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Springer eBooks

Influences of lattice dimension d and character of spin-interactions on the thermodynamics of Ising models

H. Nencka-Ficek

Published Apr 6, 2008• 0 citations
Publisher
DOI
10.1007/3-540-12291-5_85

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The ground-state properties and low-lying excitations of spin systems with competing interactions are studied for various finite two-dimensional S= 1 / 2 Heisenberg systems of 8 spins and corresponding Ising systems of up to 60 spins. The response of the spin systems on the conflict in the interactions is studied using numerical exact results. In the Ising model the competition of the interactions influences the ground state and the excitations essentially. Contrary to conventional Ising systems in disordered strongly frustrated Ising systems the low-energy density of states is smooth, the gap between the ground state and the first excitations vanishes. The low-lying excitations are characterised by a large portion of multiple spin excitations. Thermodynamic consequences are indicated. In the Heisenberg case the influence of the conflict in the interactions is less obvious, the character of the ground state and of the first excitation has similarities with that of conventional antiferromagnets, the spins have an enlarged number of possibilities to evade the conflict in the interactions.
The phase diagram and the thermodynamics of the axial decorated Ising model on a square lattice with m-dimensional bond spins are studied for the m=1 (Ising), m=2 (XY), m=3 (Heisenberg) and m= infinity cases. Competing interactions between the site and decorating bond spins have been considered. The expressions for the partition function, the thermodynamics and the pair correlation functions of the decorated Ising chain have been analytically obtained. The effect of the dimensionality of the decorating bond spin on the thermodynamic behaviour of both models is analyzed.
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.