Efimov effect for two particles on a semi-infinite line

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.

Discrete scale-invariant boson-fermion duality in one dimension

“…We showed that, in addition to the unbroken phase, there exist two additional phases in the scale-invariant two-body problem: one is the discrete scale invariant phase in which two distinct geometric series of two-body bound states appear, and the other is the discrete scale invariant phase in which a single geometric series of two-body bound states appear. This is in sharp contrast to the two-body problem of identical particles on the half line [6], where there arises at most a single geometric series of two-body bound states.…”

“…This paper is aimed at studying breakdown of continuous scale invariance to discrete scale invariance under the (2) family of two-body contact interactions. To simplify the analysis, we shall focus on two-body problems on the half line, which is exactly solvable and known to exhibit discrete scale invariance for the case of identical particles [6]. We shall show that, in a certain subspace of the parameter space of (2), there arise two distinct channels that admit geometric sequences of two-body bound states.…”