The Ising model (or Lenz–Ising model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a lattice (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. Neighboring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases. The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition.See , Chapters VI-VII. Though it is a highly simplified model of a magnetic material, the Ising model can still provide qualitative and sometimes quantitative results applicable to real physical systems, and in general, it can be seen as a specialization of Stanley's n-vector model for .
The Ising model was invented by the physicist , who gave it as a problem to his student Ernst Ising. The one-dimensional Ising model was solved by alone in his 1924 thesis;Ernst Ising, Contribution to the Theory of Ferromagnetism it has no phase transition. The two-dimensional square-lattice Ising model is much harder and was only given an analytic description much later, by . It is usually solved by a transfer-matrix method, although there exists a very simple approach relating the model to a non-interacting fermionic quantum field theory.
In dimensions greater than four, the phase transition of the Ising model is described by mean-field theory. The Ising model for greater dimensions was also explored with respect to various tree topologies in the late 1970s, culminating in an exact solution of the zero-field, time-independent model for closed Cayley trees of arbitrary branching ratio, and thereby, arbitrarily large dimensionality within tree branches. The solution to this model exhibited a new, unusual phase transition behavior, along with non-vanishing long-range and nearest-neighbor spin-spin correlations, deemed relevant to large neural networks as one of its possible .
The Ising problem without an external field can be equivalently formulated as a graph maximum cut (Max-Cut) problem that can be solved via combinatorial optimization.
Definition Consider a set of lattice sites, each with a set of adjacent sites (e.g. a graph) forming a -dimensional lattice. For each lattice site there is a discrete variable such that , representing the site's spin. A spin configuration, is an assignment of spin value to each lattice site.
For any two adjacent sites there is an interaction . Also a site has an external magnetic field interacting with it. The energy of a configuration is given by the Hamiltonian function
where the first sum is over pairs of adjacent spins (every pair is counted once). The notation indicates that sites and are nearest neighbors. The magnetic moment is given by . Note that the sign in the second term of the Hamiltonian above should actually be positive because the electron's magnetic moment is antiparallel to its spin, but the negative term is used conventionally.See , Chapter 16. The Ising Hamiltonian is an example of a pseudo-Boolean function; tools from the analysis of Boolean functions can be applied to describe and study it.
The configuration probability is given by the Boltzmann distribution with inverse temperature :
where , and the normalization constant
is the partition function. For a function of the spins ("observable"), one denotes by
the expectation (mean) value of .
The configuration probabilities represent the probability that (in equilibrium) the system is in a state with configuration .
Discussion The minus sign on each term of the Hamiltonian function is conventional. Using this sign convention, Ising models can be classified according to the sign of the interaction: if, for a pair i, j
The system is called ferromagnetic or antiferromagnetic if all interactions are ferromagnetic or all are antiferromagnetic. The original Ising models were ferromagnetic, and it is still often assumed that "Ising model" means a ferromagnetic Ising model.
In a ferromagnetic Ising model, spins desire to be aligned: the configurations in which adjacent spins are of the same sign have higher probability. In an antiferromagnetic model, adjacent spins tend to have opposite signs.
The sign convention of H(σ) also explains how a spin site j interacts with the external field. Namely, the spin site wants to line up with the external field. If:
Simplifications Ising models are often examined without an external field interacting with the lattice, that is, h = 0 for all j in the lattice Λ. Using this simplification, the Hamiltonian becomes
When the external field is zero everywhere, h = 0, the Ising model is symmetric under switching the value of the spin in all the lattice sites; a nonzero field breaks this symmetry.
Another common simplification is to assume that all of the nearest neighbors ⟨ij⟩ have the same interaction strength. Then we can set Jij = J for all pairs i, j in Λ. In this case the Hamiltonian is further simplified to
Connection to graph maximum cut A subset S of the vertex set V(G) of a weighted undirected graph G determines a cut of the graph G into S and its complementary subset G\S. The size of the cut is the sum of the weights of the edges between S and G\S. A maximum cut size is at least the size of any other cut, varying S.
For the Ising model without an external field on a graph G, the Hamiltonian becomes the following sum over the graph edges E(G)
.
Here each vertex i of the graph is a spin site that takes a spin value . A given spin configuration partitions the set of vertices into two -depended subsets, those with spin up and those with spin down . We denote by the -depended set of edges that connects the two complementary vertex subsets and . The size of the cut to bipartite the weighted undirected graph G can be defined as
where denotes a weight of the edge and the scaling 1/2 is introduced to compensate for double counting the same weights .
