Zero-range Limit Research Articles

Close encounters of the sticky kind: Brownian motion at absorbing boundaries.

Encounter-based models of diffusion provide a probabilistic framework for analyzing the effects of a partially absorbing reactive surface, in which the probability of absorption depends upon the amount of surface-particle contact time. In this paper we develop a class of encounter-based models that deal with absorption at sticky boundaries. Sticky boundaries occur in a diverse range of applications, including cell biology, colloidal physics, finance, and human crowd dynamics. They also naturally arise in active matter, where confined active particles tend to spontaneously accumulate at boundaries even in the absence of any particle-particle interactions. We begin by constructing a one-dimensional encounter-based model of sticky Brownian motion (BM), which is based on the zero-range limit of nonsticky BM with a short-range attractive potential well near the origin. In this limit, the boundary-contact time is given by the amount of time (occupation time) that the particle spends at the origin. We calculate the joint probability density or propagator for the particle position and the occupation time, and then identify an absorption event as the first time that the occupation time crosses a randomly generated threshold. We illustrate the theory by considering diffusion in a finite interval with a partially absorbing sticky boundary at one end. We show how various quantities, such as the mean first passage time (MFPT) for single-particle absorption and the relaxation to steady state at the multiparticle level, depend on moments of the random threshold distribution. Finally, we determine how sticky BM can be obtained by taking a particular diffusion limit of a sticky run-and-tumble particle (RTP).

Regularization of energy-dependent pointlike interactions in 1D quantum mechanics

We construct a family of Hermitian potentials in 1D quantum mechanics that converges in the zero-range limit to a δ interaction with an energy-dependent coupling. It does not belong to the standard four-parameter family of pointlike interactions in 1D, obtained by requiring hermiticity. But we show that although our Hamiltonian is Hermitian for the standard inner product when the range of the potential is finite, the eigenstates become orthogonal for a different inner product in the zero-range limit. This inner product attributes a finite probability (and not probability density) for the particle to be exactly located at the position of the potential. Such pointlike interactions can then be used to construct potentials with a finite support with an energy-dependent coupling.

Three-boson stability for boosted interactions towards the zero-range limit

We study the three-boson bound-state mass and wave functions for ground and excited states within the three-body relativistic framework with Kamada and Glöcke boosted potentials in the limit of a zero-range interaction. We adopt a nonrelativistic short-range separable potential, with Yamaguchi and Gaussian form factors, and drive them towards the zero-range limit by letting the form factors' momentum scales go to large values while keeping the two-body binding fixed. We show that the three-boson relativistic masses and wave functions are model-independent towards the zero-range limit, and the Thomas collapse is avoided, while the nonrelativistic limit kept the Efimov effect. Furthermore, the stability in the zero-range limit is a result of the reduction of boosted potential with the increase of the virtual pair center of mass momentum within the three-boson system. Finally, we compare the present results with Light-Front and Euclidean calculations.

Role of Effective Range in the Bulk Viscosity of Resonantly Interacting s- and p-Wave Fermi Gases.

We investigate the role of the effective range on the bulk viscosity of s- and p-wave Fermi gases. At resonance, the presence of the effective range breaks the scale invariance of the system, and hence results in a nonzero bulk viscosity. However, we show that the effective range plays a very different role in the two cases. In the s-wave case, the role of the effective range is perturbative, and its contribution to the bulk viscosity vanishes in the limit of zero effective range. On the other hand, the effective range in p-wave Fermi gases leads to a nonzero bulk viscosity, even in the zero-range limit. We employ a general diagrammatic approach to compute the bulk viscosity spectral function that includes the effects of the effective range. We then compute the analytic expressions for the spectral function in the high temperature limit, at low and high frequencies. We also derive the sum rules for the bulk viscosity spectral function for both s- and p-wave gases.

Clustering of Four-Component Unitary Fermions.

