Zero-range Limit Research Articles

Quasiparticle properties of an impurity in a Fermi gas

We report on a study of a spin-down impurity strongly coupled to a spin-up Fermi sea (a so-called Fermi polaron) with the diagrammatic Monte-Carlo (DiagMC) technique. Conditions of zero temperature and three dimensions are considered for an ultracold atomic gas with resonant interactions in the zero-range limit. A Feynman diagrammatic series is developed for the one-body and two-body propagators providing information about the polaron and molecule channel respectively. The DiagMC technique allows us to reach diagram orders that are high enough for extrapolation to infinite order. The robustness of the extracted results is examined by checking various resummation techniques and by running the simulations with various choices for the propagators and vertex functions. It turns out that dressing the lines in the diagrams as much as possible is not always the optimal choice. We also identify classes of dominant diagrams for the one-body and two-body self-energy in the region of strong interaction. These dominant diagrams turn out to be the leading processes of the strong-coupling limit. The quasiparticle energies and $Z$-factor are obtained as a function of the interaction strength. We find that the DiagMC results for the molecule and polaron properties are very similar to those obtained with a variational ansatz. Surprisingly, this variational ansatz gives very good predictions for the quasiparticle residue even when this residue is significantly smaller than one.

Recent Advances in Description of Few Two-Component Fermions

Overview of the recent advances in description of the few two-component fermions is presented. The model 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 $m_1$: it turns out that two $L^P = 1^-$ three-body bound states emerge if mass ratio $m/m_1$ increases up to the critical value $\mu_c \approx 13.607$, above which the Efimov effect takes place. The topics considered include rigorous treatment of the few-fermion problem in the zero-range interaction limit, low-dimensional results, the four-body energy spectrum, crossover of the energy spectra for $m/m_1$ near $\mu_c $, and properties of potential-dependent states. At last, enlisted are the problems, whose solution is in due course.

Integral equations for the four-body problem

We consider the four-boson and 3+1 fermionic problems with a model Hamiltonian which encapsulates the mechanism of the Feshbach resonance involving the coherent coupling of two atoms in the open channel and a molecule in the closed channel. The model includes also the pair-wise direct interaction between atoms in the open channel and in the bosonic case, the direct molecule-molecule interaction in the closed channel. Interactions are modeled by separable potentials which makes it possible to reduce the four-body problem to the study of a single integral equation. We take advantage of the rotational symmetry and parity invariance of the Hamiltonian to reduce the general eigenvalue equation in each angular momentum sector to an integral equation for functions of three real variables only. A first application of this formalism in the zero-range limit is given elsewhere [Y. Castin, C. Mora, L. Pricoupenko, Phys. Rev. Lett. 105, 223201 (2010)].

Trapped two-component Fermi gases with up to six particles: Energetics, structural properties, and molecular condensate fraction

We investigate small equal-mass two-component Fermi gases under external spherically symmetric confinement in which atoms with opposite spins interact through a short-range two-body model potential. We employ a non-perturbative microscopic framework, the stochastic variational approach, and determine the system properties as functions of the interspecies s-wave scattering length a, the orbital angular momentum L of the system, and the numbers N1 and N2 of spin-up and spin-down atoms (with N1-N2 =0 or 1 and N < 7, where N=N1+N2). At unitarity, we determine the energies of the five- and six-particle systems for various ranges r0 of the underlying two-body model potential and extrapolate to the zero-range limit. These energies serve as benchmark results that can be used to validate and assess other numerical approaches. We also present structural properties such as the pair distribution function and the radial density. Furthermore, we analyze the one-body and two-body density matrices. A measure for the molecular condensate fraction is proposed and applied. Our calculations show explicitly that the natural orbitals and the momentum distributions of atomic Fermi gases approach those characteristic for a molecular Bose gas if the s-wave scattering length a, a>0, is sufficiently small.

