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 must be BRST-exact. 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.