Identical Wells, Symmetry Breaking, and the Near-Unitary Limit

TL;DR

This paper explores how symmetry breaking affects energy level splitting in one-dimensional atomic systems near the unitary limit, using geometric and algebraic methods linked to graph theory.

Contribution

It introduces a geometric analogy with graph theory and develops an algebraic framework to calculate energy splitting due to symmetry breaking near the unitary limit.

Findings

Symmetry breaking explains energy level splitting.

Graph theory analogy aids understanding of symmetry effects.

Algebraic methods enable calculation of energy differences.

Abstract

Energy level splitting from the unitary limit of contact interactions to the near unitary limit for a few identical atoms in an effectively one-dimensional well can be understood as an example of symmetry breaking. At the unitary limit in addition to particle permutation symmetry there is a larger symmetry corresponding to exchanging the $N!$ possible orderings of $N$ particles. In the near unitary limit, this larger symmetry is broken, and different shapes of traps break the symmetry to different degrees. This brief note exploits these symmetries to present a useful, geometric analogy with graph theory and build an algebraic framework for calculating energy splitting in the near unitary limit.