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We study the finite-temperature properties of the half-filled two-bandHubbard 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 afinite-temperature extension of the zero-temperature gauge variable previouslydeveloped by Hassan and de' Medicihttp://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 isenhanced by the Hund's rule. We identify paramagnetic solutions that have zeroquasi-particle weight with bad metallic phases, and the first-order transitiontemperature between it and the Fermi liquid phase as a coherence temperaturethat signals the crossover to the bad metallic state. When orbital anisotropyis present, we found an intermediate transition to an orbital-selective badmetal (OSBM), where the narrow band becomes a bad metal while the wide bandremains a renormalised Fermi liquid. The temperatures $T_\textrm{coh}$ and$T_\textrm{OSBM}$ at which the system transitions to the bad metal phases canbe orders of magnitude less than the Fermi temperature associated with thenon-interacting band. The parameter dependence of the temperature at which theOSBM is destroyed can be understood in terms of a ferromagnetic Kondo-Hubbardlattice model. In general, Hund's rule coupling enhances the bad metallicphases, reduce interorbital charge fluctuations and increase spin fluctuations.The qualitative difference found in the ground state whether the Hund's rule ispresent or not, related to the degeneracy of the low energy manifold, is alsomaintained for finite temperatures.

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