Renormalization of entanglement in quantum spin models

In recent years, the characterization of certain many-body condensed matter systems has received some attention from quantum information theory. Entanglement as one of the most intriguing feature of quantum theory has attracted much attention due to its nonlocal connotation that is regarded as a valuable resource in quantum information theory. To find the connection between entanglement and quantum phase transition, the behavior of entanglement in the vicinity of quantum critical point in many systems has been revisited. We have examined the behavior of entanglement in different steps of renormalization group (RG) for different couplings close to critical point. The one dimensional S = 1/2 Ising model in transverse field (ITF) and anisotropic Heisenberg model (XXZ) model have been considered by implementing the quantum renormalization group (QRG) approach . For ITF model we have calculated the renormalized Hamiltonian in standard QRG scheme using two sites blocking. The present scheme allows us to have the analytic RG equations, which give a better understanding of the behavior of system by running of coupling constants. The evolution of entanglement and its first derivative in each block has been studied in RG steps. After few RG steps the entanglement acts as an order parameter, i.e., for the paramagnetic phase the entanglement is zero and for the ferromagnetic phase the entanglement is equal one. Moreover, the QRG scheme with 3-sites blocking has been used to get the renormalized Hamiltonian in a self similar manner for the XXZ model. In each block we evaluate the amount of entanglement of one spin with the other two remaining ones and also between two arbitrary spins. For the former Shannon entropy and for the latter concurrence has been considered as a measure of entanglement. Similar to the case of ITF model, the evolution of entanglement and its first derivative with RG steps has been studied. The entanglement shows a critical behavior such as discontinuity at the quantum phase transition point and after some RG steps, the concurrence acts as an order parameter like the entanglement in ITF model.