Heisenberg Symmetry and Collective Modes of One Dimensional Unitary Correlated Fermions

TL;DR

This paper uncovers a Heisenberg symmetry in one-dimensional unitary fermions, enabling exact mappings and revealing unique collective modes, contrasting with two-dimensional behaviors.

Contribution

It introduces the Heisenberg symmetry in 1D fermionic systems and derives exact mappings and collective mode predictions, extending understanding of unitary fermions.

Findings

Exact map between interacting and non-interacting systems in 1D

Prediction of collective modes at integral harmonic frequencies in 1D

Contrast with 2D breathing modes at 2ω

Abstract

The correlated fermionic many-particle system, near infinite scattering length, reveals an underlying Heisenberg symmetry in one dimension, as compared to an $SO(2,1)$ symmetry in two dimensions. This facilitates an exact map from the interacting to the non-interacting system, both with and without a harmonic trap, and explains the short-distance scaling behavior of the wave-function. Taking advantage of the phenomenological Calogero-Sutherland-type interaction, motivated by the density functional approach, we connect the ground-state energy shift, to many-body correlation effect. For the excited states, modes at integral values of the harmonic frequency $\omega$, are predicted in one dimension, in contrast to the breathing modes with frequency $2\omega$ in two dimensions.