Omnipresent bound state of two holes in antiferromagnetic Bethe lattices

TL;DR

This paper investigates whether two holes bind in the $t$--$J$ model on Bethe lattices with Ising coupling, revealing that binding depends on symmetry, particle statistics, and the ratio of $J_z$ to $t$, with specific regimes of binding and unbinding.

Contribution

It provides a conclusive analysis of hole binding in the $t$--$J$ model on Bethe lattices, highlighting the effects of symmetry, particle statistics, and coupling strength, which was previously unresolved.

Findings

$s$-wave holes unbind below $J_z \,\simeq\, 0.3 t$

Bosonic holes show strong superlinear binding at low $J_z / t$

Fermionic $p$ and $d$ waves bind for all $J_z / t$ and are degenerate

Abstract

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 $\sim t (J_z / t)^{3/2}$.