Abinitio nuclear physics tackles the problem of strongly interacting four-component fermions. The same setting could foreseeably be probed experimentally in ultracold atomic systems, where two- and three-component experiments have led to major breakthroughs in recent years. Both due to the problem's inherent interest and as a pathway to nuclear physics, in this Letter we study four-component fermions at unitarity via the use of quantum MonteCarlo methods. We explore novel forms of the trial wave function and find one which leads to a ground state of the eight-particle system whose energy is almost equal to that of two four-particle systems. We investigate the clustering properties involved and also extrapolate to the zero-range limit. In addition to being experimentally testable, our results impact the prospects of developing nuclear physics as a perturbation around the unitary limit.

Mass-imbalanced fermionic mixture in a harmonic trap

The mass-imbalanced fermionic mixture is studied, where $N\le5$ identical fermions interact resonantly with an impurity, a distinguishable atom. The shell structure is explored, and the physics of a dynamic light-impurity is shown to be different from that of the static heavy-impurity case. The energies in a harmonic trap at unitarity are calculated and extrapolated to the zero-range limit. In doing so, the scaling factor of the ground state, as well as of a few excited states, is calculated. In the $2 \le N \le 4$ systems, pure $(N+1)$ Efimov states exist for large enough mass ratio. However, no sign for a six-body Efimov state in the $(5+1)$ system is found in the mass ratio explored, $M/m \le 12$.

Universality in the Neutron– $$^{\mathbf {19}}$$ 19 C Scattering Using Finite-Range Separable Interactions

We report a study on the low-energy properties of the elastic $s-$wave scattering of a neutron ($n$) in the carbon isotope $^{19}$C near the critical condition for the occurrence of an excited Efimov state in the three-body $n-n-^{18}$C system. For the separation energy of the two halo neutrons in $^{20}$C we use the available experimental data. We also investigate to which extent the universal scaling laws, strictly valid in the zero-range limit, will survive when using finite-range interactions. By allowing to vary the $n-^{18}$C binding energy, a scaling behavior for the real and imaginary parts of the $s-$wave phase-shift $\delta_0$ is verified, emerging some universal characteristics given by the pole-position of $k\cot(\delta_0^R)$ and effective-range parameters.

Neutron-19C scattering: Emergence of universal properties in a finite range potential

The low-energy properties of the elastic $s-$wave scattering for the $n-^{19}$C are studied near the critical condition for the occurrence of an excited Efimov state in $n-n-^{18}$C. It is established to which extent the universal scaling laws, strictly valid in the zero-range limit, survive when finite range potentials are considered. By fixing the two-neutrons separation energy in $^{20}$C with available experimental data, it is studied the scaling of the real ($\delta_0^R$) and imaginary parts of the $s-$wave phase-shift with the variation of the $n-^{18}$C binding energy. We obtain some universal characteristics given by the pole-position of $k\cot(\delta_0^R)$ and effective-range parameters. By increasing the $n-^{18}$C binding energy, it was verified that the excited state of $^{20}$C goes to a virtual state, resembling the neutron-deuteron behavior in the triton. It is confirmed that the analytical structure of the unitary cut is not affected by the range of the potential or mass asymmetry of the three-body system.

Infinite matter properties and zero-range limit of non-relativistic finite-range interactions

We discuss some infinite matter properties of two finite-range interactions widely used for nuclear structure calculations, namely Gogny and M3Y interactions. We show that some useful informations can be deduced for the central, tensor and spin–orbit terms from the spin–isospin channels and the partial wave decomposition of the symmetric nuclear matter equation of state. We show in particular that the central part of the Gogny interaction should benefit from the introduction of a third Gaussian and the tensor parameters of both interactions can be deduced from special combinations of partial waves. We also discuss the fact that the spin–orbit of the M3Y interaction is not compatible with local gauge invariance. Finally, we show that the zero-range limit of both families of interactions coincides with the specific form of the zero-range Skyrme interaction extended to higher momentum orders and we emphasize from this analogy its benefits.

