Dimensional crossover and finite-range effects in a quasi-two-dimensional gas of fermionic dimers

TL;DR

The paper investigates dimensional crossover and finite-range effects in a strongly interacting, quasi-two-dimensional Fermi gas of dimers by combining FN-DMC calculations with analytical GP and perturbative approaches. It demonstrates that the system can be effectively described as a molecular Bose gas in quasi-2D, and it evaluates where mean-field and beyond-mean-field theories apply by comparing to 2D Bose gas predictions. A key contribution is the analytical modeling of the transverse density profile and its broadening as interactions strengthen, linking a 3D description to 2D behavior. The results provide a benchmark for bosonic descriptions of fermionic dimers and yield insights into the crossover from 2D to 3D in strongly correlated quantum gases, with potential implications for understanding stability and excitations in molecular Bose systems.

Abstract

We investigate the ground-state properties of ultracold two-component Fermi gases in the presence of a transverse harmonic potential, focusing on the strongly interacting regime in which pairs of fermions form tightly bound molecules. Using the fixed-node diffusion Monte Carlo method, we calculate the equation of state and density profiles for the full fermionic system, which allows us to address the importance of finite-range corrections arising from the internal fermionic structure of the composite bosons. We interpret the results in terms of a molecular Bose gas in quasi-two-dimensional confinement and compare them with theoretical predictions for a weakly interacting two-dimensional Bose gas, identifying the range of validity of mean-field and beyond-mean-field descriptions. We also develop an analytical theory for the transverse density profile, capturing its broadening with increasing interaction strength. This work provides a benchmark for an effective bosonic description of strongly bound fermionic dimers and offers new insights into the three- to two-dimensional crossover.

Figures (4)

• Figure 1: Sketch of the system. Fermionic spin-up and spin-down atoms, represented as spheres with an arrow, are confined to a two-dimensional $(x,y)$ plane indicated by the blue rectangle. The finite width of the blue rectangle illustrates the harmonic confinement along the $z$ direction. Pairs of spin-up and spin-down fermions form tightly bound dimers, which can be approximately treated as composite bosons when dimer size $a_F$ is small compared to the mean interparticle distance, $n_F^{-1/2}$.
• Figure 2: Ground-state energy in units of harmonic oscillator level spacing. Panel a) energy per dimer obtained from FN-DMC calculation (blue circles), compared to the binding energy of two atoms interacting via square-well (red solid line) or contact (green dashed line) potentials as a function of the ratio between the 3D fermionic scattering length $a_F$ and oscillator length $l_F$. Panel b) Energy after subtraction of the two–body binding energy in free space $\varepsilon_b(0)$ (red crosses) or in trapped geometry $\varepsilon_b(\omega)$ (blue circles). Ground–state energy of the harmonic oscillator $\hbar\omega/2$ is shown with a green dashed line. Panel c) Energy of 2D motion as a function of the dimensionless 2D bosonic gas parameter $n_B a_{2D}^2$, compared with the mean–field (MF) (\ref{['Eq:EoS:2D:MF']}) and beyond–mean–field (BMF) (\ref{['Eq:EoS:2D:BMF']}) theoretical predictions for the 2D Bose gas energy. The FN DMC calculations are done using $N_F=66$ fermions.
• Figure 3: Energy per boson, after subtracting both the two–body binding energy, $\varepsilon_b(\omega)$, and the harmonic–oscillator ground-state energy, expressed in units of the mean–field (MF) equation of state (\ref{['Eq:EoS:2D:MF']}). Symbols show DMC results obtained using different prescriptions for extracting $a_{2D}$ from the 3D parameters. Red circles -- both bosonic and fermionic ranges set to zero; blue stars -- finite bosonic range and zero fermionic range; green squares -- finite fermionic range and zero bosonic range; pink diamonds -- both ranges are finite. Analytic curves show different theoretical descriptions. For a strictly 2D Bose gas, MF (\ref{['Eq:EoS:2D:MF']}) and BMF (\ref{['Eq:EoS:2D:BMF']}) predictions are shown as dashed and solid black lines. Quasi-2D corrections are included on top of the BMF theory by including the negative correction obtained from the variational Gross–Pitaevskii theory (\ref{['Eq:Q2Denergy']}) (dash-dotted line) and adiabatic perturbation theory (\ref{['olsh']}) (dash-dot-dotted line).
• Figure 4: Dimensional crossover from two- to three dimensions. Panel (a): Width $\sigma$ of the Gaussian density profile, in units of the fermionic oscillator length $l_{\mathrm{F}}$, as a function of the 3D fermionic scattering length $a_F$ in units of $l_{\mathrm{F}}$. The black dotted line corresponds to the case of constant $\sigma = l_B$, the solid black line shows the numerical solution obtained from the minimization of the Gross–Pitaevskii functional, the dotted green line is the analytical solution in Eq. \ref{['GP']} and the dashed red line is the solution obtained from adiabatic perturbation theory Merloti, that coincides with VGP to lowest order, as shown in the EM. Blue circles represent $\sigma$ values extracted from a Gaussian fit to the DMC density profile along the $z$ direction. The insets display the DMC density profiles along $z$ at $a_F/l_{F}=0.09$ (b) and $a_F/l_{F}=0.53$ (c), compared with the variational Gross–Pitaevskii (VGP) profiles and with the harmonic–oscillator ground–state density.