Many physical systems can be represented by a regular arrangement of molecules in a lattice structure. Knowing how neighboring molecules in the lattice interact with one another can give great insight into a material's macroscopic behavior. A very popular and effective means of investigating these microscopic interactions is the Ising Model. This model, suggested first by Wilhelm Lenz in 1920 and later expanded by Ernst Ising in 1925, is based on the assumptions that each molecule in a lattice structure can be represented by its spin value (+l or -1) and that only nearest neighbors contribute to the total interaction energy. The Ising Model, which was initially used in the study of ferromagnetic systems, can now be used to study a variety of physical systems. Some of these include antiferromagnetic crystals, binary alloys, DNA, and lattice gasses.
Many physical systems can be represented by a regular arrangement of molecules in a lattice structure. Knowing how neighboring molecules in the lattice interact with one another can give great insight into a material's macroscopic behavior. A very popular and effective means of investigating these microscopic interactions is the Ising Model. This model, suggested first by Wilhelm Lenz in 1920 and later expanded by Ernst Ising in 1925, is based on the assumptions that each molecule in a lattice structure can be represented by its spin value (+l or -1) and that only nearest neighbors contribute to the total interaction energy. The Ising Model, which was initially used in the study of ferromagnetic systems, can now be used to study a variety of physical systems. Some of these include antiferromagnetic crystals, binary alloys, DNA, and lattice gasses.
A simple mathematical scheme is derived here. With this scheme, three problems of lattice-spin systems are solved exactly. The first one is the problem of solving, thermodynamically, a linear chain of the Lenz-Ising model in a zero magnetic field with nearest, as well as next-nearest, neighbor couplings. The problem turns out to be equivalent to the problem of a linear chain with only nearest-neighbor couplings but in a finite magnetic field. The second one is to solve an imperfect 1-dimensional Heisenberg-Dirac model, similar to the partially solved ``Ising-Heisenberg'' model of Lieb-Mattis-Schultz, in a zero magnetic field. The problem is solved completely in the sense that all the elementary excitations of this model are shown in terms of some pseudofermions and the spectra are given as ε1(q)=[(Aq+Bq)2+Eq2]12,ε2(q)=[(Aq−Bq)2+Eq2]12,where Aq, Bq, and Eq are three different functions of sin (q), cos (q), and coupling strengths involved. The third one, the XY model, is used to study the contribution of ``inhomogeneity'' in the coupling strengths to the system, as compared to the anisotropicity contribution to it.
The laws of nature encompass the small, the large, the few, and the many. In this thesis we are concerned with classical (i.e. not quantum) many-body systems, which refers to any microscopic or macroscopic system that contains a large number of interacting entities. The nearest-neighbor Ising model, originally developed in 1920 by Wilhelm Lenz, forms a cornerstone in our theoretical understanding of collective effects in classical many-body systems, and is to date a paradigm for statistical physics. Despite its elegant and simplistic description, exact analytical results in dimensions equal and larger than two are difficult to obtain. Therefore, much work has been done to con- struct methods that allow for approximate, yet accurate, analytical solutions. One of these methods is the Bethe-Guggenheim approximation, originally developed independently by Hans Bethe and Edward Guggenheim in 1935. This approximation goes beyond the well- known mean field approximation, and explicitly accounts for pair correlations between the spins in the Ising model. In this thesis, we embark on a journey to exploit the full capacity of the Bethe-Guggenheim approximation, in non-uniform and non-equilibrium settings. After we formally introduce the original Bethe-Guggenheim approximation for uniform systems, we will extend its scope to non-uniform systems, and derive a Cahn-Hilliard free energy functional. Here we find that the one-dimensional equilibrium concentration profile undergoes a delocalization-induced broadening transition at interaction strengths near and above the thermal energy. The broadening transition arises from a decreasing amplitude of capillary wave fluctuations, and is not accounted for in the mean field approximation. Finally, going beyond equilibrium properties, we also study the kinetics of the Ising model in and out of equilibrium. First, based on firm theoretical and experimental evidence, we con- struct an Ising-like minimal model for interacting cellular adhesion molecules. Here we find a dynamical critical point where adhesion cluster dissolution and formation are the fastest, and undergo a qualitative change in dynamics. Second, we demonstrate the existence of a finite-time dynamical phase transition for disordering quenches in the nearest-neighbor Ising model. Starkly different from the mean field approximation, the time at which the dynamical phase transition occurs, is bounded from below by a speed limit. Altogether, in this thesis, we unveil the non-trivial and a priori non-intuitive effects of pair correlations in the classical nearest-neighbor Ising model, which are taken into account in the Bethe-Guggenheim approximation, and neglected in the mean field approximation.
