Quantum Ising Systems, Edge Modes, and Rabi Lattice

The thesis contains theoretical study of number of systems in low dimension, which are related to quantum Ising model. The main results are emergence of Majorana fermions in condensed matter system. We discuss an array of cavity-QED system, describing atom-photon (spin-boson) interacting problem, namely Rabi lattice model, and calculated its phase diagram. More importantly we showed that in any dimension, the system has Ising dynamics in strong atom-photon coupling limit. For 1D, a special relation between end-to-end spin-spin correlation is presented, which is very much different form the bulk spin-spin correlation. The entanglement between cavities is calculated and shown to have maximum value at the phase transition point. Other problems we studied were the frustrated quantum Ising model and its phase diagram as well as existence of Majorana like edge modes in different phases of the model. The fermionic version of the frustrated quantum Ising problem is also presented. On a different but related note, we present some beautiful results on Falicov Kimball (FK) model, a fermionic model similar to Hubbard model, on a square lattice. Using a canonical representation for electrons (invented by Brijesh Kumar), the FK model is mapped to classical Ising model (at infinite Hubbard interaction limit) on 2D lattice. We performed classical Monte Carlo simulation to study magnetization as a function of Hubbard interaction and also as a function of temperature. The motivation was to study (quantum) charge fluctuation (thus, quantum fluctuation of top of classical Ising thermodynamics) due to finite Hubbard interaction term. We also devote a chapter for basic numerical techniques, such as Density matrix Renormalization Group method, Exact diagonalization for spin system and classical Monte Carlo simulation (for classical 2D Ising model), which we have used in this thesis.