Thermal entanglement of a coupled electronic spins system: interplay between an external magnetic field, nuclear field and spin–orbit interaction

We have studied the thermal entanglement as a function of the temperature for a two-qubits Heisenberg spins system; we have included Dzyaloshinskii–Moriya interaction (DM), an external magnetic field (EMF) and hyperfine interaction due to the nuclear field of the surrounding nuclei. A critical value for the EMF was found, around \(B^{(c)}_{\mathrm{ext},z} \sim 39\) mT, which characterizes two regimes of behavior of the thermal entanglement. Our results show that the DM term acts as a facilitator for the entanglement because it prolongs the nonzero thermal entanglement for larger temperatures. We found that the concurrence as a function of the temperature has a local maximum, for values of the magnetic field larger than the critical field. We also show that the critical temperature \(T_\mathrm{c}\) follows a polynomial growth as a function of the DM term, with characteristic behavior \(T_{\mathrm{c}} \sim \beta _{0}^{2}\), and the hyperfine field implies a critical temperature as a function of the field variance, \(\sigma \) of the form \(T_{\mathrm{c}} \sim \sigma ^{2}\). We show that in this system, the entanglement measure by the concurrence and the one-spin polarization observable exhibit opposite behavior, providing a method to obtain the entanglement from the measurement of an observable.