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I have been reading about the Efimov effect, which is that a system of three identical bosons has a particular excitation spectrum, including a bound state, independent of the underlying interaction and even if there aren't any two-body bound states. This paper mentions that Efimov effect can only be seen in $3$-dimensions. Can anyone explain in succinctly terms why occurs this? Is there a similar phenomena in, say, $1$-dimensional systems?

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The linked abstract does not say anything about the dimensionality of the Efimov effect. In the paper itself, however, we read the following:

"So far multi-magnon bound states have been observed with different experimental techniques, but mostly in quasi-one-dimensional compounds. Although the Efimov effect emerges only in three dimensions, the same spectroscopic measurements can be used to observe the emergent Efimov states of magnons."

The references are to the early 2000-s, The three-body problem with short-range interactions by Nielsen et al. and Universality in few-body systems with large scattering length by Braaten-Hammer. But the idea goes back to a 1980 paper of Lim and Maurone Nonexistence of the Efimov effect in two dimensions, who explain the reason as follows:

"Now in 3D and as attested by Efimov's construction the Efimov effect arises because of the appearance of an attractive three body force of $1/R^2$ character, cutoff at the lower end at $R\sim r_0$ by the nonresonant (or short-range) portion of the pair force, and at the higher end at $R\sim a$ by the resonant part (as determined by the pole term in the amplitude, corresponding to the bound or virtual state of two particles); $r_0$ and $a$ are related to the range of the two body force and the scattering length, respectively. Our analysis indicates that in the 2D trimer an attractive long-range force does not appear. This excludes any possibility of an Efimov effect in 2D."

"Any possibility" is a little too categorical, and applies to the standard bosonic setup only. In fact, the first author of the OP linked paper also co-authored Super Efimov effect of resonantly interacting fermions in two dimensions and Semi-super Efimov effect of two-dimensional bosons at a three-body resonance.

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