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Entanglement robustness in Heisenberg spin chains coupled to dissipative environment at finite temperature

We consider a finite one-dimensional Heisenberg XYZ spin chain under the influence of dissipative Lindblad environment obeying the Born-Markovian constrain in presence of an external magnetic field. We apply both closed and open boundary conditions at zero and finite temperature. We present an exact numerical solution for the Lindblad master equation of the system in the Liouville space. we find that, in the free spin chain (in absence of any environment), the entanglement at all ranges evolve in time in a non-uniform oscillatory form that changes significantly depending on the initial state, system size and the spatial anisotropy. The oscillatory behavior is suppressed once the system is coupled to the environment. Furthermore, the asymptotic behavior of the entanglement, nearest neighbor and beyond, in the system under the influence of the environment at zero temperature is very sensitive to the x-y spatial anisotropy, which causes them to reach either a zero or a finite sustainable steady state value regardless of the initial state of the system. The anisotropy in the $z-$direction may enhance the entanglement depending on the interplay with the magnetic field applied in the same direction. As the temperature is raised, the steady state of the short range entanglements is found to be robust within very small non-zero temperature range, which depends critically on the spatial anisotropy of the system. The entanglement at each range depends differently on the spatial anisotropy. Moreover, the end to end entanglement transfer time and speed through the open boundary chain vary significantly based on the degree of anisotropy and the temperature of the environment.