Ballistic quench-induced correlation waves in ultracold gases

We investigate the wave packet dynamics of a pair of particles that undergoes a rapid change of scattering length. The short-range interactions are modeled in the zero-range limit, where the quench is accomplished by switching the boundary condition of the wave function at vanishing particle separation. This generates a correlation wave that propagates rapidly to nonzero particle separations. We have derived universal, analytic results for this process that lead to a simple phase-space picture of the quench-induced scattering. Intuitively, the strength of the correlation wave relates to the initial contact of the system. We find that, in one spatial dimension, the $k^{-4}$ tail of the momentum distribution contains a ballistic contribution that does not originate from short-range pair correlations, and a similar conclusion can hold in other dimensionalities depending on the quench protocol. We examine the resultant quench-induced transport in an optical lattice in 1D, and a semiclassical treatment is found to give quantitatively accurate estimates for the transport probabilities.

Partial-wave decomposition of the finite-range effective tensor interaction

We perform a detailed analysis of the properties of the finite-range tensor term associated with the Gogny and M3Y effective interactions. In particular, by using a partial wave decomposition of the equation of state of symmetric nuclear matter, we show how we can extract their tensor parameters directly from microscopic results based on bare nucleon-nucleon interactions. Furthermore, we show that the zero-range limit of both finite-range interactions has the form of the N3LO Skyrme pseudo-potential, which thus constitutes a reliable approximation in the density range relevant for finite nuclei. Finally, we use Brueckner-Hartree-Fock results to fix the tensor parameters for the three effective interactions.

Trapped unitary two-component Fermi gases with up to ten particles

The properties of two-component Fermi gases with zero-range interactions are universal. We use an explicitly correlated Gaussian basis set expansion approach to investigate small equal-mass two-component Fermi gases under spherically symmetric external harmonic confinement. At unitarity, we determine the ground state energy for systems with up to ten particles interacting through finite-range two-body potentials for both even and odd number of particles. We extrapolate the energies to the zero-range limit using a novel scheme that removes the linear and, in some cases, also the quadratic dependence of the ground state energies on the two-body range. Our extrapolated zero-range energies are compared with results from the literature. We also calculate the two-body Tan contact and structural properties.

Controllable resonant tunnelling through single-point potentials: A point triode

Abstract A zero-thickness limit of three-layer heterostructures under two bias voltages applied externally, where one of which is supposed to be a gate parameter, is studied. As a result, an effect of controllable resonant tunnelling of electrons through single-point potentials is shown to exist. Therefore the limiting structure may be termed a “point triode” and considered in the theory of point interactions as a new object. The simple limiting analytical expressions adequately describe the resonant behaviour in the transistor with realistic parameter values and thus one can conclude that the zero-range limit of multi-layer structures may be used in fabricating nanodevices. The difference between the resonant tunnelling across single-point potentials and the Fabry–Perot interference effect is also emphasized.

Some aspects of universality in Efimov physics

The Efimov effect is the appearance of an infinite tower of bound states, En3, that accumulate in a geometrical way at zero energy in systems of three-identical bosons having an infinite large two-body scattering length a. In the zero-range limit the geometrical factor is En+13/En3 = e−2π/s0 with s0 ≈ 1.00624 an universal number. The Efimov effect is the signature of the discrete scale invariance (DSI) present in these systems close to the unitary limit (1/a = 0). The DSI constrains also the spectrum in N > 3 systems. In this contribution the two-level tree structure observed in bosons systems with N ≥ 4 will be discussed using potential models.

Recent advances in description of few two-component fermions

Overview of the recent advances in description of the few two-component fermions is presented. Themodel of zero-range interaction is generally considered to discuss the principal aspects of the few-body dynamics. Particular attention is paid to detailed description of two identical fermions of mass m and a distinct particle of mass m1: it turns out that two LP = 1− three-body bound states emerge if mass ratio m/m1 increases up to the critical value µc ≈ 13.607, abovewhich the Efimov effect takes place. The topics considered include rigorous treatment of the few-fermion problem in the zero-range interaction limit, lowdimensional results, the four-body energy spectrum, crossover of the energy spectra for m/m1 near µc, and properties of potential-dependent states. At last, enlisted are the problems, whose solution is in due course.

