Exploring quantum phases of the Zig-Zag chain

Includes bibliographical references (pages 37-39)The spin liquid quantum mechanical state is at the forefront of condensed matter physics\ud and motivates these introductory studies into quantum magnetism on a one-dimensional\ud spin chain. After reviewing some context and history, the approach (exact diagonalization)\ud and the model (Heisenberg Hamiltonian in one dimension with nearest neighbor (nn),\ud next-nearest neighbor (nnn), and a four-site interaction known as a ring exchange (ring))\ud are described with simple examples for the reader to study. Methods of observation of\ud quantum spin chains are reviewed, including the structure factor which is an important observable.\ud The entanglement entropy is discussed and demonstrated with hand examples.\ud Then, the structure factor is used to demonstrate a phase transition in a Heisenberg model\ud with only nearest neighbor and next-nearest neighbor interactions. Entanglement entropy\ud is used to investigate and compare rectangular and triangular ladders (nn and nnn with and\ud without ring). Entanglement entropy is used to create a surface plot phase diagram for a nn, nnn, ring model for a wide range of interaction energies. Proposals for future studies are\ud presented in the conclusions, including varied boundary conditions, implementing larger\ud system sizes, and investigating interesting regions of the entropy phase diagram, among\ud other possibilities