Zero-range Interaction Research Articles

Few-fermion systems in one dimension: Ground- and excited-state energies and contacts

Using the lattice Monte Carlo method, we compute the energy and Tan's contact in the ground state as well as the first excited state of few- to many-fermion systems in a one-dimensional periodic box. We focus on unpolarized systems of N=4,6,...,12 particles, with a zero-range interaction, and a wide range of attractive couplings. In addition, we provide extrapolations to the infinite-volume and thermodynamic limits.

Solving for the particle-number-projected HFB wavefunction

Recently we proposed a particle-number-conserving theory for nuclear pairing (Jia, 2013) [19] through the generalized density matrix formalism. The relevant equations were solved for the case when each single-particle level has a distinct set of quantum numbers and could only pair with its time-reversed partner (BCS-type Hamiltonian). In this work we consider the more general situation when several single-particle levels could have the same set of quantum numbers and pairing among these levels is allowed (HFB-type Hamiltonian). The pair condensate wavefunction (the HFB wavefunction projected onto good particle number) is determined by the equations of motion for density matrix operators instead of the variation principle. The theory is tested in the simple two-level model with factorizable pairing interactions, and semi-realistic models with the zero-range delta interaction and the realistic Bonn-CD interaction.

Stability of closed gaps for the alternating Kronig–Penney Hamiltonian

We consider the Kronig–Penney model for a quantum crystal with equispaced periodic delta-interactions of alternating strength. For this model all spectral gaps at the centre of the Brillouin zone are known to vanish, although so far this noticeable property has only been proved through a very delicate analysis of the discriminant of the corresponding ODE and the associated monodromy matrix. We provide a new, alternative proof by showing that this model can be approximated, in the norm resolvent sense, by a model of regular periodic interactions with finite range for which all gaps at the centre of the Brillouin zone are still vanishing. In particular this shows that the vanishing gap property is stable in the sense that it is present also for the “physical” approximants and is not only a feature of the idealised model of zero-range 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.

Energy lower bound for the unitary N + 1 fermionic model

We consider the stability problem for a unitary N + 1 fermionic model, i.e., a system of N identical fermions interacting via zero-range interactions with a different particle, in the case of infinite two-body scattering length. Starting from the two-body boundary condition, we construct an explicit expression for the expectation value of the energy. Then we investigate its boundedness from below and exhibit a sufficient condition on the mass ratio, which guarantees the stability of the model.

Stochastic pumping of particles with zero-range interactions

We consider a model of a stochastic pump in which many particles jump between sites along a ring. The particles interact with each other via zero-range interactions. We argue that for slow driving the dynamics can be approximated by a nonlinear diffusion equation. This nonlinear equation is then used to derive a current decomposition formula, expressing the current as a sum of two contributions. The first is from the momentary steady state while the second arises due to the variation of the density with time, and is identified as the pumped current. The dynamics is found to satisfy the no-pumping theorem for cyclic processes, in agreement with recent results in discrete systems.

Incorporating exact two-body propagators for zero-range interactions intoN-body Monte Carlo simulations

Ultracold atomic gases are, to a very good approximation, described by pairwise zero-range interactions. This paper demonstrates that $N$-body systems with two-body zero-range interactions can be treated reliably and efficiently by the finite temperature and ground state path integral Monte Carlo approaches, using the exact two-body propagator for zero-range interactions in the pair product approximation. Harmonically trapped one- and three-dimensional systems are considered. A new propagator for the harmonically trapped two-body system with infinitely strong zero-range interaction, which may also have applications in real time evolution schemes, is presented.

A variational approach to repulsively interacting three-fermion systems in a one-dimensional harmonic trap

We study a three-body system with zero-range interactions in a one-dimensional harmonic trap. The system consists of two spin-polarized fermions and a third particle which is distinct from two others (2+1 system). First we assume that the particles have equal masses. For this case the system in the strongly and weakly interacting limits can be accurately described using wave function factorized in hyperspherical coordinates. Inspired by this result we propose an interpolation ansatz for the wave function for arbitrary repulsive zero-range interactions. By comparison to numerical calculations, we show that this interpolation scheme yields an extremely good approximation to the numerically exact solution both in terms of the energies and also in the spin-resolved densities. As an outlook, we discuss the case of mass imbalanced systems in the strongly interacting limit. Here we find spectra that demonstrate that the triply degenerate spectrum at infinite coupling strength of the equal mass case is in some sense a singular case as this degeneracy will be broken down to a doubly degenerate or non-degenerate ground state by any small mass imbalance.