Diagrammatic Monte Carlo study of the Fermi polaron in two dimensions

We study the properties of the two-dimensional Fermi polaron model in which an impurity attractively interacts with a Fermi sea of particles in the zero-range limit. We use a diagrammatic Monte Carlo (DiagMC) method which allows us to sample a Feynman diagrammatic series to very high order. The convergence properties of the series and the role of multiple particle-hole excitations are discussed. We study the polaron and molecule energy as a function of the coupling strength, revealing a transition from a polaron to a molecule in the ground state. We find a value for the critical interaction strength which complies with the experimentally measured one and predictions from variational methods. For all considered interaction strengths, the polaron $Z$ factor from the full diagrammatic series almost coincides with the one-particle-hole result. We also formally link the DiagMC and the variational approaches for the polaron problem at hand.

INTRINSIC RESONANT TUNNELING PROPERTIES OF THE ONE-DIMENSIONAL SCHRÖDINGER OPERATOR WITH A DELTA DERIVATIVE POTENTIAL

Restricting ourselves to a simple rectangular approximation but using properly a two-scale regularization procedure, additional resonant tunneling properties of the one-dimensional Schrödinger operator with a delta derivative potential are established, which appear to be lost in the zero-range limit. These "intrinsic" properties are complementary to the main already proved result that different regularizations of Dirac's delta function produce different limiting self-adjoint operators. In particular, for a given regularizing sequence, a one-parameter family of connection condition matrices describing bound states is constructed. It is proposed to consider the convergence of transfer matrices when the potential strength constant is involved into the regularization process, resulting in an extension of resonance sets for the transmission across a δ′-barrier.

Small two-component Fermi gases in a cubic box with periodic boundary conditions

The properties of two-component Fermi gases become universal if the interspecies $s$-wave scattering length ${a}_{s}$ and the average interparticle spacing are much larger than the range of the underlying two-body potential. Using an explicitly correlated Gaussian basis set expansion approach, we determine the eigenenergies of two-component Fermi gases in a cubic box with periodic boundary conditions as functions of the interspecies $s$-wave scattering length and the effective range of the two-body potential. The universal properties of systems consisting of up to four particles are determined by extrapolating the finite-range energies to the zero-range limit. We determine the eigenenergies of states with vanishing and finite momenta. In the weakly attractive BCS regime, we analyze the energy spectra and degeneracies using first-order degenerate perturbation theory. Excellent agreement between the perturbative energy shifts and the numerically determined energies is obtained. For the infinitely large scattering length case, we compare our results---where available---with those presented in the literature.

Single-point potentials with total resonant tunneling

Two rectangular models described by the one-dimensional Schr\"odinger equation with sharply localized potentials are suggested. The potentials have a multilayer thin structure being composed from adjacent barriers and wells. Their peculiar tunneling properties are studied in considerable detail. Particularly, in the zero-range limit when the potentials are squeezed to a single point, sharp peaks with total transmission are observed at certain (positive and negative) quantized values of the potential strength constant forming infinite discrete sets. Beyond these sets, the barrier-well structures behave as a perfectly reflecting wall. The transcendental equations with respect to potential strengths, the solutions of which determine transmission (resonance) sets, are derived. In this regard, both the models are exactly solvable. The energy dependence of an incident particle is shown to reveal a resonance behavior, being completely different from that observed in a typical double-barrier structure.

Two trapped particles interacting by a finite-range two-body potential in two spatial dimensions

We examine the problem of two particles confined in an isotropic harmonic trap, which interact via a finite-ranged Gaussian-shaped potential in two spatial dimensions. We derive an approximative transcendental equation for the energy and study the resulting spectrum as a function of the interparticle interaction strength. Both the attractive and repulsive systems are analyzed. We study the impact of the potential's range on the ground-state energy. Complementary, we also explicitly verify by a variational treatment that in the zero-range limit the positive delta potential in two dimensions only reproduces the non-interacting results, if the Hilbert space in not truncated. Finally, we establish and discuss the connection between our finite-range treatment and regularized zero-range results from the literature.