Omnipresent bound state of two holes in antiferromagnetic Bethe lattices

For decades, it has remained an open question, whether two dopants in the $t$--$J$ model bind in the strongly correlated regime of small spin couplings versus hopping, $J \ll t$. Here, we investigate this problem in Bethe lattice structures with Ising coupling $J_z$, and mainly focus on the case of a coordination number equal to 4. The special geometry circumvents subtle effects in regular lattices, but importantly still contains the non-trivial dependency on the rotational symmetry and particle statistics. It, furthermore, allows us to reach numerical convergence, and we conclusively answer whether binding occurs or not. In particular, we find that the rotationally symmetric $s$ waves unbind below $J_z \simeq 0.3 t$, which we unveil is tied to Pauli blocking of symmetrical hole configurations. This is further substantiated by the fact that holes in a bosonic spin environment shows strong -- superlinear -- binding at low $J_z / t$. Additionally, higher rotational fermionic $p$ and $d$ waves partially overcome the blocking, are perfectly degenerate, and bind for all $J_z / t$. In the strongly correlated regime of $J_z \ll t$, this binding occurs sublinearly with scaling $\sim t (J_z / t)^{3/2}$.