Orbital-selective bad metals due to Hund's rule and orbital anisotropy: a finite-temperature slave-spin treatment of the two-band Hubbard model

We study the finite-temperature properties of the half-filled two-band Hubbard model in the presence of Hund's rule coupling and orbital anisotropy. We use the mean-field treatment of the $Z_2$ slave-spin theory with a finite-temperature extension of the zero-temperature gauge variable previously developed by Hassan and de' Medici http://dx.doi.org/10.1103/PhysRevB.81.035106}{Phys. Rev. B {\bf 81}, 035106 (2010)}]. We consider the instability of the Fermi liquid phases and how it is enhanced by the Hund's rule. We identify paramagnetic solutions that have zero quasi-particle weight with bad metallic phases, and the first-order transition temperature between it and the Fermi liquid phase as a coherence temperature that signals the crossover to the bad metallic state. When orbital anisotropy is present, we found an intermediate transition to an orbital-selective bad metal (OSBM), where the narrow band becomes a bad metal while the wide band remains a renormalised Fermi liquid. The temperatures $T_\textrm{coh}$ and $T_\textrm{OSBM}$ at which the system transitions to the bad metal phases can be orders of magnitude less than the Fermi temperature associated with the non-interacting band. The parameter dependence of the temperature at which the OSBM is destroyed can be understood in terms of a ferromagnetic Kondo-Hubbard lattice model. In general, Hund's rule coupling enhances the bad metallic phases, reduce interorbital charge fluctuations and increase spin fluctuations. The qualitative difference found in the ground state whether the Hund's rule is present or not, related to the degeneracy of the low energy manifold, is also maintained for finite temperatures.