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Springer eBooks

Ising and XY Models of Interacting Spins

James K. Freericks

Published Jan 1, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1007/978-3-031-87845-9_10

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Oxford University Press eBooks•Dec 18, 2003•0 citations
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.
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.
The simplest possible quantum degree of freedom has only two discrete possible values and is called an Ising spin. This could be a spin system having only two possible values, namely “up” or “down,” or this could be a quantum particle that can occupy only two possible positions. To simplify the problem to its bare essentials, a two-state system is studied in zero-dimension space and in Euclidean time; the two-state system occupies a single point and, as it evolves in time, it can point either up or down. Furthermore, time evolution is simplified by discretizing time into a finite time lattice. The two-state degree of freedom in zero-dimension space propagating on a time lattice is called the Ising model. The one-dimensional Ising model is a toy model that is studied to develop the essential ideas of quantum mechanics. The concepts of state space, Hamiltonian, evolution kernel, and path integration have a discrete and simple representation in the Ising model. The Ising model can be viewed as a statistical mechanical system in thermal equilibrium, with the Ising spins occupying a one-dimensional space lattice, and hence is called the one-dimensional Ising model. The Ising model can be defined on higher dimensional lattices and forms one of the bedrocks of theoretical statistical mechanics. In Sections 8.1 to 8.5 the degree of freedom, state space, Hamiltonian, path integral and correlator for the Ising model are discussed. In Section 8.6, the technique of spin decimation is illustrated using an Ising model with coupling depending on the lattice site.
Though the idea to use numerical techniques, in order to solve complex three-dimensional problems, has become quite old, computational techniques have gained immense importance in past few decades because of the advent of new generation fast and efficient computers and development of algorithms as parallel computing.Many mathematical problems have no exact solutions.Depending on the complexity of the equations, one needs to use approximate methods.But there are problems, which are beyond our limits, and need support of computers.Ernst Ising published his PhD dissertation in the form of a scientific report in 1925.He used a string of magnetic moments; spin up (+1/2) and spin down (À1/2), and applied periodic boundary conditions to prove that magnetic phase transition does not exist in one dimensions.Lars Onsager, latter, exactly solved the phase transition problem in two dimensions in 1944.It is going to be a century-old problem now.A variety of potential applications of Ising model are possible now a days; classified as Ising universality class models.It has now become possible to solve phase transition problems in complex three-dimensional geometries.Though the area of spinotronics still needs more engagements of computational techniques, its limited use have provided good insights at molecular scale in recent past.This chapter gives a brief introduction to Ising model and its applications, highlighting the developments in the field of magnetism relevant to the area of solid state physics.
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.
This chapter examines the question of spin systems, the simplest realization of Luttinger liquids. The peculiarities of one dimension make the solution of this problem quite surprising and different from what our intuition expects based on a classical vision of the spins. The spin 1/2 chain and its physical properties are discussed, along with the use of one-dimensional interacting fermions to solve the spin chain problem. The ideal spin 1/2 chain can be perturbed in many ways. Each of these perturbations can lead to some quite rich physics. This chapter considers the simplest of them, perturbations that apply to strictly one-dimensional chains, and examines coupling between chains.