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We consider a finite two-dimensional Heisenberg triangular spin lattice atdifferent degrees of anisotropy coupled to a dissipative Lindblad environmentobeying the Born-Markovian constrain at finite temperature. We show howapplying an inhomogeneous magnetic field to the system may significantly affectthe entanglement distribution and properties among the spins in the asymptoticsteady state of the system. Particularly, applying an inhomogeneous field withan inward (growing) gradient toward the central spin is found to considerablyenhance the nearest neighbor entanglement and its robustness to the thermaldissipative decay effect in the completely anisotropic (Ising) system, whereasall the beyond nearest neighbor entanglements vanish entirely. Applying thesame field to a partially anisotropic (XYZ) system, does not only enhance thenearest neighbor entanglements and their robustness but also all the beyondnearest neighbor ones. Nevertheless, the inhomogeneity of the field shows noeffect on the asymptotic behavior of the entanglement in the isotropic (XXX)system, which vanishes under any system configuration. Moreover, the sameinhomogeneous field exhibits the most influential impact, compared with theother ones, on the the spin dynamics as well. Although in the isotropic systemthe spins relax to a separable (disentangled) steady state with all the spinsreaching a common spin state regardless of the field inhomogeneity, the spinsin the steady state of the completely anisotropic system reach differentdistinguished spin states depending on their positions in the lattice. However,in the XYZ system, though the anisotropy is lower, the spin states become evenmore distinguished, accompanying the long range quantum correlation across thesystem, which is a sign of a critical behavior taking place at this combinationof system anisotropy and field inhomogeneity.

Entanglement robustness and spin relaxation in thermal states of two-dimensional dissipative spin system in an inhomogeneous magnetic field