Energy lower bound for the unitary N + 1 fermionic model (original) (raw)

STABILITY FOR A SYSTEM OF N FERMIONS PLUS A DIFFERENT PARTICLE WITH ZERO-RANGE INTERACTIONS

Reviews in Mathematical Physics, 2012

We study the stability problem for a non-relativistic quantum system in dimension three composed by N ≥ 2 identical fermions, with unit mass, interacting with a different particle, with mass m, via a zero-range interaction of strength α ∈ R. We construct the corresponding renormalised quadratic (or energy) form F α and the so-called Skornyakov-Ter-Martirosyan symmetric extension H α , which is the natural candidate as Hamiltonian of the system. We find a value of the mass m * (N ) such that for m > m * (N ) the form F α is closed and bounded from below. As a consequence, F α defines a unique self-adjoint and bounded from below extension of H α and therefore the system is stable. On the other hand, we also show that the form F α is unbounded from below for m < m * (2). In analogy with the well-known bosonic case, this suggests that the system is unstable for m < m * (2) and the so-called Thomas effect occurs.

Validity of the scattering-length approximation in strongly interacting Fermi systems

Physical Review A, 2011

We investigate the energy spectrum of systems of two, three and four spin-1 2 fermions with short range attractive interactions both exactly, and within the scattering length approximation. The formation of molecular bound states and the ferromagnetic transition of the excited scattering state are examined systematically as a function of the 2-body scattering length. Use of the scattering length approximation increases the scattering energy, and causes the ferromagnetic transition at a much lower ratio of scattering length to interatomic spacing. We show that an adiabatic ferromagnetic transition occurs at an interaction strength in agreement with experiment if one includes bound states in the description.

Lower bound on fermion binding energies

Physical Review B, 2001

We derive a new lower bound for the ground state energy E F (N, S) of N fermions with total spin S in terms of binding energies E F (N −1, S ±1/2) of (N −1) fermions. Numerical examples are provided for some simple short-range or confining potentials.

Improved lower bounds for the ground-state energy of many-body systems

Physical Review A, 2001

New lower bounds for the binding energy of a quantum-mechanical system of interacting particles are presented. The new bounds are expressed in terms of two-particle quantities and improve the conventional bounds of the Hall-Post type. They are constructed by considering not only the energy in the twoparticle system, but also the structure of the pair wave function. We apply the formal results to various numerical examples, and show that in some cases dramatic improvement over the existing bounds is reached.

Publisher's Note: Effective theory for trapped few-fermion systems [Phys. Rev. A 76, 063613 (2007)]

Physical Review A, 2008

We apply the general principles of effective field theories to the construction of effective interactions suitable for few-and many-body calculations in a no-core shell model framework. We calculate the spectrum of systems with three and four two-component fermions in a harmonic trap. In the unitary limit, we find that three-particle results are within 10% of known semi-analytical values even in small model spaces. The method is very general, and can be readily extended to other regimes, more particles, different species (e.g., protons and neutrons in nuclear physics), or more-component fermions (as well as bosons). As an illustration, we present calculations of the lowest-energy threefermion states away from the unitary limit and find a possible inversion of parity in the ground state in the limit of trap size large compared to the scattering length. Furthermore, we investigate the lowest positive-parity states for four fermions, although we are limited by the dimensions we can currently handle in this case.

The quantum N-body problem

Journal of Mathematical Physics, 2000

This selective review is written as an introduction to the mathematical theory of the Schro ¤ d inger ¥ equation for N ¦ particles. § Characteristic for these systems are the cluster properties of the potential in configuration space, which are expressed in a simple © geometric language. The methods developed over the last 40 years to deal with this primary aspect are described by giving full proofs of a number of basic and by now classical results. The central theme is the interplay between the spectral theorÿ of N ¦-body Hamiltonians and the space-time and phase-space analysis of bound states and scattering states.

Universal Properties of a Trapped Two-Component Fermi Gas at Unitarity

Physical Review Letters, 2007

We treat the trapped two-component Fermi system, in which unlike fermions interact through a two-body short-range potential having no bound state but an infinite scattering length. By accurately solving the Schrödinger equation for up to N = 6 fermions, we show that no manybody bound states exist other than those bound by the trapping potential, and we demonstrate unique universal properties of the system: Certain excitation frequencies are separated by 2 ω, the wavefunctions agree with analytical predictions and a virial theorem is fulfilled. Further calculations up to N = 30 determine the excitation gap, an experimentally accessible universal quantity, and it agrees with recent predictions based on a density functional approach.

Stability of inhomogeneous multicomponent Fermi gases

Physical Review A, 2008

Two-component equal-mass Fermi gases, in which unlike atoms interact through a short-range two-body potential and like atoms do not interact, are stable even when the interspecies s-wave scattering length becomes infinitely large. Solving the many-body Schrödinger equation within a hyperspherical framework and by Monte Carlo techniques, this paper investigates how the properties of trapped two-component gases change if a third or fourth component are added. If all interspecies scattering lengths are equal and negative, our calculations suggest that both three-and four-component Fermi gases become unstable for a certain critical set of parameters. The relevant length scale associated with the collapse is set by the interspecies scattering length and we argue that the collapse is, similar to the collapse of an attractive trapped Bose gas, a many-body phenomenon. Furthermore, we consider a three-component Fermi gas in which two interspecies scattering lengths are negative while the other interspecies scattering length is zero. In this case, the stability of the Fermi system is predicted to depend appreciably on the range of the underlying two-body potential. We find parameter combinations for which the system appears to become unstable for a finite negative scattering length and parameter combinations for which the system appears to be made up of weakly-bound trimers that consist of one fermion of each species.

Singular perturbation theory for interacting fermions in two dimensions

Physical Review B, 2005

We consider a system of interacting fermions in two dimensions beyond the second-order perturbation theory in the interaction. It is shown that the mass-shell singularities in the self-energy, arising already at the second order of the perturbation theory, manifest a non-perturbative effect: an interaction with the zero-sound mode. Resumming the perturbation theory for a weak, shortrange interaction and accounting for a finite curvature of the fermion spectrum, we eliminate the singularities and obtain the results for the quasi-particle self-energy and the spectral function to all orders in the interaction with the zero-sound mode. A threshold for emission of zero-sound waves leads a non-monotonic variation of the self-energy with energy (or momentum) near the mass shell. Consequently, the spectral function has a kink-like feature. We also study in detail a non-analytic temperature dependence of the specific heat, C(T ) ∝ T 2 . It turns out that although the interaction with the collective mode results in an enhancement of the fermion self-energy, this interaction does not affect the non-analytic term in C(T ) due to a subtle cancellation between the contributions from the real and imaginary parts of the self-energy. For a short-range and weak interaction, this implies that the second-order perturbation theory suffices to determine the non-analytic part of C(T ). We also obtain a general form of the non-analytic term in C(T ), valid for the case of a generic Fermi liquid, i.e., beyond the perturbation theory.