\nonumber \\
  & & + \sum_{\langle ij \rangle} V_{hh} \,h_i^\dagger 
  h_i^{\vphantom\dagger} h_j^\dagger h_j^{\vphantom\dagger}
  \label{tJ2}\end{aligned}$$ where $\mu_h = 2E_{\rm \scriptscriptstyle G} - U + V + t^2/(2V) -
t^2/[2(2E_{\rm \scriptscriptstyle G}+V)]$ is the hole “chemical potential”, $t_h = t^2/(2V)$, $V_{hh} =  t^2/V - {1 \over
  2}t^2/(2E_{\rm \scriptscriptstyle G}+V)$, and ${\cal P}_{ij} = ({3 \over 2}+ 
2\vec{S}_{ia}\cdot\vec{S}_{ja})({3 \over 2} +
2\vec{S}_{ib}\cdot\vec{S}_{jb})$ is the operator which interchanges the spin states at sites $i$ and $j$. Like in a conventional doped anti-ferromagnet, the presence of the ${\cal P}_{ij}$ operator in the hopping term leads to difficulties of hole motion in an antiferromagnetic spin background. Naive successive hopping of a single hole in an antiferromagnetic state results in a generalization of the well-known “string” of misaligned spins in its wake.

Introducing the $a-b$ hopping term (Eq. \[nnn\]) affects the system in several ways. There are renormalizations of the coupling constants in Eq. \[tJ2\] and Eq. \[tJ1\], of order $t_{ab}^2/V$, $t_{ab}^2/(V+4E_G)$. Since, by assumption, $t_{ab} \ll t$, these are negligible. New exchange couplings are also generated between next-nearest-neighbor $a$ and $b$ spins, which were not previously present. Because they are small, unfrustrating, and break no additional symmetries, these are also negligible. The most important effect is 