For decades, the Holy Grail of statistical mechanics has been a mathematical problem known as the Ising model. Introduced in the 1920s by German physicist Ernst Ising, the Ising model is a powerful tool for studying phase transitions: the abrupt changes of state that occur, for instance, when ice melts or cooling iron becomes magnetic. Although they've learned much from approximate solutions and computer simulations, physicists have long sought an exact mathematical solution to the Ising model, which would provide much more information about such still-mysterious transitions.
For decades, the Holy Grail of statistical mechanics has been a mathematical problem known as the Ising model. Introduced in the 1920s by German physicist Ernst Ising, the Ising model is a powerful tool for studying phase transitions: the abrupt changes of state that occur, for instance, when ice melts or cooling iron becomes magnetic. Although they've learned much from approximate solutions and computer simulations, physicists have long sought an exact mathematical solution to the Ising model, which would provide much more information about such still-mysterious transitions.
Among the many applications of statistical mechanics, some of the most intriguing and challenging theoretical problems arise in connection with phase transitions. These are abrupt changes of state such as occur, for example, when a liquid is transformed into a vapour, a ferromagnet loses its magnetization upon heating to its Curie temperature, or at the onset at sufficiently low temperatures of superfluidity or superconductivity. It is within the theory of phase transitions, too, that the mathematical relationships between statistical mechanics and relativistic field theories are most powerful. Indeed, the idea of spontaneous symmetry breaking, which lies at the heart of the theory of phase transitions, is the crucial ingredient that turns the gauge theories of chapter 8 into a real working model of the fundamental forces of nature, to be discussed in the next chapter.
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.
Among the many applications of statistical mechanics, some of the most intriguing and challenging theoretical problems arise in connection with phase transitions. These are abrupt changes of state such as occur, for example, when a liquid is transformed into a vapour, a ferromagnet loses its magnetization upon heating to its Curie temperature, or at the onset at suciently low temperatures of super uidity or superconductivity. It is within the theory of phase transitions, too, that the mathematical relationships between statistical mechanics and relativistic eld theories are most powerful. Indeed, the idea of spontaneous symmetry breaking, which lies at the heart of the theory of phase transitions, is the crucial ingredient that turns the gauge theories of Chapter 8 into a real working model of the fundamental forces of nature, to be discussed in the next chapter.
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.
Abstract In undergraduate statistical mechanics courses the Ising model always plays an important role because it is the simplest non-trivial model used to describe magnetic systems. The one-dimensional model is easily solved analytically, while the two-dimensional one can be solved exactly by the Onsager solution. For this reason, numerical simulations are usually used to solve the two-dimensional model. Keeping in mind that the two-dimensional model is the platform for studying phase transitions, it is usually an exercise in computational undergraduate courses because its numerical solution is relatively simple to implement and its critical exponents are perfectly known. The purpose of this article is to present a detailed numerical study of the second-order phase transition in the two-dimensional Ising model at an undergraduate level, allowing readers not only to compare the mean-field solution, the exact solution and the numerical one through a complete study of the order parameter, the correlation function and finite-size scaling, but to present the techniques, along with hints and tips, for solving it themselves. We present the elementary theory of phase transitions and explain how to implement Markov chain Monte Carlo simulations and perform them for different lattice sizes with periodic boundary conditions. Energy, magnetization, specific heat, magnetic susceptibility and the correlation function are calculated and the critical exponents determined by finite-size scaling techniques. The importance of the correlation length as the relevant parameter in phase transitions is emphasized.
Abstract We have now seen how powerful the tools of statistical mechanics can be in the study of complex systems in thermal equilibrium and how they can be used to determine exactly the behaviour of some simplified, but nevertheless interesting, models of phase transitions. However, these exact solutions are sometimes lengthy and difficult and there is no guarantee that any particular model will yield exact results. For example, we were able to solve the two-dimensional, but not the three-dimensional, Ising model, and that only in zero magnetic field. So, it is natural to consider whether we can use a computer to simulate these systems. In this chapter we shall see that this is indeed possible; a great deal of our understanding of critical phenomena is based wholly or partly on the results of such numerical work. However, as we shall also see, such simulations are not entirely straightforward and a number of tricks are needed to obtain accurate results. This is partly a consequence of the very physics that makes the phase transitions interesting to start with. One should probably think of numerical simulation as a whole subject, almost an art, of its own; we shall have time only to scratch at the surface and discuss a few of the most important ideas.