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arXiv:2201.10869 · American Journal of Physics

Efimov effect for two particles on a semi-infinite line

Satoshi Ohya

Published Sep 22, 2022• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1119/5.0086802

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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 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.
We study a one-dimensional quantum problem of two particles interacting with a third one via a scale-invariant subcritically attractive inverse square potential, which can be realized, for example, in a mixture of dipoles and charges confined to one dimension. We find that above a critical mass ratio, this version of the Calogero problem exhibits the generalized Efimov effect, the emergence of discrete scale invariance manifested by a geometric series of three-body bound states with an accumulation point at zero energy.
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.
Scale invariance in quantum mechanics can be broken in several ways. A well-known example is the breakdown of continuous scale invariance to discrete scale invariance, whose typical realization is the Efimov effect of three-body problems. Here we discuss yet another discrete symmetry to which continuous scale invariance can be broken: discrete phase invariance. We first revisit the one-body problem on the half line in the presence of an inverse-square potential -- the simplest example of nontrivial scale-invariant quantum mechanics -- and show that continuous scale invariance can be broken to discrete phase invariance in a small window of coupling constant. We also show that discrete phase invariance manifests itself as circularly distributed simple poles on Riemann sheets of the S-matrix. We then present three examples of few-body problems that exhibit discrete phase invariance. These examples are the one-body Aharonov-Bohm problem, a two-body problem of nonidentical particles in two dimensions, and a three-body problem of nonidentical particles in one dimension, all of which contain a codimension-two ``magnetic'' flux in configuration spaces.
In systems of ultracold atoms, the quantum Efimov effect can appear where identical bosons form an infinite tower of bound trimer states in the resonant limit, at the bound dimer dissociation threshold. The most characteristic feature of this effect is that their energy spectrum obey a geometric scaling law, which is universal in the sense that it emerges irrespective of the nature of the two body forces. Using a model potential, constructed to resemble the two body interaction between alkali atoms, which was fine tuned to control the scattering length, energy eigenvalues for the two- and threebody problem were calculated numerically. The results where fitted to the analytic theory and the appearance of the first Efimov state was positioned at a scattering length of -9.23rvdW , which is in good agreement with the universal value -9.2rvdW .
Abstract The quantum mechanical three-body problem is a source of continuing interest due to its complexity and not least due to the presence of fascinating solvable cases. The prime example is the Efimov effect where infinitely many bound states of identical bosons can arise at the threshold where the two-body problem has zero binding energy. An important aspect of the Efimov effect is the effect of spatial dimensionality; it has been observed in three dimensional systems, yet it is believed to be impossible in two dimensions. Using modern experimental techniques, it is possible to engineer trap geometry and thus address the intricate nature of quantum few-body physics as function of dimensionality. Here we present a framework for studying the three-body problem as one (continuously) changes the dimensionality of the system all the way from three, through two, and down to a single dimension. This is done by considering the Efimov favorable case of a mass-imbalanced system and with an external confinement provided by a typical experimental case with a (deformed) harmonic trap.