  We investigate the Kitaev-Heisenberg (KH) model at finite temperature using
the exact environment full update (eeFU), introduced in Phys. Rev. B 99, 035115
(2019), which represents the purification of a thermal density matrix on an
infinite hexagonal lattice by an infinite projected entangled pair state
(iPEPS). We show that thanks to a dynamical mapping from a hexagonal to a
rhombic lattice, the eeFU on the hexagonal lattice is as efficient as the
simple full update (FU) algorithm. Critical temperatures for coupling constants
in the stripy and the antiferromagnetic phase are estimated. They are an order
of magnitude less than the couplings in the Hamiltonian. By a duality
transformation, these results can be mapped to, respectively, the ferromagnetic
and zigzag phases. For the special case of the pure Kitaev model, which is
tractable by quantum Monte-Carlo but the most challenging for tensor networks,
the algorithm is benchmarked against the Monte-Carlo results. It recovers
accurately the crossover to spin ordering and qualitatively the one to flux
ordering.
