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Ziran zazhi

Advances on the Ising Model

Zhang Zhi-dong

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

The Ising model and its applications,the procedure for deriving the exact solution of the two-dimensional Ising model,the difficulty for solving explicitly the three-dimensional(3D)Ising model have been introduced briefly.The progresses in studying physical properties and critical phenomena of the 3D Ising model by various approximation methods such as the mean field theory and its extensions,renormalization group theory and Monte Carlo simulations etc.have been reviewed.Finally,our recent advances in the conjectures and the putative exact solution of the 3D Ising model are introduced.

Related papers

The nature of three dimensions: Non-local behavior in the three-dimensional (3D) Ising model

Zhang Zhi-dong

In this work, we first reveal the nature of three dimensions: the existence of non-local behavior. Taking the three-dimensional (3D) Ising model as an example, we analyze the transfer matrices as well as the partition function to discuss in details the origin of non-local behavior in three dimensions. We then point out that fail to deal with non-local behavior is one of the main disadvantages of several theoretical approaches (such as conventional low-temperature expansions, conventional high-temperature expansions, perturbative renormalization group and Monte Carlo methods, etc), all of which take local environments into account only. This work provides a deep understanding on the natural difference between 3D bulk materials and 2D materials, and shows that how to treat the non-local behavior and the interaction between spins along the third dimension is a key point for solving exactly the problems of the 3D systems with the nearest neighboring interactions. Finally, we will briefly review recent advance in the 3D Ising model and related topics.

Journal of Physics Conference Series•May 17, 2017•7 citations

Theory and Simulation of the Ising Model

Ashkan Shekaari, Mahmoud Jafari

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.

arXiv:2105.00841•Apr 24, 2021•3 citations

Fifty Years of the Exact Solution of the Two-Dimensional Ising Model by Onsager

Somendra M. Bhattacharjee, Avinash Khare

The exact solution of the two-dimensional Ising model by Onsager in 1944 represents one of the landmarks in theoretical physics. On the occassion of the fifty years of the exact solution, we give a historical review of this model. After briefly discussing the exact solution by Onsager, we point out some of the recent developments in this field. The exact solution by Onsager has inspired several developments in various other fields. Some of these are also briefly mentioned.

arXiv:cond-mat/9511003•Nov 1, 1995•26 citations

Exact solutions and related topics

Hidetoshi Nishimori, Gerardo Ortíz

Abstract Only a limited number of models of phase transitions and critical phenomena can be solved exactly. These examples nevertheless play important roles in many aspects including the verification of the accuracy of approximation theories such as the mean-field theory and renormalization group. Mathematical methods to solve such examples are interesting in their own right and constitute an important subfield of mathematical physics. In particular the exact solution of the two-dimensional Ising model occupies an outstanding status as one of the founding studies of the modern theory of phase transitions and critical phenomena. The present chapter shows simple but typical examples of exact solutions of classical spin systems such as the one-dimensional Ising model with various boundary conditions, the n-vector model, the spherical model, the one-dimensional quantum $XY$ model, and the two-dimensional Ising model. An account on the Yang-Lee theory will also be given as a set of basic rigorous results on phase transitions.

•Dec 2, 2010•0 citations

Duality of the Two-dimensional Ising Model

Giuseppe Mussardo

Abstract In this chapter we will begin our study of the Ising model on the two-dimensional lattice. In two dimensions the model has a phase transition, with critical exponents that have different values from those obtained in the mean field approximation. For this reason, it provides an important example of critical phenomena. As we will see in great detail in this chapter and in the next, among all exactly solved models of statistical mechanics, the two-dimensional Ising model is not only the one that has been most studied but it is also the model that has given a series of deep mathematical and physical results. Many solutions of the model stand out for the ingenious methods used, such as the theory of determinants, combinatorial approaches, Grassmann variables, or elliptic functions. Many results have deeply influenced the understanding of critical phenomena and have strongly stimulated new fields of research. Ideas that have matured within the study of the two-dimensional Ising model, such as the duality between its highand low-temperature phases, have been readily generalized to other systems of statistical mechanics and have also found important and fundamental applications in other important areas such as, for instance, quantum field theory. Equally fundamental is the discovery that in the vicinity of the critical point, the dynamics of the model can be described from the relativistic Dirac equation for Majorana fermions.

