Strong-coupling ansatz for the one-dimensional Fermi gas in a harmonic potential

TL;DR

This paper introduces a strong-coupling ansatz for the one-dimensional Fermi gas in a harmonic trap, accurately capturing the physics near infinite repulsion and providing analytical solutions for impurity states, with implications for quantum technologies.

Contribution

The authors propose a novel ansatz for strongly interacting 1D Fermi gases in harmonic potentials, deriving an effective spin-chain model and analytical impurity eigenstates, advancing understanding of correlated quantum systems.

Findings

Ansatz matches exact numerical results for few- and many-body cases.

Derived an analytical impurity eigenstate within the spin-chain model.

Ground-state wavefunction involves Pascal's triangle structure.

Abstract

A major challenge in modern physics is to accurately describe strongly interacting quantum many-body systems. One-dimensional systems provide fundamental insights since they are often amenable to exact methods. However, no exact solution is known for the experimentally relevant case of external confinement. Here, we propose a powerful ansatz for the one-dimensional Fermi gas in a harmonic potential near the limit of infinite short-range repulsion. For the case of a single impurity in a Fermi sea, we show that our ansatz is indistinguishable from numerically exact results in both the few- and many-body limits. We furthermore derive an effective Heisenberg spin-chain model corresponding to our ansatz, valid for any spin-mixture, within which we obtain the impurity eigenstates analytically. In particular, the classical Pascal's triangle emerges in the expression for the ground-state wavefunction. As well as providing an important benchmark for strongly correlated physics, our results are relevant for emerging quantum technologies, where a precise knowledge of one-dimensional quantum states is paramount.