S-wave scattering and the zero-range limit of the finite square well in arbitrary dimensions

We examine the zero-range limit of the finite square well in arbitrary dimensions through a systematic analysis of the reduced, s-wave, two-body, time-independent Schrödinger equation. A natural consequence of our investigation is the requirement of a delta function multiplied by a regularization operator to model the zero-range limit of the finite-square well when the dimensionality is greater than one. The case of two dimensions turns out to be surprisingly subtle, and needs to be treated separately from all other dimensions.

On δ′-like potential scattering on star graphs

We discuss the potential scattering on the noncompact star graph. The Schrödinger operator with the short-range potential localized in a neighborhood of the graph vertex is considered. We study the asymptotic behavior of the corresponding scattering matrix in the zero-range limit. It has been known for a long time that in dimension 1 there is no non-trivial Hamiltonian with the distributional potential δ′, i.e. the δ′ potential acts as a totally reflecting wall. Several authors have, in recent years, studied the scattering properties of the regularizing potentials αε−2Q(x/ε) approximating the first derivative of the Dirac delta function. A non-zero transmission through the regularized potential has been shown to exist as ε → 0. We extend these results to star graphs with the point interaction, which is an analog of the δ′ potential on the line. We prove that generically such a potential on the graph is opaque. We also show that there exists a countable set of resonant intensities for which a partial transmission through the potential occurs. This set of resonances is referred to as the resonant set and is determined as the spectrum of an auxiliary Sturm–Liouville problem associated with Q on the graph.

Short-range interaction in three-dimensional quantum mechanics

We show that it is possible to define shape-independent three-dimensional short-range quantum interactions in two-parameter form for non-spherical angular momentum channels through double rescaling of potential strength. Unlike the special case of l = 0 , where the zero-range limit of the system is renormalizable, the effective ranges diverge for l ≠ 0 channels, and the system becomes trivial at zero-size limit. It is also shown that the two-parameter representation with finite interaction range is useful in analyzing phase shifts and describing resonances with accuracy in non-spherical scatterings.

Point interactions of the dipole type defined through a three-parametric power regularization

A family of point interactions of the dipole type is studied in one dimension using a regularization by rectangles in the form of a barrier and a well separated by a finite distance. The rectangles and the distance are parametrized by a squeezing parameter ε → 0 with three powers μ, ν and τ describing the squeezing rates for the barrier, the well and the distance, respectively. This parametrization allows us to construct a whole family of point potentials of the dipole type including some other point interactions, such as e.g. δ-potentials. Varying the power τ, it is possible to obtain in the zero-range limit the following two cases: (i) the limiting δ′-potential is opaque (the conventional result obtained earlier by some authors) or (ii) this potential admits a resonant tunneling (the opposite result obtained recently by other authors). The structure of resonances (if any) also depends on a regularizing sequence. The sets of the {μ, ν, τ}-space where a non-zero (resonant or non-resonant) transmission occurs are found. For all these cases in the zero-range limit the transfer matrix is shown to be of the form with real parameters χ and g depending on a regularizing sequence. Those cases when χ ≠ 1 and g ≠ 0 mean that the corresponding δ′-potential is accompanied by an effective δ-potential.

Universal relations for a trapped four-fermion system with arbitrarys-wave scattering length

A two-component four-fermion system with equal masses, interspecies s-wave scattering length a and vanishing intraspecies interactions under external spherically symmetric harmonic confinement is considered. Using a correlated Gaussian basis set expansion approach, we determine the energies and various structural properties of the energetically lowest-lying gas-like state throughout the crossover for various ranges of the underlying two-body potential. Extrapolating to the zero-range limit, our numerical results show explicitly that the total energy, the trap energy as well as certain aspects of the pair distribution function and of the momentum distribution are related through the so-called integrated contact intensity I(a). Furthermore, it is shown explicitly that the total energy and the trap energy are related through a generalized virial theorem that accounts for a non-zero range.

