Efimov effect for two particles on a semi-infinite line

TL;DR

This paper presents an exactly solvable one-dimensional model demonstrating the Efimov effect, where two bosons on a semi-infinite line form a geometric sequence of bound states due to scale invariance breaking.

Contribution

It introduces a simple, exactly solvable toy model in one dimension showing the Efimov effect with novel boundary conditions and scale invariance breaking mechanisms.

Findings

Existence of a geometric sequence of two-body bound states.

Exact reflection amplitude exhibits log-periodicity.

Scale invariance is broken, leading to discrete scale invariance.

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 $\delta$-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 a log-periodicity. This article is intended for students and non-specialists interested in discrete scale invariance.