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Physical Review A

Binding properties of the (2+1)-fermion system with zero-range interspecies interaction

Alessandro Michelangeli, Christian Schmidbauer

Published May 6, 2013• 17 citations
Publisher

Abstract

We study analytically and numerically the binding properties, in particular the ground state, of the so-called $(2+1)$-fermion system, which is a three-dimensional system of two identical fermions interacting with a third particle of different species through a zero-range interaction. We model the system with a specific self-adjoint point interaction Hamiltonian recently constructed in the mathematical literature. First we characterize the internal symmetry of the bound states in the attractive case. Then we show that the system confines in a precise regime of masses and that its ground-state energy stays finite as the mass becomes close to the critical point of the collapse of the system.

DOI
10.1103/physreva.87.053601

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We study analytically and numerically the binding properties, in particular the ground state, of the so-called (2+1)(2+1)(2+1)-fermion system, which is a three-dimensional system of two identical fermions interacting with a third particle of different species through a zero-range interaction. We model the system with a specific self-adjoint point interaction Hamiltonian recently constructed in the mathematical literature. First we characterize the internal symmetry of the bound states in the attractive case. Then we show that the system confines in a precise regime of masses and that its ground-state energy stays finite as the mass becomes close to the critical point of the collapse of the system.
We qualify the main features of the spectrum of the Hamiltonian of point interaction for a three-dimensional quantum system consisting of three point-like particles, two identical fermions, plus a third particle of different species, with two-body interaction of zero range. For arbitrary magnitude of the interaction, and arbitrary value of the mass parameter (the ratio between the mass of the third particle and that of each fermion) above the stability threshold, we identify the essential spectrum, localise and prove the finiteness of the discrete spectrum, qualify the angular symmetry of the eigenfunctions, and prove the monotonicity of the eigenvalues with respect to the mass parameter. We also demonstrate the existence of bound states in a physically relevant regime of masses.
Abstract We introduce a 3D model, and we study its stability, consisting of two distinct pairs of identical fermions coupled with a two-body interaction between fermions of different species, whose effective range is essentially zero (a so called (2+2)-fermionic system with zero-range interaction). The interaction is modelled by implementing the celebrated (and ubiquitous in the literature of this field) Bethe–Peierls contact condition with given two-body scattering length within the Krein–Višik–Birman theory of extensions of semi-bounded symmetric operators, in order to make the Hamiltonian a well-defined (self-adjoint) physical observable. After deriving the expression for the associated energy quadratic form, we show analytically and numerically that the energy of the model is bounded below, thus describing a stable system.
For sufficiently large mass ratios the attractive exchange force caused by a single light atom interacting with a few heavy identical fermions can overcome their Fermi degeneracy pressure and bind them into an N+1 N + 1 cluster. Here, by using a mean-field approach valid for large N N , we find that N+1 N + 1 clusters can attract each other and form a self-bound charge density wave, the properties of which we fully characterize. Our work shows that there are no fundamental obstacles for having self-bound states in fermionic mixtures with zero-range interactions.
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 renormalized quadratic (or energy) form [Formula: see text] 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 [Formula: see text] is closed and bounded from below. As a consequence, [Formula: see text] 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 [Formula: see text] 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.
A quantum-mechanical three-body problem for two identical fermions of mass m and a distinct particle of mass m 1 in the universal limit of zero-range two-body interaction is studied. For the unambiguous formulation of the problem in the interval μ r < m / m 1 ≤ μ c ( μ r ≈ 8.619 and μ c ≈ 13.607 ) an additional parameter b determining the wave function near the triple-collision point is introduced; thus, a one-parameter family of self-adjoint Hamiltonians is defined. The dependence of the bound-state energies on m/m 1 and b in the sector of angular momentum and parity L P = 1 − is calculated and analysed with the aid of a simple model.