Monte Carlo Calculations for Fermi Gases in the Unitary Limit with a Zero-Range Interaction

An ultracold Fermi gas with a zero-range attractive potential in the unitary limit is investigated using variational and diffusion Monte Carlo methods. Previous calculations have used a finite range interactions and extrapolate the results to zero-range. Here we extend the quantum Monte Carlo method to directly use a zero-range interaction without extrapolation. We employ a trial wave function with the correct boundary conditions, and modify the sampling procedures to handle the zero-range interaction. The results are reliable and have low variance.

Density dependence of the pairing interaction and pairing correlation in unstable nuclei

This work aims at a global assessment of the effect of the density dependence of the zero-range pairing interaction. Systematic Skyrme-Hartree-Fock-Bogoliubov calculations with the volume, surface and mixed pairing forces are carried out to study the pairing gaps in even-even nuclei over the whole nuclear chart. Calculations are also done in coordinate representation for unstable semi-magic even-even nuclei. The calculated pairing gaps are compared with empirical values from four different odd-even staggering formulae. Calculations with the three pairing interactions are comparable for most nuclei close to $\beta$-stability line. However, the surface interaction calculations predict neutron pairing gaps in neutron-rich nuclei that are significantly stronger than those given by the mixed and volume pairing. On the other hand, calculations with volume and mixed pairing forces show noticeable reduction of neutron pairing gaps in nuclei far from the stability.

Covariance analysis for energy density functionals and instabilities

We present the covariance analysis of two successful nuclear energy density functionals (EDFs), (i) a non-relativistic Skyrme functional built from a zero-range effective interaction, and (ii) a relativistic nuclear EDF based on density dependent meson–nucleon couplings. The covariance analysis is a useful tool for understanding the limitations of a model, the correlations between observables and the statistical errors. We show, for our selected test nucleus Pb, that when the constraint on a property A included in the fit is relaxed, correlations with other observables B become larger; on the other hand, when a strong constraint is imposed on A, the correlations with other properties become very small. We also provide a brief review, partly connected with the covariance analysis, of some instabilities displayed by several EDFs currently used in nuclear physics.

Fermion-dimer scattering using an impurity lattice Monte Carlo approach and the adiabatic projection method

We present lattice Monte Carlo calculations of fermion-dimer scattering in the limit of zero-range interactions using the adiabatic projection method. The adiabatic projection method uses a set of initial cluster states and Euclidean time projection to give a systematically improvable description of the low-lying scattering cluster states in a finite volume. We use L\"uscher's finite-volume relations to determine the s-wave, p-wave, and d-wave phase shifts. For comparison, we also compute exact lattice results using Lanczos iteration and continuum results using the Skorniakov-Ter-Martirosian equation. For our Monte Carlo calculations we use a new lattice algorithm called impurity lattice Monte Carlo. This algorithm can be viewed as a hybrid technique which incorporates elements of both worldline and auxiliary-field Monte Carlo simulations.

Regularization of zero-range effective interactions in finite nuclei

The problem of the divergences which arise in beyond mean-field calculations, when a zero-range effective interaction is employed, has not been much considered so far. Some of us have proposed, quite recently, a scheme to regularize a zero-range Skyrme-type force when it is employed to calculate the total energy, at second-order perturbation theory level, in uniform matter. Although this scheme looked promising, the extension for finite nuclei is not straightforward. We introduce such procedure in the current paper, by proposing a regularization procedure that is similar, in spirit, to the one employed to extract the so-called V_{\rm low-k} from the bare force. Although this has been suggested already by B.G. Carlsson and collaborators, the novelty of our work consists in setting on equal footing uniform matter and finite nuclei; in particular, we show how the interactions that have been regularized in uniform matter behave when they are used in a finite nucleus with the corresponding cutoff. We also address the problem of the validity of the perturbative approach in finite nuclei for the total energy.

Energetics of an interacting electron system on a surface

We present some properties of an interacting N-fermion system on a cylindrical surface focusing in the quantum mechanical size effects of the ground-state observables. We use an effective zero-range two-fermion interaction which allows for an analytical Hartree–Fock approach. Within this model, quantum size effects can be understood as a consequence of filling states in the Fermi sea in which naturally arises a clear dependence on the Fermi momentum and the cylinder radius. Particularly, the model shows that the difference between the surface and bulk energies does not depend on the strength interaction. A relationship between the chemical potential and the bulk energy is analytically derived. The approach is also suitable for carbon nanotubes.

