Trimers in the resonant 2+1 fermionic problem on a narrow Feshbach resonance : Crossover from Efimovian to Hydrogenoid spectrum

TL;DR

This paper investigates the spectrum of three-body bound states in a fermionic system near a narrow Feshbach resonance, revealing a transition from Efimovian to hydrogenoid spectra as the mass ratio varies.

Contribution

It provides a detailed analytical and numerical analysis of the three-body spectrum, identifying the crossover from Efimovian to hydrogenoid behavior based on the mass ratio.

Findings

Trimer states appear at a critical mass ratio with Efimovian scaling.

The entire spectrum becomes geometric near the critical point.

For larger mass ratios, the spectrum transitions to hydrogenoid levels.

Abstract

We study the quantum three-body free space problem of two same-spin-state fermions of mass $m$ interacting with a different particle of mass $M$, on an infinitely narrow Feshbach resonance with infinite s-wave scattering length. This problem is made interesting by the existence of a tunable parameter, the mass ratio $\alpha=m/M$. By a combination of analytical and numerical techniques, we obtain a detailed picture of the spectrum of three-body bound states, within {\sl each} sector of fixed total angular momentum $l$. For $\alpha$ increasing from 0, we find that the trimer states first appear at the $l$-dependent Efimovian threshold $\alpha_c^{(l)}$, where the Efimov exponent $s$ vanishes, and that the {\sl entire} trimer spectrum (starting from the ground trimer state) is geometric for $\alpha$ tending to $\alpha_c^{(l)}$ from above, with a global energy scale that has a finite and non-zero limit. For further increasing values of $\alpha$, the least bound trimer states still form a geometric spectrum, with an energy ratio $\exp(2\pi/|s|)$ that becomes closer and closer to unity, but the most bound trimer states deviate more and more from that geometric spectrum and eventually form a hydrogenoid spectrum.