Scaling and universality in two dimensions: three-body bound states with short-ranged interactions

TL;DR

This paper investigates universal three-body bound states in two-dimensional systems with short-range interactions, deriving analytical formulas and mapping the conditions for the existence of multiple stable states based on mass ratios and two-body energies.

Contribution

It provides analytical expressions and universal phase diagrams for three-body bound states in 2D, highlighting the dependence on mass ratios and two-body energies, which was not previously characterized.

Findings

Three-body energy is a universal function of two-body energies and mass ratio.

Maximum of two stable bound states for mass ratio > 0.22.

Number of bound states varies with energy and mass ratio, mapped in a phase diagram.

Abstract

The momentum space zero-range model is used to investigate universal properties of three interacting particles confined to two dimensions. The pertinent equations are first formulated for a system of two identical and one distinct particle and the two different two-body subsystems are characterized by two-body energies and masses. The three-body energy in units of one of the two-body energies is a universal function of the other two-body energy and the mass ratio. We derive convenient analytical formulae for calculations of the three-body energy as function of these two independent parameters and exhibit the results as universal curves. In particular, we show that the three-body system can have any number of stable bound states. When the mass ratio of the distinct to identical particles is greater than 0.22 we find that at most two stable bound states exist, while for two heavy and one light mass an increasing number of bound states is possible. The specific number of stable bound states depends on the ratio of two-body bound state energies and on the mass ratio and we map out an energy-mass phase-diagram of the number of stable bound states. Realizable systems of both fermions and bosons are discussed in this framework.