A microscopic model beyond mean-field: from giant resonance properties to the fit of new effective interactions

Self-consistent mean-field models are able to reproduce well the overall properties of nuclei for a wide range of masses. Nevertheless, they are intrinsically unsuitable for the description of some important observables like the single-particle strength distribution or, in connection with collective states, their damping width and their gamma decay to the ground state or to low lying states. For this reason, a completely microscopic approach beyond mean- field has been implemented recently, based on the Skyrme functional.When beyond mean-field theories are handled, the mean-field-fitted effective interaction should be refitted at the desired level of approximation. If zero-range interactions are used, divergences arise. We present some steps towards the refitting of Skyrme interactions, for its application in finite nuclei.

Pairing properties of hot nuclei within the finite temperature Hartree Fock+BCS approximation

In this work, finite temperature Hartree Fock BCS (FT+HFBCS) calculations are performed in Sn isotopes with the zero-range Skyrme type interactions. Vanishing of pairing correlations due to the increase in temperature are investigated with various types of Skyrme interactions. In addition, dependence of the critical temperature with respect to the several properties of these energy density functionals are analysed.

Spectral flow of trimer states of two heavy impurities and one light condensed boson

The spectral flow of three-body (trimer) states consisting of two heavy (impurity) particles sitting in a condensate of light bosons is considered. Assuming that the condensate is weakly interacting and that an impurity and a boson have a resonant zero-range two-body interaction, we use the Born-Oppenheimer approximation to determine the effective three-body potential. We solve the resulting Schrödinger equation numerically and determine the trimer binding energies as a function of the coherence length of the light bosonic condensate particles. The binding energy is found to be suppressed by the presence of the condensate when the energy scale corresponding to the coherence length becomes of order the trimer binding energy in the absence of the condensate. We find that the Efimov scaling property is reflected in the critical values of the condensate coherence length at which the trimers are pushed into the continuum.

Mapping the two-component atomic Fermi gas to the nuclear shell-model

The physics of a two-component cold fermi gas is now frequently addressed in laboratories. Usually this is done for large samples of tens to hundreds of thousands of particles. However, it is now possible to produce few-body systems (1-100 particles) in very tight traps where the shell structure of the external potential becomes important. A system of two-species fermionic cold atoms with an attractive zero-range interaction is analogous to a simple model of nucleus in which neutrons and protons interact only through a residual pairing interaction. In this article, we discuss how the problem of a two-component atomic fermi gas in a tight external trap can be mapped to the nuclear shell model so that readily available many-body techniques in nuclear physics, such as the Shell Model Monte Carlo (SMMC) method, can be directly applied to the study of these systems. We demonstrate an application of the SMMC method by estimating the pairing correlations in a small two-component Fermi system with moderate-to-strong short-range two-body interactions in a three-dimensional harmonic external trapping potential.

Universality and scaling in theN-body sector of Efimov physics

Universal behaviour has been found inside the window of Efimov physics for systems with $N=4,5,6$ particles. Efimov physics refers to the emergence of a number of three-body states in systems of identical bosons interacting {\it via} a short-range interaction becoming infinite at the verge of binding two particles. These Efimov states display a discrete scale invariance symmetry, with the scaling factor independent of the microscopic interaction. Their energies in the limit of zero-range interaction can be parametrized, as a function of the scattering length, by a universal function. We have found, using a particular form of finite-range scaling, that the same universal function can be used to parametrize the energies of $N\le6$ systems inside the Efimov-physics window. Moreover, we show that the same finite-scale analysis reconciles experimental measurements of three-body binding energies with the universal theory.

One-dimensional Dirac operators with zero-range interactions: Spectral, scattering, and topological results

The spectral and scattering theory for 1-dimensional Dirac operators with mass m and with zero-range interactions are fully investigated. Explicit expressions for the wave operators and for the scattering operator are provided. These new formulae take place in a representation which links, in a suitable way, the energies −∞ and +∞, and which emphasizes the role of ±m. Finally, a topological version of Levinson's theorem is deduced, with the threshold effects at ±m automatically taken into account.