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 (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.

arXiv:2201.10869American Journal of PhysicsSep 22, 20222 citations

Charged analogs of the Efimov effect

Yusuke Nishida

Nov 1, 20220 citations

Analytic representation of the Efimov effect in the helium trimer

LAWRENCE L. JUN. LOHR, S. M. Blinder

Exact solutions for the low-temperature helium dimer and trimer, $^{4}\mathrm{He}_{2}$ and $^{4}\mathrm{He}_{3}$ , are derived, based on our $\ensuremath{\delta}$ function model for the interatomic potential. For the trimer, the Faddeev equations are shown to be separable in hyperspherical coordinates, with the $S$ -wave alone giving an exact solution. The parameters ${\ensuremath{\lambda}}_{0}$ and ${r}_{0}$ are fitted to accurate computations on the dimer and trimer. Excited states of the trimer are shown to exhibit the Efimov effect, whereby artificially reducing the strength of the two-body potential causes an infinite number of weakly-bound levels to condense out of the continuum. All the features anticipated by Efimov are quantitatively reproduced within our model. Since short-range details of the intermolecular forces are not relevant, our results can be considered to be universally applicable.Exact solutions for the low-temperature helium dimer and trimer, $^{4}\mathrm{He}_{2}$ and $^{4}\mathrm{He}_{3}$, are derived, based on our $\ensuremath{\delta}$ function model for the interatomic potential. For the trimer, the Faddeev equations are shown to be separable in hyperspherical coordinates, with the $S$-wave alone giving an exact solution. The parameters ${\ensuremath{\lambda}}_{0}$ and ${r}_{0}$ are fitted to accurate computations on the dimer and trimer. Excited states of the trimer are shown to exhibit the Efimov effect, whereby artificially reducing the strength of the two-body potential causes an infinite number of weakly-bound levels to condense out of the continuum. All the features anticipated by Efimov are quantitatively reproduced within our model. Since short-range details of the intermolecular forces are not relevant, our results can be considered to be universally applicable.

Physical Review AJun 2, 20045 citations

Exact solutions for the low-temperature helium dimer and trimer,

^{4}\mathrm{He}_{2}

and

^{4}\mathrm{He}_{3}

, are derived, based on our $\ensuremath{\delta}$ function model for the interatomic potential. For the trimer, the Faddeev equations are shown to be separable in hyperspherical coordinates, with the

S

-wave alone giving an exact solution. The parameters ${\ensuremath{\lambda}}_{0}$ and

{r}_{0}

are fitted to accurate computations on the dimer and trimer. Excited states of the trimer are shown to exhibit the Efimov effect, whereby artificially reducing the strength of the two-body potential causes an infinite number of weakly-bound levels to condense out of the continuum. All the features anticipated by Efimov are quantitatively reproduced within our model. Since short-range details of the intermolecular forces are not relevant, our results can be considered to be universally applicable.