Crossover From Strong to Weak Pairing States in $t-J-U$ Model Studied by A Slave Spin Method

We investigate the superconductivity in the $t-J-U$ model within a slave-spin formalism. We show that the BCS mean-field theory implemented with the slave spin formalism naturally predicts two distinct gaps which are the pairing gap of the spinons $\Delta_f$ and the Cooper pairing gap of the electrons $\Delta_{SC} = Z\Delta_f$, where $Z$ is the quasiparticle weight. Furthermore, we find that the nature of the superconducting state depends crucially on the interaction providing the pairing mechanism. For spin interactions, the bandwidth of the spinon hopping term is renormalized by $Z$ but the its pairing term is not. As a result, if $U$ exceeds the critical value for the Mott insulating state at half-filling, $Z$ develops a strong doping dependence, leading to a doping-driven crossover from strong to weak pairing states. In the strong pairing state, while $\Delta_f$ is enhanced as $x\to 0$, $\Delta_{SC}\sim x$ due to the renormalization of $Z$. In the weak pairing state, $Z$ does not change with $x$ significantly. Therefore, $\Delta_{SC}$ is mainly controlled by $\Delta_f$, and both of them go to zero at larger doping. The crossover from strong to weak pairing states is well captured by the slave spin formalism within reasonable range of parameters just at the mean-field level, indicating the slave spin formalism is a powerful tool to study correlated materials. For charge interactions, we find that the bandwidth of the spinon hopping term and its pairing term are renormalized by $Z$ and $Z^2$ respectively. Consequently, both $\Delta_f$ and $\Delta_{SC}$ are suppressed at small $x$ in large U, and no crossover will occur. The implication of our results for the superconducting states in correlated materials will be discussed.