Two qubits can be entangled in two distinct temperature regions

We have found that for a wide range of two-qubit Hamiltonians the canonical-ensemble thermal state is entangled in two distinct temperature regions. In most cases the ground state is entangled; however we have also found an example where the ground state is separable and there are still two regions. This demonstrates that the qualitative behavior of entanglement with temperature can be much more complicated than might otherwise have been expected; it is not simply determined by the entanglement of the ground state, even for the simple case of two qubits. Furthermore, we prove a finite bound on the number of possible entangled regions for two qubits, thus showing that arbitrarily many transitions from entanglement to separability are not possible. We also provide an elementary proof that the spectrum of the thermal state at a lower temperature majorizes that at a higher temperature, for any Hamiltonian, and use this result to show that only one entangled region is possible for the special case of Hamiltonians without magnetic fields.