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

Area of Science:

• Quantum Many-Body Physics
• Condensed Matter Physics
• Atomic, Molecular, and Optical Physics

Background:

• Accurately describing strongly interacting quantum many-body systems is a major challenge in modern physics.
• One-dimensional (1D) systems offer fundamental insights due to their amenability to exact methods.
• An exact solution for 1D Fermi gases with external confinement remains elusive.

Purpose of the Study:

• To propose a powerful ansatz for the 1D Fermi gas in a harmonic potential near infinite short-range repulsion.
• To provide an exact solution for the experimentally relevant case of external confinement.
• To derive an effective spin-chain model and obtain impurity eigenstates analytically.

Main Methods:

• Development of a novel ansatz for the 1D Fermi gas.
• Comparison of ansatz results with numerically exact solutions for few- and many-body limits.
• Derivation of an effective Heisenberg spin-chain model.

Main Results:

• The proposed ansatz yields results indistinguishable from numerically exact data for a single impurity in a Fermi sea.
• An effective Heisenberg spin-chain model is derived for any spin-mixture.
• Impurity eigenstates are obtained analytically, revealing the emergence of Pascal's triangle in the ground-state wave function.

Conclusions:

• The developed ansatz provides a benchmark for strongly correlated physics in confined 1D systems.
• The findings are relevant for emerging quantum technologies requiring precise knowledge of 1D quantum states.
• This work offers an analytical solution for a previously intractable problem in quantum many-body physics.