Violation of the Bell inequality for thermal states of interaction qubits investigated via a multi-qubit Heisenberg model

We study the violations of the Bell inequality for thermal states of qubits in a multi-qubit Heisenberg model as a function of temperature and external magnetic field. Unlike the behaviours of the entanglement, the violation cannot be obtained by increasing the temperature or the magnetic field. The threshold temperatures of the violation are found be less than that of the entanglement. We also consider a realistic cavity-QED model which is a special case of the multi-qubit Heisenberg model.