The identities
where the total sum in the first term does not depend on , imply that minimizing in is equivalent to minimizing . Defining the edge weight thus turns the Ising problem without an external field into a graph Max-Cut problem maximizing the cut size , which is related to the Ising Hamiltonian as follows,
Questions A significant number of statistical questions to ask about this model are in the limit of large numbers of spins: In a typical configuration, are most of the spins +1 or −1, or are they split equally? If a spin at any given position i is 1, what is the probability that the spin at position j is also 1? If β is changed, is there a phase transition? On a lattice Λ, what is the fractal dimension of the shape of a large cluster of +1 spins?
Basic properties and history
The most studied case of the Ising model is the translation-invariant ferromagnetic zero-field model on a d-dimensional lattice, namely, Λ = Zd, Jij = 1, h = 0.
No phase transition in one dimension In his 1924 PhD thesis, Ising solved the model for the d = 1 case, which can be thought of as a linear horizontal lattice where each site only interacts with its left and right neighbor. In one dimension, the solution admits no phase transition. Namely, for any positive β, the correlations ⟨σiσj⟩ decay exponentially in |i − j|:
and the system is disordered. On the basis of this result, he incorrectly concluded that this model does not exhibit phase behaviour in any dimension.
Phase transition and exact solution in two dimensions The Ising model undergoes a phase transition between an ordered and a disordered phase in 2 dimensions or more. Namely, the system is disordered for small β, whereas for large β the system exhibits ferromagnetic order:
This was first proven by Rudolf Peierls in 1936, using what is now called a Peierls argument.
The Ising model on a two-dimensional square lattice with no magnetic field was analytically solved by . Onsager obtained the correlation functions and free energy of the Ising model and announced the formula for the spontaneous magnetization for the 2-dimensional model in 1949 but did not give a derivation. gave the first published proof of this formula, using a limit formula for Fredholm determinants, proved in 1951 by Szegő in direct response to Onsager's work.
Correlation inequalities A number of correlation inequalities have been derived rigorously for the Ising spin correlations (for general lattice structures), which have enabled mathematicians to study the Ising model both on and off criticality.
Griffiths inequality
Given any subset of spins and on the lattice, the following inequality holds,
where .
With , the special case results.
This means that spins are positively correlated on the Ising ferromagnet. An immediate application of this is that the magnetization of any set of spins is increasing with respect to any set of coupling constants .
Simon-Lieb inequality The Simon-Lieb inequality states that for any set disconnecting from (e.g. the boundary of a box with being inside the box and being outside),
This inequality can be used to establish the sharpness of phase transition for the Ising model.
FKG inequality
This inequality is proven first for a type of positively-correlated percolation model, of which includes a representation of the Ising model. It is used to determine the critical temperatures of planar Potts model using percolation arguments (which includes the Ising model as a special case).
Historical significance While the laws of chemical bonding made it clear to nineteenth century chemists that atoms were real, among physicists the debate continued well into the early twentieth century. Atomists, notably James Clerk Maxwell and Ludwig Boltzmann, applied Hamilton's formulation of Newton's laws to large systems, and found that the statistical behavior of the atoms correctly describes room temperature gases. But classical statistical mechanics did not account for all of the properties of liquids and solids, nor of gases at low temperature.
Once modern quantum mechanics was formulated, atomism was no longer in conflict with experiment, but this did not lead to a universal acceptance of statistical mechanics, which went beyond atomism. Josiah Willard Gibbs had given a complete formalism to reproduce the laws of thermodynamics from the laws of mechanics. But many faulty arguments survived from the 19th century, when statistical mechanics was considered dubious. The lapses in intuition mostly stemmed from the fact that the limit of an infinite statistical system has many zero-one laws which are absent in finite systems: an infinitesimal change in a parameter can lead to big differences in the overall, aggregate behavior.
No phase transitions in finite volume In the early part of the twentieth century, some believed that the partition function could never describe a phase transition, based on the following argument:
The partition function is a sum of e−βE over all configurations. The exponential function is everywhere analytic as a function of β. The sum of analytic functions is an analytic function.
This argument works for a finite sum of exponentials, and correctly establishes that there are no singularities in the free energy of a system of a finite size. For systems which are in the thermodynamic limit (that is, for infinite systems) the infinite sum can lead to singularities. The convergence to the thermodynamic limit is fast, so that the phase behavior is apparent already on a relatively small lattice, even though the singularities are smoothed out by the system's finite size.