Consistent gauge theories for the slave particle representation of the strongly correlated $t$-$J$ model

Abstract

We aim to clarify the confusion and inconsistency in our recent works [1,2], and to address the incompleteness therein. In order to avoid the ill-defined nature of the free propagator of the gauge field in the ordered states of the $t$-$J$ model, we adopted a gauge fixing that was not of the Becchi-Rouet-Stora-Tyutin (BRST) exact form in our previous work [2]. This led to the situation where Dirac's second-class constraints, namely, the slave particle number constraint and the Ioffe-Larkin current constraint, were not rigorously obeyed. Here we show that a consistent gauge fixing condition that enforces the exact constraints is BRST-exact in our theory. An example is the Lorenz gauge. On the other hand, we prove that although the free propagator of the gauge field in the Lorenz gauge is ill-defined, the full propagator is still well-defined. This implies that the strongly correlated $t$-$J$ model can be exactly mapped to a perturbatively controllable theory within the slave particle representation.