•Aug 27, 2009•0 citations

Aspects of the Ising and tricritical Ising models

Philip Giokas

This thesis is concerned with several aspects of the Ising and tri-critical Ising models in two-dimensions. These are much-studied models relevant in both condensed matter physics as descriptions of the critical phenomena of two- dimensional systems and in String theory as building blocks of the string world sheet theory. The first part of the thesis is concerned with the derivation of differential equations for the critical four-point function in the Ising model. We present a method which provides the well-known standard solutions by a new and efficient route. The second part of the thesis is concerned with off-critical behaviour, and in particular the numerical study of perturbations of conformal field theory through the truncated conformal space approach. We show that the coupling constant undergoes significant renormalization in this scheme, and in particular the Ising model can be found as a fixed point for a finite value of the bare coupling constant. The renormalization group equations we find are of general use in the TCSA approach. The final part of the thesis considers off-critical boundary conditions in the tri-critical Ising model. We study them using a variant of the mean-field method and find a qualitative description of the space of boundary conditions that is in accord with the exact conformal field theory description. This is both a test of the method and its applicability in new domains, and also shows that previously published results are in error.

•Jan 1, 2013•0 citations
The Ising model and its applications,the procedure for deriving the exact solution of the two-dimensional Ising model,the difficulty for solving explicitly the three-dimensional(3D)Ising model have been introduced briefly.The progresses in studying physical properties and critical phenomena of the 3D Ising model by various approximation methods such as the mean field theory and its extensions,renormalization group theory and Monte Carlo simulations etc.have been reviewed.Finally,our recent advances in the conjectures and the putative exact solution of the 3D Ising model are introduced.
In this work, we first reveal the nature of three dimensions: the existence of non-local behavior. Taking the three-dimensional (3D) Ising model as an example, we analyze the transfer matrices as well as the partition function to discuss in details the origin of non-local behavior in three dimensions. We then point out that fail to deal with non-local behavior is one of the main disadvantages of several theoretical approaches (such as conventional low-temperature expansions, conventional high-temperature expansions, perturbative renormalization group and Monte Carlo methods, etc), all of which take local environments into account only. This work provides a deep understanding on the natural difference between 3D bulk materials and 2D materials, and shows that how to treat the non-local behavior and the interaction between spins along the third dimension is a key point for solving exactly the problems of the 3D systems with the nearest neighboring interactions. Finally, we will briefly review recent advance in the 3D Ising model and related topics.
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 Only a limited number of models of phase transitions and critical phenomena can be solved exactly. These examples nevertheless play important roles in many aspects including the verification of the accuracy of approximation theories such as the mean-field theory and renormalization group. Mathematical methods to solve such examples are interesting in their own right and constitute an important subfield of mathematical physics. In particular the exact solution of the two-dimensional Ising model occupies an outstanding status as one of the founding studies of the modern theory of phase transitions and critical phenomena. The present chapter shows simple but typical examples of exact solutions of classical spin systems such as the one-dimensional Ising model with various boundary conditions, the n-vector model, the spherical model, the one-dimensional quantum XYXYXY model, and the two-dimensional Ising model. An account on the Yang-Lee theory will also be given as a set of basic rigorous results on phase transitions.
Abstract In this chapter we will begin our study of the Ising model on the two-dimensional lattice. In two dimensions the model has a phase transition, with critical exponents that have different values from those obtained in the mean field approximation. For this reason, it provides an important example of critical phenomena. As we will see in great detail in this chapter and in the next, among all exactly solved models of statistical mechanics, the two-dimensional Ising model is not only the one that has been most studied but it is also the model that has given a series of deep mathematical and physical results. Many solutions of the model stand out for the ingenious methods used, such as the theory of determinants, combinatorial approaches, Grassmann variables, or elliptic functions. Many results have deeply influenced the understanding of critical phenomena and have strongly stimulated new fields of research. Ideas that have matured within the study of the two-dimensional Ising model, such as the duality between its highand low-temperature phases, have been readily generalized to other systems of statistical mechanics and have also found important and fundamental applications in other important areas such as, for instance, quantum field theory. Equally fundamental is the discovery that in the vicinity of the critical point, the dynamics of the model can be described from the relativistic Dirac equation for Majorana fermions.
This thesis is concerned with several aspects of the Ising and tri-critical Ising models in two-dimensions. These are much-studied models relevant in both condensed matter physics as descriptions of the critical phenomena of two- dimensional systems and in String theory as building blocks of the string world sheet theory. The first part of the thesis is concerned with the derivation of differential equations for the critical four-point function in the Ising model. We present a method which provides the well-known standard solutions by a new and efficient route. The second part of the thesis is concerned with off-critical behaviour, and in particular the numerical study of perturbations of conformal field theory through the truncated conformal space approach. We show that the coupling constant undergoes significant renormalization in this scheme, and in particular the Ising model can be found as a fixed point for a finite value of the bare coupling constant. The renormalization group equations we find are of general use in the TCSA approach. The final part of the thesis considers off-critical boundary conditions in the tri-critical Ising model. We study them using a variant of the mean-field method and find a qualitative description of the space of boundary conditions that is in accord with the exact conformal field theory description. This is both a test of the method and its applicability in new domains, and also shows that previously published results are in error.