In this work, we study the numerical optimization of nearest-neighbor concurrence of bipartite one and two dimensional lattices, as well as non bipartite two dimensional lattices. These systems are described in the framework of a tight-binding Hamiltonian while the optimization of concurrence was performed using genetic algorithms. Our results show that the concurrence of the optimized lattice structures is considerably higher than that of non optimized systems. In the case of one dimensional chains the concurrence is maximized when the system begins to dimerize, i.e. it undergoes a structural phase transition (Peierls distortion). This result is consistent with the idea that entanglement is maximal or shows a singularity near quantum phase transitions and that quantum entanglement cannot be freely shared between many objects (monogamy property). Moreover, the optimization of concurrence in two-dimensional bipartite and non bipartite lattices is achieved when the structures break into smaller subsystems, which are arranged in geometrically distinguishable configurations. This behavior is again related to the monogamy property.
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