The fractal geometry and the mapping of Efimov states to Bloch states

TL;DR

This paper reveals the fractal geometry underlying Efimov states, linking their discrete scale invariance in momentum space to Bloch states, and connects Efimov physics to atomic collapse phenomena.

Contribution

It introduces a novel geometric and mathematical framework for understanding Efimov states through fractal structures and mapping to Bloch states, providing new insights into three-body quantum systems.

Findings

Efimov states exhibit a fractal structure as a logarithmic spiral in momentum space.

The scattering amplitude's discrete scale invariance is characterized by a Weierstrass function and a zeta function pole structure.

Mapping to Bloch states simplifies the three-body problem to a single-particle lattice model.

Abstract

Efimov states are known to have a discrete real space scale invariance, working in momentum space we identify the relevant discrete scale invariance for the scattering amplitude defining its Weierstrass function as well. Through the use of the mathematical formalism for discrete scale invariance for the scattering amplitude we identify the scaling parameters from the pole structure of the corresponding zeta function, it's zeroth order pole is fixed by the Efimov physics. The corresponding geometrical fractal structure for Efimov physics in momentum space is identified as a ray across a logarithmic spiral. This geometrical structure also appears in the physics of atomic collapse in the relativistic regime connecting it to Efimov physics. Transforming to logarithmic variables in momentum space we map the three-body scattering amplitude into Bloch states and the ladder of energies of the Efimov states are simply obtained interms of the Bohr-Sommerfeld quantization rule. Thus through the mapping the complex problem of three-body short range interaction is transformed to that of a non-interacting single particle in a discrete lattice.