{3}$ in a large periodic cubic box. The energy shift due to ${D}_{F}$ is proportional to ${D}_{F}/{\mathrm{\ensuremath{\Omega}}}^{2}$, where $\mathrm{\ensuremath{\Omega}}$ is the volume of the box. We also calculate the shifts of energy and pressure of spin-polarized Fermi gases due to a nonzero ${D}_{F}$ and the three-body recombination rate of spin-polarized ultracold atomic Fermi gases at finite temperatures.

Adiabatic expressions for the wave function of an electron in a finite-range potential and an intense low-frequency laser pulse

The wave function of an electron interacting with a finite-range potential and an intense low-frequency laser pulse is analyzed within the adiabatic approximation. The closed analytic form for the wave function, which includes the rescattering corrections, is obtained with quasiclassical accuracy for an electron in both the initial bound and continuum states. We discuss the parametrizations of amplitudes of fundamental strong-field processes in terms of laser and binding-potential parameters. Based on the analytic results for the adiabatic wave functions, we develop the perturbation theory in an additional weak field. The modification of high-order harmonic generation amplitude caused by a weak extreme ultraviolet pulse is discussed in the first order of the perturbation theory.

Delay time and persistent oscillations for a shifted quantum shutter

We derive an exact analytical solution to the time-dependent Schrödinger equation based on a resonant state expansion, to explore the time-evolution of cutoff plane waves scattered by finite range potentials, within a shifted quantum shutter model. The latter allows to control both, the position of the shutter and an hypothetical detector, which are initially separated by a distance ΔX. The dynamical advance-time (negative delay-time), ΔT, is explored for the particular case of a delta potential well, by measuring the difference of the first maxima of the time-diffraction pattern of the probability density, corresponding to the free and delta potential cases. We show that in general, ΔT exhibits a monotonic behaviour as a function of ΔX, and derive a simple formula for the timescale by using the symmetry and rescaling properties of the dynamical solution. We demonstrate that below a critical value of the shutter-detector separation, ΔX c , the monotonic behaviour of ΔT is hindered due to an oscillatory phenomena of the probability density, known as persistent oscillations, that drastically distorts the maxima used in the measurement process. These persistent oscillations are periodic Rabi-type oscillations that arise from an interplay between the incidence energy of the initial quantum wave and the bound state of the system, and their frequency play an important role in the dynamics of the delay time.

Three-body structure of B19 : Finite-range effects in two-neutron halo nuclei

The structure and $B(E1)$ transition strength of $^{19}$B are investigated in a $^{17}\text{B}+n+n$ model, triggered by a recent experiment showing that $^{19}$B exhibits a well pronounced two-neutron halo structure. Preliminary analysis of the experimental data was performed by employing contact $n$-$n$ interactions, which are known to underestimate the $s$-wave content in other halo nuclei such as $^{11}$Li. In the present work, the three-body hyperspherical formalism with finite-range two-body interactions is used to describe $^{19}$B. In particular, two different finite-range $n$-$n$ interactions will be used, as well as a simple central Gaussian potential whose range is progressively reduced. The purpose is to determine the main properties of the nucleus and investigate how they change when using contact-like $n$-$n$ potentials. Special attention is also paid to the dependence on the prescription used to account for three-body effects, i.e., a three-body force or a density-dependent $n$-$n$ potential. We have found that the three-body model plus finite-range potentials provide a description of $^{19}$B consistent with the experimental data. The results are essentially independent of the short-distance details of the two-body potentials, giving rise to an $(s_{1/2})^2$ content of about 55%, clearly larger than the initial estimates. Very little dependence has been found as well on the prescription used for the three-body effects. The total computed $B(E1)$ strength is compatible with the experimental result, although we slightly overestimate the data around the low-energy peak of the $dB(E1)/d\varepsilon$ distribution. Finally, we show that a reduction of the $n$-$n$ interaction range produces a significant reduction of the $s$-wave contribution, which then should be expected in calculations using contact interactions.

The Feynman–Kac Representation and Dobrushin–Lanford–Ruelle States of a Quantum Bose-Gas

This paper focuses on infinite-volume bosonic states for a quantum particle system (a quantum gas) in Rd. The kinetic energy part of the Hamiltonian is the standard Laplacian (with a boundary condition at the border of a ‘box’). The particles interact with each other through a two-body finite-range potential depending on the distance between them and featuring a hard core of diameter a>0. We introduce a class of so-called FK-DLR functionals containing all limiting Gibbs states of the system. As a justification of this concept, we prove that for d=2, any FK-DLR functional is shift-invariant, regardless of whether it is unique or not. This yields a quantum analog of results previously achieved by Richthammer.

