Qualitative understanding of the sign of $t^{\prime }$ asymmetry in the extended $t-J$ model and relevance for pairing properties

Numerical calculations illustrate the effect of the sign of the next-nearest-neighbor hopping term $t^{\prime }$ on the two-hole properties of the $t-t^{\prime }-J$ model. Working mainly on two-leg ladders, in the $-1.0<~t^{\prime }/t<~1.0$ regime, it is shown that introducing $t^{\prime }$ in the $t-J$ model is equivalent to effectively renormalizing J, namely $t^{\prime }$ negative (positive) is equivalent to an effective $t-J$ model with smaller (bigger) J. This effect is present even at the level of a $2\times 2$ plaquette toy model, and was observed also in calculations on small square clusters. Analyzing the transition probabilities of a hole pair in the plaquette toy model, it is argued that the coherent propagation of such hole pair is enhanced by a constructive interference between both t and $t^{\prime }$ for $t^{\prime }>0.\mathrm{}$ This interference is destructive for $t^{\prime }<0.\mathrm{}$