Universal three-body dynamics in binary mixtures of ultra-cold atoms

The universal three-body dynamics in ultra-cold binary Fermi and Fermi-Bose mixtures is studied. A comprehensive universal description of the rotational-vibrational spectrum for two identical particles of mass m and the third particle of mass m 1 in the zero-range limit of the interaction between different particles is given for arbitrary values of the mass ratio m/m 1 and the total angular momentum L. For those L and m/m 1, which correspond to the finite number of vibrational states, all the binding energies are described by the universal function depending on two scaled variables. The detailed study is performed for L = 1; in particular, it is shown that a two-hump structure in the mass-ratio dependencies of the elastic scattering cross section and the three-body recombination rate are connected with arising of the bound states.

Three-body recombination of identical bosons with a large positive scattering length at nonzero temperature

For identical bosons with a large scattering length, the dependence of the three-body recombination rate on the collision energy is determined in the zero-range limit by universal functions of a single scaling variable. There are six scaling functions for angular momentum zero and one scaling function for each higher partial wave. We calculate these universal functions by solving the Skorniakov-Ter-Martirosian equation. The results for the three-body recombination as a function of the collision energy are in good agreement with previous results from solving the three-body Schr\"odinger equation for $^{4}\mathrm{He}$ atoms. The universal scaling functions can be used to calculate the three-body recombination rate at nonzero temperature. We obtain an excellent fit to the data from Grimm and co-workers for $^{133}\mathrm{Cs}$ atoms with a large positive scattering length.

Universal description of the rotational-vibrational spectrum of three particles with zero-range interactions

A comprehensive universal description of the rotational-vibrational spectrum for two identical particles of mass m and a third particle of mass m 1 in the zero-range limit of the interaction between different particles is given for arbitrary values of the mass ratio m/m 1 and the total angular momentum L. It is found that the number of vibrational states is determined by the functions L c(m/m 1) and L b(m/m 1). Explicitly, if the two-body scattering length is positive, the number of states is finite for L c(m/m 1) ≤ L ≤ L b(m/m 1), zero for L > L b(m/m 1), and infinite for L < L c(m/m 1). If the two-body scattering length is negative, the number of states is zero for L ≥ L c(m/m 1) and infinite for L < L c(m/m 1). For the finite number of vibrational states, all the binding energies are described by the universal function ɛLN(m/m 1) = (ξ, η), where ξ = (N − 1/2)/√L(L + 1), η = √m/[m 1 L(L + 1)], and N is the vibrational quantum number. This scaling dependence is in agreement with the numerical calculations for L > 2 and only slightly deviates from those for L = 1, 2. The universal description implies that the critical values L c(m/m 1) and L b(m/m 1) increase as 0.401 √m/m 1 and 0.563 √m/m 1, respectively, while the number of vibrational states for L ≥ L c(m/m 1) is within the range N ≤ N max ≈ 1.1√L(L + 1) + 1/2.

Resonant tunnelling through short-range singular potentials

A three-parameter family of point interactions constructed from sequences of symmetric barrier–well–barrier and well–barrier–well rectangles is studied in the limit, when the rectangles are squeezed to zero width but the barrier height and the well depth become infinite (the zero-range limit). The limiting generalized potentials are referred to as the second derivative of Dirac's delta function ±λδ″(x) with a renormalized coupling constant λ > 0 or simply as ±δ″-like point interactions. As a result, a whole family of self-adjoint extensions of the one-dimensional Schrödinger operator is shown to exist, which results in full and partial resonant tunnelling through this class of singular potentials. The resonant tunnelling occurs for countable sets of interaction strength values in the λ-space which are the roots of several transcendental equations. The comparison with the previous results for δ′-like point interactions is also discussed.

Unitary gas in an isotropic harmonic trap: Symmetry properties and applications

We consider N atoms trapped in an isotropic harmonic potential, with s-wave interactions of infinite scattering length. In the zero-range limit, we obtain several exact analytical results: mapping between the trapped problem and the free-space zero-energy problem, separability in hyperspherical coordinates, SO(2,1) hidden symmetry, and relations between the moments of the trapping potential energy and the moments of the total energy.