Decorrelation of a class of Gibbs particle processes and asymptotic properties of U-statistics

AbstractWe study a stationary Gibbs particle process with deterministically bounded particles on Euclidean space defined in terms of an activity parameter and non-negative interaction potentials of finite range. Using disagreement percolation, we prove exponential decay of the correlation functions, provided a dominating Boolean model is subcritical. We also prove this property for the weighted moments of a U-statistic of the process. Under the assumption of a suitable lower bound on the variance, this implies a central limit theorem for such U-statistics of the Gibbs particle process. A by-product of our approach is a new uniqueness result for Gibbs particle processes.

Complex Density of Continuum States in Resonant Quantum Tunneling

We introduce a complex-extended continuum level density and apply it to one-dimensional scattering problems involving tunneling through finite-range potentials. We show that the real part of the density is proportional to a real "time shift" of the transmitted particle, while the imaginary part reflects the imaginary time of an instantonlike tunneling trajectory. We confirm these assumptions for several potentials using the complex scaling method. In particular, we show that stationary points of the potentials give rise to specific singularities of both real and imaginary densities which represent close analogues of excited-state quantum phase transitions in bound systems.

Transient quantum beats, Rabi oscillations, and delay time of modulated matter waves

Transient phenomena of phase modulated cut-off wavepackets are explored by deriving an exact general solution to Schr\"odinger's equation for finite range potentials involving arbitrary initial quantum states. We show that the dynamical features of the probability density are governed by a virtual \textit{self-induced two-level system} with energies $E_{+}$, and $E_{-}$, due to the phase modulation of the initial state. The asymptotic probability density exhibits Rabi-oscillations characterized by a frequency $\Omega=(E_{+}-E_{-})/\hbar$, which are independent of the potential profile. It is also found that for a system with a bound state, the interplay between the virtual levels with the latter causes a \textit{quantum beat} effect with a beating frequency, $\Omega$. We also find a regime characterized by a \textit{time-diffraction} phenomenon that allows to measure unambiguously the delay-time, which can be described by an exact analytical formula. It is found that the delay-time agrees with the phase-time only for the case of strictly monochromatic waves.

Pairing and molecule formation along the BCS-BEC crossover for finite range potentials

We address the thermodynamics of the zero temperature BCS-BEC crossover of a balanced two component mixture of a finite range interacting fermions within a mean field approach. For the analysis we consider two wide types of finite range potentials describing the interaction between different Fermi species: purely attractive, as the square well, the exponential and the Yukawa potentials, and others having both a repulsive wall at the origin as well as a finite attractive tail, as Van der Waals and dipolar potentials. We study the energy gap, the chemical potential and the pressure as functions of the scattering length for all the interacting potentials considered. The system spatial structure is further analyzed via the the pair functions associated to each potential. We find that for those with pure attractive contributions the crossover can be recognized either, when the chemical potential changes its sign or when the scattering length diverges. In contrast, the pair functions linked to potentials with both repulsive and attractive terms, show wide crossover regions without sharp distinctions at the sign change of the chemical potential nor at unitarity.

Confinement-induced resonance (CIR) in classical vs. quantum scattering under 2D harmonic confinement

Using complex analysis, we have investigated classical quasi-one-dimensional atom–atom scattering under 2D harmonic confinement with two different interaction potentials (Yukawa and Lennard-Jones) and found that the confinement-induced resonance (CIR) that occurs in the quantum system seems to have a classical analogue. We observed CIR in our classical results in the sense that a clear minimum appeared in the transmission coefficient for the different interaction potentials. We also investigated the changes in the value and position of this minimum by varying the characteristic parameters of the system including the angular momentum $$L_z$$ along the longitudinal axis. In the quantum case, it has already been shown that for the zero range Huang potential, CIR occurs only for $$L_z = 0$$. Our results indicate that classical CIR can also occur for $$L_z \ne 0$$ using finite range potentials. We have endeavoured to provide extensive physical arguments to explain the observed results and to support the proposed analogy between the classical and quantum regimes where applicable.

Pseudopotentials for two-dimensional ultracold scattering in the presence of synthetic spin-orbit coupling

We derive a pseudopotential in two dimensions (2D) with the presence of a 2D Rashba spin-orbit-coupling (SOC), following the same spirit of frame transformation in {[}Phys. Rev. A 95, 020702(R) (2017){]}. The frame transformation correctly describes the non-trivial phase accumulation and partial wave couplings due to the presence of SOC and gives rise to a different pseudopotential than the free-space one, even when the length scale of SOC is significantly larger than the two-body potential range. As an application, we apply our pseudopotential with the Lippmann-Schwinger equation to obtain an analytical scattering matrix. To demonstrate the validity, we compare our results with a numerical scattering calculation of finite-range potential and show perfect agreement over a wide range of scattering energy and SOC strength. Our results also indicate that the differences between our pseudopotential and the original free-space pseudopotential are essential to reproduce scattering observables correctly.