We have provided a concise introduction to the Ising model as one of the most important models in statistical mechanics and in studying the phenomenon of phase transition. The required theoretical background and derivation of the Hamiltonian of the model have also been presented. We finally have discussed the computational method and details to numerically solve the two- and three-dimensional Ising problems using Monte Carlo simulations. The related computer codes in both Python and Fortran, as well as a simulation trick to visualize the spin lattice, have also been provided.
We use a recently developed quantum simulation approach to study the properties of a three-dimensional Ising model consisting of S = 1/2 quantum spins localized at the sites of a simple cubic lattice. We assume nearest-neighbor interaction between spins with an exchange interaction that can be either ferromagnetic or antiferromagnetic. It is found that the computational method quickly converges towards the expected equilibrium spin configurations. The resulting spontaneous magnetization curves corresponding to the two types of magnetic interactions under consideration were found to be almost identical to the ones obtained via quantum mean field theory at all temperatures. The derived total energies, total free energies, magnetic entropies and specific heats per mole of spins show no sizeable differences from known theoretical values. Furthermore, the results of the simulations for two different 3D Ising systems containing 4×4×4 and 20×20×20 spins localized at the sites of a simple cubic lattice were found to be almost identical to each other. This finding suggests that the self-consistent algorithm approach of the current simulation method allows one to obtain the physical bulk properties of a large magnetic system by relying on simulations of a much smaller spin system sample. Therefore, the method presently considered appears to be not only very accurate as gauged by comparison to mean field theory, but also able to greatly increase the speed of simulations.
The two-dimensional Ising model for a system of interacting spins (or for the ordering of an AB alloy) on a square lattice is one of the very few nontrivial many-body problems that is exactly soluble and shows a phase transition. Although the exact solution in the absence of an external magnetic field was first given almost twenty years ago in a famous paper by Onsager1 using the theory of Lie algebras, the flow of papers on both approximate and exact methods has remained strong to this day.2 One reason for this has been the interest in testing approximate methods on an exactly soluble problem. A second reason, no doubt, has been the considerable formidability of the Onsager method. The simplification achieved by Bruria Kaufman3 using the theory of spinor representations has diminished, but not removed, the reputation of the Onsager approach for incomprehensibility, while the subsequent application of this method by Yang4 to the calculation of the spontaneous magnetization has, if anything, helped to restore this reputation.
So far, we have primarily focused on the spin or angular momentum of an individual quantum particle. In this chapter, we examine the simplest models of spins interacting with each other. The Ising model was created by Wilhelm Lenz—he gave the problem to his Ph. D. student Ernst Ising, who solved the one-dimensional model. Hence, a statistical-mechanical model for a simplified model of quantum spins was solved even before the quantum theory for spin was fully developed! In this chapter, we investigate a number of different spin models. They are all exactly solvable, but some are more complicated than others. One of the key techniques we use is fermionization, which relates the spins to fermions. This allows us to introduce the ideas of second quantization and of fermions. Specifically, we look at the Ising model, the XY model, the transverse-field Ising model, the Lipkin-Meshkov-Glick model, and the Dicke model.