Haldane phase in the bond-alternating spin-1/2 XXZ chain: DMRG and fermionization studies

Abstract

In this thesis, the subject of study is the Haldane phase in bond - alternating XXZ chain with S =1/2. We firstly mapped our model to the fermionic chain by the use of standard Jordan-Wigner Transformation, which leads to the famous interacting spinless Su-Schrieffer-Heeger (SSH) fermionic model with modifications. Firstly, we studied trivial quantum phase transition (QPT) under the magnetic field in exactly soluble non-interacting limit, which corresponds to the bond-alternating XX chain . Excitation spectrum, magnetization, magnetic susceptibility are used to characterize QPT and compared with the numerical results. Correlation functions for all components of spins are calculated exactly. Secondly, we studied symmetry - protected topological phase transition in the given non-interacting SSH model and characterized it by calculating topological winding number. Fermionized string order parameter as a function of spin couplings is obtained and the Haldane phase diagram in the XX limit is confirmed. The correspondence of the Haldane phase in the spin chain in XX limit and topological insulating phase of fermionic non-interacting SSH model is shown. Finally, by the use of entanglement spectrum as an order parameter, we numerically obtained Haldane phase diagram for XXZ model with bond-alternation. For numerical investigation, we used density matrix renormalization group theory in matrix product states formulation (MPS-DMRG) for a system with open boundary conditions (OBC) .