Hyperspherical Analysis of Dimer-Dimer Scattering in One-Dimensional Systems

TL;DR

This study addresses 1D four-body dimer-dimer scattering in ultracold fermionic systems by applying the adiabatic hyperspherical representation (AHR) augmented with slow variable discretization (SVD). The sinh-cosh potential serves as a tractable model, enabling analytic benchmarks in integrable regimes and revealing non-integrable behavior, including tetramer-induced resonances. The authors develop a robust hyperspherical formulation, exploit symmetry to construct an efficient antisymmetrized basis, and use an $R$-matrix approach to extract energy-dependent scattering lengths $a_D(k)$, confirming universal relations such as $a_{\rm dd}=\tfrac{1}{2}a_{\rm aa}$ in the shallow-dimer limit. The results demonstrate the accuracy and flexibility of AHR+SVD for low-dimensional few-body scattering and provide a foundation for exploring universal physics in ultracold gases, including extensions to mass-imbalanced systems and excited-state dynamics.

Abstract

We present a comprehensive analysis of four-body scattering in one-dimensional (1D) quantum systems using the adiabatic hyperspherical representation (AHR). Focusing on dimer-dimer collisions between two species of fermions interacting via the sinh-cosh potential, we implement the slow variable discretization (SVD) method to overcome numerical challenges posed by sharp avoided crossings in the potential curves. Our numerical approach is benchmarked against exact analytical results available in integrable regimes, demonstrating excellent agreement. We further explore non-integrable regimes where no analytical solutions exist, revealing novel features such as resonant enhancement of the scattering length associated with tetramer formation. These results highlight the power and flexibility of the AHR+SVD framework for accurate few-body scattering calculations in low-dimensional quantum systems, and establish a foundation for future investigations of universal few-body physics in ultracold gases.

Figures (7)

• Figure 1: (a) Schematic illustration of the four-particle configuration. As the system is one-dimensional, only the horizontal displacement is physically meaningful; vertical offsets are used solely for visual clarity. (b) Mapping between the geometric configuration of particles and the hyperangular coordinates $\theta$ and $\phi$. (c) The four-body potential $V(R,\Omega)$ plotted as a function of angles at fixed hyperradius $R=10$. The pairwise two-body interaction is modeled using the $\sinh$-$\cosh$ potential with parameters $s=s^{\prime}=1.5$.
• Figure 2: Hyperspherical potential curves for $s=s^{\prime}=1.5$ in symmetry sectors: (a) $\left(p_{\theta},p_{\phi}\right)=\left(+1,+1\right)$ and (b) $\left(p_{\theta},p_{\phi}\right)=\left(+1,-1\right)$. Solid horizontal lines indicate dimer-dimer thresholds; dashed lines mark dimer--atom--atom thresholds; dash-dotted lines represent the atom-atom-atom-atom threshold at zero energy.
• Figure 3: Same as Fig. \ref{['fig:UR1']}, but for symmetry sectors: (a) $\left(p_{\theta},p_{\phi}\right)=\left(-1,+1\right)$ and (b) $\left(p_{\theta},p_{\phi}\right)=\left(-1,-1\right)$.
• Figure 4: Hyperspherical potential curves for $\left(p_{\theta},p_{\phi}\right)=\left(+1,+1\right)$ and $s=1.5$ with different $s^{\prime}$: (a) $s^{\prime}=1.9$ and (b) $s^{\prime}=0.9$. In panel (b), the second-lowest channel approaches a trimer--atom threshold due to weakened repulsion between like particles, indicating trimer formation.
• Figure 5: Channel functions for (a) dimer-dimer channel (b) dimer-atom-atom channel (c) Four-atom channel and (d) trimer-atom channel. The color scale indicates the probability density of the wave function, shown in arbitrary units.