Phase Diagram of the Anisotropic Heisenberg Spin Ladder

In this thesis, we deal with 1-dimensional anisotropic spin models more specifically with the chain and ladder geometries. These models are fundamental to the study of condensed matter theory and quantum magnetism because of their simplicity. Research into the chain model dates back to 1930s with Hans Bethe. One method to study these systems is through numerical analysis including renormalization group techniques. These allow for the diagonalization of the Hamiltonian without sacrificing computation to a large virtual Hilbert space.

In this thesis, we present results for both the spin chain and spin ladder geometries in open boundary conditions. This work uses the density matrix renormalization group technique to calculate system energy. Initially we will present a study into the spin energy gap of the spin-\sfrac{1}{2} anisotropic (XXZ) Heisenberg chain in open boundary conditions (OBCs). The energy gap is shown to be reduced to half in the ground sector when compared to periodic boundary conditions (PBCs) due to edge effects. Secondly, a full phase diagram for the spin-\sfrac{1}{2} anisotropic Heisenberg ladder is presented which shows the emergence of a rich schematic with a variety of phases. Lastly, weak coupling limit maps are compared to quantum field theory phase transition predictions. This research was motivated by the lack of comprehensive phase diagram results for ladders and by real materials being investigated at the University of Kent.