Entanglement Entropy and Entanglement Hamiltonian as Characterizations of Phases and Phase Transitions

In this thesis, we study the entanglement properties of quantum systems to characterize quantum phases and phase transitions. We focus on the free fermion lattice systems and we use numerical calculation to verify our ideas. Behavior of the entanglement entropy is used to distinguish different phases, in addition the area law of the entanglement entropy is studied. We propose that beside the entanglement entropy, there is physical information in the entanglement Hamiltonian of the reduced density matrix of a chosen subsystem. We verify our ideas by studying different free fermion models. The verification is made by comparing the results we obtain from studying the behavior of the entanglement Hamiltonian with the known previous results. As starting point, to show that entanglement Hamiltonian eigenmodes have physical information, we employ the XX spin chain model. Real space renormalization group method predicts that the ground state is the product state of singlet states and thus those singlet that cross the boundary make the entanglement. We use the entanglement Hamiltonian to show that its single particle eigenmode shows the location of the entangled singlet spins. This is done in the case of ground state at T = 0. We also studied the entanglement properties of the highly excited eigenstate of the system. We use modified version of real space renormalization group for excited state and we show that in T ≠ 0 case where singlet and triplet state with total S[subscript Z] = 0 make entanglement, entanglement Hamiltonian eigenmode shows the location of the entangled spins. We distinguish one eigenmode of the entanglement Hamiltonian as the maximally entangled mode. This mode corresponds to the smallest entanglement energy and thus contributes the most to the entanglement entropy. In addition, we use two one-dimensional free fermion models, namely the random dimer model and power law random banded model to show that for a localized-delocalized phase transition, behavior of the maximally entangled mode is similar to the behavior of the eigenmode of the original Hamiltonian at the Fermi level. We quantify this by comparing their overlaps and the inverse participation ratio of eigenmodes. The behavior of the entanglement entropy as a well-known quantity is studied in disordered free fermion models. In random dimer model and power law random banded model where the correlated disorder yields to the localized-delocalized phase transition, we show that entanglement entropy saturates in localized phase and diverges in delocalized phase. In addition it violates the area law in delocalized phase. Entanglement entropy of Anderson model in one, two, and three dimensions is also studied and we observed that area law is correct even for the delocalized phase of the Anderson model in three dimensions, provided that system size is larger than the mean free path. The study of a single impurity, one non-zero on-site energy, in the Anderson model is also examined. We concluded that this single impurity changes only the subleading term of the entanglement entropy which is proportional to the inverse of the subsystem size. This subleading term has non-oscillation and oscillating part.