Non-Fermi Liquid Behavior of the $t$-$J$ Model in the Strange Metal Phase: $U(1)$ Gauge Theory Consistent with Local Constraints

Abstract: In the slave particle representation with $U(1)$ gauge symmetry, local constraints on physical states characterized by various mean field solutions belong to Dirac's second-class ones. Although constrained systems are extensively investigated, realistic methods to solve the gauge theory problem with second-class constraints are yet to be developed. We formulate a Becchi-Rouet-Stora-Tyutin (BRST) quantization theory, called consistent $U(1)$ gauge theory, that is consistent with both first- and second-class local constraints for strongly correlated condensed matter systems. In our consistent $U(1)$ gauge theory, the redundant gauge degrees of freedom are removed by proper gauge fixing conditions while the constraints are exactly retained and the gauge invariance is guaranteed by the BRST symmetry. Furthermore, the gauge fixing conditions endow the gauge field with dynamics. This turns the strongly correlated electron model into a weakly coupled slave boson model, so most of the system's physical properties can be calculated by the conventional quantum many-body perturbation method. We focus on the property of the strange metal phase in the $t$-$J$ model. The electron momentum distribution and the spectral function are calculated, and the non-Fermi liquid behavior agrees with the angle-resolved photoemission spectroscopy measurements for cuprate materials. We also study the electromagnetic responses of the strange metal state. The observed non-Fermi liquid anomalies are captured by our calculations. Especially, we find that the Hall resistivity decreases as temperature increases, and the sign of the Hall resistivity varies from negative to positive when the dopant concentration varies from optimal doping to underdoping in the strange metal regime.