Scattering properties of point dipole interactions

The scattering properties of a three-parameter family of point dipole-like interactions constructed from a sequence of barrier-well rectangles are studied in the zero-range limit. Besides the real (unrenormalized) δ'-interaction, the derivative of Dirac's delta function, a whole family of point dipole interactions with a renormalized coupling constant are analysed. Depending on the parameter values, all these interactions being self-adjoint extensions of the one-dimensional Schrodinger operator are shown to be divided into four types: (i) interactions will full transparency, (ii) non-transparent interactions, (iii) partially transparent interactions acting effectively as a δ-interaction and (iv) interactions with partial transparency at discrete resonant values of the coupling constant.

Description of nuclear systems within the relativistic Hartree–Fock method with zero-range self-interactions of the scalar field

An exact method is suggested to treat the nonlinear self-interactions (NLSI) in the relativistic Hartree–Fock (RHF) approach for nuclear systems. We consider here the NLSI constructed from the relativistic scalar nucleon densities including products of six and eight fermion fields. This type of NLSI corresponds to the zero-range limit of the standard cubic and quartic self-interactions of the scalar field. The method to treat the NLSI uses the Fierz transformation, which enables one to express the exchange (Fock) components in terms of the direct (Hartree) ones. The method is applied to nuclear matter and finite nuclei. It is shown that, in the RHF formalism, the NLSI, which are explicitly isovector-independent, generate scalar, vector and tensor nucleon self-energies with a strong isovector dependence. This strong isovector structure of the self-energies is due to the exchange terms of the RHF method. Calculations are carried out with a parametrization containing five free parameters. The model allows a description of both types of systems compatible with experimental data.

Spectral properties of a short-range impurity in a quantum dot

The spectral properties of the quantum mechanical system consisting of a quantum dot with a short-range attractive impurity inside the dot are studied in the zero-range limit. The Green function of the system is obtained in an explicit form. In the case of a spherically symmetric quantum dot, the dependence of the spectrum on the impurity position and strength of the impurity potential is analyzed in detail. The recovering of the confinement potential of the dot from the spectroscopy data is proven; the consequences of the hidden symmetry breaking by the impurity are considered. The effect of the positional disorder is analyzed.

On the existence of resonances in the transmission probability for interactions arising from derivatives of Dirac s delta function

The scattering properties of regularizing finite-range potentials constructed in the form of squeezed rectangles, which approximate the first and second derivatives of the Dirac delta function δ(x), are studied in the zero-range limit. Particularly, for a countable set of interaction strength values, a non-zero transmission through the point potential δ'(x), defined as the weak limit (in the standard sense of distributions) of a special dipole-like sequence of rectangles, is shown to exist when the rectangles are squeezed to zero width. A tripole sequence of rectangles, which gives in the weak limit the distribution δ''(x), is demonstrated to exhibit the total transmission for a countable sequence of the rectangle's width that tends to zero. However, this tripole sequence does not admit a well-defined point interaction in the zero-range limit, making sense only for a finite range of the regularizing rectangular-like potentials.

Rigorous approach to the problem of ultraviolet divergencies in dilute Bose gases.

In this Letter we consider a system of N pairwise finite-range interacting atoms and prove rigorously that in the zero-range interaction limit all the eigenstates and eigenenergies of the Hamiltonian converge to those corresponding to N atoms interacting via the Fermi-Huang regularized pseudopotential. Next, we show that the latter eigensystem (if treated exactly) is invariant under a nontrivial transformation of the interaction potential. Finally, we realize that most of the approximate schemes of many-body physics do not exhibit this invariance: We use this property to resolve all inconsistencies of the Hartree-Fock-Bogoliubov variational formalism known thus far.

Role of short- and long-range interactions in quantum Bose fluids

Information of the effects that the range of the interaction has on the ground-state, excitations, and phase transitions in strongly correlated quantum fluids is obtained by studying the Yukawa Bose fluid using the microscopic Jastrow–Feenberg variational method. Our results for the ground-state structure and energetics agree with Monte Carlo data. The excitation spectrum is seen to transform from a free-particle spectrum in the zero-range limit into a plasmon spectrum in the infinite-range (Coulomb) limit through a phase with linear dispersion at low momenta. Our results for the liquid–solid phase transition indicate that the symmetry of the forming crystal depends from the range of the interaction.