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arXiv:2306.12157 · Letters in Mathematical Physics

Rigorous derivation of the Efimov effect in a simple model

Davide Fermi, Daniele Ferretti, Alessandro Teta

Published Nov 3, 2023• 6 citations
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Abstract

Abstract We consider a system of three identical bosons in $$\mathbb {R}^3$$ R 3 with two-body zero-range interactions and a three-body hard-core repulsion of a given radius $$ a > 0$$ a > 0 . Using a quadratic form approach, we prove that the corresponding Hamiltonian is self-adjoint and bounded from below for any value of a . In particular, this means that the hard-core repulsion is sufficient to prevent the fall to the center phenomenon found by Minlos and Faddeev in their seminal work on the three-body problem in 1961. Furthermore, in the case of infinite two-body scattering length, also known as unitary limit, we prove the Efimov effect, i.e., we show that the Hamiltonian has an infinite sequence of negative eigenvalues $$E_n$$ E n accumulating at zero and fulfilling the asymptotic geometrical law $$\;E_{n+1} / E_n \; \rightarrow \; e^{-\frac{2\pi }{s_0}}\,\; \,\text {for} \,\; n\rightarrow +\infty $$ E n + 1 / E n → e - 2 π s 0 for n → + ∞ holds, where $$s_0\approx 1.00624$$ s 0 ≈ 1.00624 .

DOI
10.1007/s11005-023-01734-3

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Abstract We consider a system of three identical bosons in R3\mathbb {R}^3R3 R 3 with two-body zero-range interactions and a three-body hard-core repulsion of a given radius a>0 a > 0a>0 a > 0 . Using a quadratic form approach, we prove that the corresponding Hamiltonian is self-adjoint and bounded from below for any value of a . In particular, this means that the hard-core repulsion is sufficient to prevent the fall to the center phenomenon found by Minlos and Faddeev in their seminal work on the three-body problem in 1961. Furthermore, in the case of infinite two-body scattering length, also known as unitary limit, we prove the Efimov effect, i.e., we show that the Hamiltonian has an infinite sequence of negative eigenvalues EnE_nEn​ E n accumulating at zero and fulfilling the asymptotic geometrical law   En+1/En  →  e−2πs0    for   n→+∞\;E_{n+1} / E_n \; \rightarrow \; e^{-\frac{2\pi }{s_0}}\,\; \,\text {for} \,\; n\rightarrow +\infty En+1​/En​→e−s0​2π​forn→+∞ E n + 1 / E n → e - 2 π s 0 for n → + ∞ holds, where s0≈1.00624s_0\approx 1.00624s0​≈1.00624 s 0 ≈ 1.00624 .
In this note we discuss the Efimov effect emerging in a three-particle quantum system with zero-range interactions. In particular, we consider two non-interacting identical bosons plus a different lighter particle such that the interaction between a boson and the light particle is resonant. We also assume the validity of the Born-Oppenheimer approximation. Under these conditions, we show that the three-particle system exhibits infinitely many negative eigenvalues which accumulate at zero and satisfy the universal geometrical law characterising the Efimov effect. The result we find is a generalisation of previous results recently obtained in [13, 24].
The Efimov effect is the appearance of an infinite tower of bound states, E n 3 , 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 E n +1 3 / E n 3 = e −2π/ s 0 with s 0 ≈ 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.
By studying the eigenvalue spectrum of the Faddeev kernel in a certain singular limit, we give an independent proof of an effect recently deduced by Efimov: When three identical particles interact via short-range pairwise potentials, the number of three-body bound states grows without limit when the pairwise scattering length aaa becomes large. [The number of bound states is then roughly $(\frac{1}{\ensuremath{\pi}})\mathrm{ln}(\ensuremath{\Lambda}|a|)$, where $\ensuremath{\Lambda}$ is a momentum cutoff]. We extend our proof to the case where only two particles are identical and show that Efimov's effect persists in the special limiting cases with two heavy and one light particle, and with two light and one heavy particle.
The Efimov effect (in a broad sense) refers to the onset of a geometric sequence of many-body bound states as a consequence of the breakdown of continuous scale invariance to discrete scale invariance. While originally discovered in three-body problems in three dimensions, the Efimov effect has now been known to appear in a wide spectrum of many-body problems in various dimensions. Here, we introduce a simple, exactly solvable toy model of two identical bosons in one dimension that exhibits the Efimov effect. We consider the situation where the bosons reside on a semi-infinite line and interact with each other through a pairwise δ-function potential with a particular position-dependent coupling strength that makes the system scale invariant. We show that, for sufficiently attractive interaction, the bosons are bound together, and a new energy scale emerges. This energy scale breaks continuous scale invariance to discrete scale invariance and leads to the onset of a geometric sequence of two-body bound states. We also study the two-body scattering off the boundary and derive the exact reflection amplitude that exhibits discrete scale invariance.
The Efimov phenomenon manifests itself as an emergent discrete scaling symmetry in the quantum three-body problem. In the unitarity limit, it leads to an infinite tower of three-body bound states with energies forming a geometric sequence. In this work, we study the evolution of these so-called Efimov states using relativistic scattering theory. We identify them as poles of the three-particle SSS matrix in the complex energy plane, and we study how they transform from virtual states through bound states to resonances when we change the interaction strength. We dial the scattering parameters toward the unitarity limit and observe the emergence of the universal scaling of energies and couplings---a behavior known from the nonrelativistic case. We additionally find that Efimov resonances follow unusual, cyclic trajectories accumulating at the three-body threshold and then disappear at some values of the two-body scattering length. We propose a partial resolution to this ``missing states'' problem.
To regularize the three-body problem, Minlos and Faddeev suggested a modification of zero-range model, which diminishes interaction at the triple-collision point. The analysis reveals that this regularization results in four alternatives depending on the regularization parameter σ. Explicitly, Efimov or Thomas effects remain for σ<σc\sigma < {{\sigma }_{c}}σ<σc​ , the additional boundary conditions of two types should be imposed at the triple-collision point for σc⩽σ<σe{{\sigma }_{c}} \leqslant \sigma < {{\sigma }_{e}}σc​⩽σ<σe​ and σe<σ<σr{{\sigma }_{e}} < \sigma < {{\sigma }_{r}}σe​<σ<σr​ , and the problem is regularized for σ⩾σr\sigma \geqslant {{\sigma }_{r}}σ⩾σr​ . Critical values σc<σe<σr{{\sigma }_{c}} < {{\sigma }_{e}} < {{\sigma }_{r}}σc​<σe​<σr​ separating different alternatives are determined both for a two-component three-body system and for three identical bosons.