Generalized Efimov Effect in One Dimension

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.

Figure 1

Hyperangular domain partitioning.

Figure 2

(a) Critical mass ratio as a function of $\alpha$ for bosons: upper blue (lower orange) line is the Numerov numerics for the plus (minus) branch, dashed lines are analytic asymptotes near $\alpha =0$ and $\alpha =1/8$ (see text). The shaded regions (i), (ii), and (iii) denote the regimes in which the Efimov effect does not occur, occurs for the minus-branch solution only, and occurs for both minus- and plus-branch solution, respectively. (b) Same as (a) but for fermions.

Figure 3

Two identical dipoles and a charge confined in one dimension.