Effective Forces Between Quantum Bound States

Recent ab initio lattice studies have found that the interactions between alpha particles (${}^4\mathrm{He}$ nuclei) are sensitive to seemingly minor details of the nucleon-nucleon force, such as interaction locality. In order to uncover the essential physics of this puzzling phenomenon without unnecessary complications, we study a simple model involving two-component fermions in one spatial dimension. We probe the interaction between two bound dimers for several different particle-particle interactions and measure an effective potential between the dimers using external point potentials which act as numerical tweezers. We find that the strength and range of the local part of the particle-particle interactions play a dominant role in shaping the interactions between the dimers and can even determine the overall sign of the effective potential.

Figure 1

Illustration of the external potential acting as numerical tweezers by holding the down-spin particles at separation distance $n$.

Figure 2

The tweezer effective potential potential versus separation distance for interactions 1 and 3 for dimer binding energy $1/40$ and tweezer couplings $u=0.100$, 0.125, 0.150.

Figure 3

The tweezer effective potential versus separation distance for interactions 1–5 for dimer binding energy $1/40$ and tweezer coupling $u=0.100$.

Figure 4

Local part of the particle-particle interaction $I_{i,i}$ versus relative particle separation $i$ for interactions 1–5 and dimer binding energy $1/40$.

Figure 5

The tweezer effective potential versus separation distance for interactions 1–5, with dimer binding energy $1/100$ and tweezer coupling $u=0.100$.