Three s-wave interacting fermions under anisotropic harmonic confinement: Dimensional crossover of energetics and virial coefficients

TL;DR

This paper provides exact solutions for three interacting fermions in various anisotropic harmonic traps, analyzing how trap geometry influences energy levels and virial coefficients, with implications for thermodynamics.

Contribution

It offers a comprehensive analysis of three-fermion systems under diverse trap geometries, including the transition between quasi- and strictly lower-dimensional regimes, and connects these to virial coefficients.

Findings

Eigenenergies depend on trap geometry and interaction strength.

Virial coefficients for anisotropic traps closely match spherical cases at relevant temperatures.

Dimensional crossover significantly affects system energetics and thermodynamics.

Abstract

We present essentially exact solutions of the Schroedinger equation for three fermions in two different spin states with zero-range s-wave interactions under harmonic confinement. Our approach covers spherically symmetric, strictly two-dimensional, strictly one-dimensional, cigar-shaped, and pancake-shaped traps. In particular, we discuss the transition from quasi-one-dimensional to strictly one-dimensional and from quasi-two-dimensional to strictly two-dimensional geometries. We determine and interpret the eigenenergies of the system as a function of the trap geometry and the strength of the zero-range interactions. The eigenenergies are used to investigate the dependence of the second- and third-order virial coefficients, which play an important role in the virial expansion of the thermodynamic potential, on the geometry of the trap. We show that the second- and third-order virial coefficients for anisotropic confinement geometries are, for experimentally relevant temperatures, very well approximated by those for the spherically symmetric confinement for all s-wave scattering lengths.