Entanglement and Criticality in Two-dimensional Heisenberg XXZ Model

Abstract

In this thesis we have investigated the quantum correlation and the entanglement quan tifiers, i.e., the concurrence, the quantum Fisher information, the trace distance and the multipartite entanglement for the two-dimensional anisotropic spin- 1 2 XXZ lattice with and without Dzyaloshinskii–Moriya interaction. It is found that for a many-body quantum system the multipartite entanglement is more advantageous than the bipartite entanglement due to the monogamy property. The entanglement quantifiers, concurrence, trace distance and multipartite entanglement decrease with an increase in the anisotropy and the Dzyaloshinskii-Moriya interaction tends to restore the spoiled entanglement. On the other hand, the quantum Fisher information shows a rather peculiar behavior. To estimate the entanglement we introduced a correlation measure from the definition of the quantum Fisher information, that behaves like the other entanglement quantifiers. The quantum renormalization group method is used to compute the stable and the unsta ble fixed points. We observe that the quantum phase transition point is independent of the chosen quantifier as the thermodynamic limit is reached. After sufficient iterations of the quantum renormalization group, we observe two different saturated values of all the above mentioned quantifiers that represent two separate phases, the N´eel phase and the spin-fluid phase. The first derivative and the scaling behavior of the renormalized quantifiers are calculated. At quantum phase transition point, the non-analytic behavior of the first derivative of the quantifiers as a function of lattice size is examined and it is found that the universal finite-size scaling law is obeyed. Furthermore, we observe that at the critical point the scaling exponent for all the quantifiers can describe the correlation length of the model