  In the present paper we study the entanglement properties of thermal (a.k.a.
Gibbs) states of quantum harmonic oscillator systems as functions of the
Hamiltonian and the temperature. We prove the physical intuition that at
sufficiently high temperatures the thermal state becomes fully separable and we
deduce bounds on the critical temperature at which this happens. We show that
the bound becomes tight for a wide class of Hamiltonians with sufficient
translation symmetry. We find, that at the crossover the thermal energy is of
the order of the energy of the strongest normal mode of the system and quantify
the degree of entanglement below the critical temperature. Finally, we discuss
the example of a ring topology in detail and compare our results with previous
work in an entanglement-phase diagram.
