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n analytic description much later by lars onsager 1944 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 3 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 barth 1981 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 applications 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 edit consider a set λ displaystyle lambda of lattice sites each with a set of adjacent sites e g a graph forming a d displaystyle d dimensional lattice for each lattice site k λ displaystyle k in lambda there is a discrete variable σ k displaystyle sigma _ k such that σ k 1 1 displaystyle sigma _ k in 1 1 representing the site s spin a spin configuration σ σ k k λ displaystyle sigma sigma _ k _ k in lambda is an assignment of spin value to each lattice site for any two adjacent sites i j λ displaystyle i j in lambda there is an interaction j i j displaystyle j_ ij also a site j λ displaystyle j in lambda has an external magnetic field h j displaystyle h_ j interacting with it the energy of a configuration σ displaystyle sigma is given by the hamiltonian function h σ i j j i j σ i σ j μ j h j σ j displaystyle h sigma sum _ langle ij rangle j_ ij sigma _ i sigma _ j mu sum _ j h_ j sigma _ j where the first sum is over pairs of adjacent spins every pair is counted once the notation i j displaystyle langle ij rangle indicates that sites i displaystyle i and j displaystyle j are nearest neighbors the magnetic moment is given by μ displaystyle mu 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 4 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 β 0 displaystyle beta geq 0 p β σ e β h σ z β displaystyle p_ beta sigma frac e beta h sigma z_ beta where β 1 k b t displaystyle beta 1 k_ text b t and the normalization constant z β σ e β h σ displaystyle z_ beta sum _ sigma e beta h sigma is the partition function for a function f displaystyle f of the spins observable one denotes by f β σ f σ p β σ displaystyle langle f rangle _ beta sum _ sigma f sigma p_ beta sigma the expectation mean value of f displaystyle f the configuration probabilities p β σ displaystyle p_ beta sigma represent the probability that in equilibrium the system is in a state with configuration σ displaystyle sigma discussion edit the minus sign on each term of the hamiltonian function h σ displaystyle h sigma is conventional using this sign convention ising models can be classified according to the sign of the interaction if for a pair i j j i j 0 displaystyle j_ ij 0 the interaction is called ferromagnetic j i j 0 displaystyle j_ ij 0 the interaction is called antiferromagnetic j i j 0 displaystyle j_ ij 0 the spins are noninteracting 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 h j 0 displaystyle h_ j 0 the spin site j desires to line up in the positive direction h j 0 displaystyle h_ j 0 the spin site j desires to line up in the negative direction h j 0 displaystyle h_ j 0 there is no external influence on the spin site simplifications edit 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 h σ i j j i j σ i σ j displaystyle h sigma sum _ langle i j rangle j_ ij sigma _ i sigma _ j 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 j ij j for all pairs i j in λ in this case the hamiltonian is further simplified to h σ j i j σ i σ j displaystyle h sigma j sum _ langle i j rangle sigma _ i sigma _ j connection to graph maximum cut edit 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 h σ i j e g j i j σ i σ j displaystyle h sigma sum _ ij in e g j_ ij sigma _ i sigma _ j here each vertex i of the graph is a spin site that takes a spin value σ i 1 displaystyle sigma _ i pm 1 a given spin configuration σ displaystyle sigma partitions the set of vertices v g displaystyle v g into two σ displaystyle sigma depended subsets those with spin up v displaystyle v and those with spin down v displaystyle v we denote by δ v displaystyle delta v the σ displaystyle sigma depended set of edges that connects the two complementary vertex subsets v displaystyle v and v displaystyle v the size δ v displaystyle left delta v right of the cut δ v displaystyle delta v to bipartite the weighted undirected graph g can be defined as δ v 1 2 i j δ v w i j displaystyle left delta v right frac 1 2 sum _ ij in delta v w_ ij where w i j displaystyle w_ ij denotes a weight of the edge i j displaystyle ij and the scaling 1 2 is introduced to compensate for double counting the same weights w i j w j i displaystyle w_ ij w_ ji the identities h σ i j e v j i j i j e v j i j i j δ v j i j i j e g j i j 2 i j δ v j i j displaystyle begin aligned h sigma sum _ ij in e v j_ ij sum _ ij in e v j_ ij sum _ ij in delta v j_ ij sum _ ij in e g j_ ij 2 sum _ ij in delta v j_ ij end aligned where the total sum in the first term does not depend on σ displaystyle sigma imply that minimizing h σ displaystyle h sigma in σ displaystyle sigma is equivalent to minimizing i j δ v j i j displaystyle sum _ ij in delta v j_ ij defining the edge weight w i j j i j displaystyle w_ ij j_ ij thus turns the ising problem without an external field into a graph max cut problem 5 maximizing the cut size δ v displaystyle left delta v right which is related to the ising hamiltonian as follows h σ i j e g w i j 4 δ v displaystyle h sigma sum _ ij in e g w_ ij 4 left delta v right questions edit 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 edit visualization of the translation invariant probability measure of the one dimensional ising model the most studied case of the ising model is the translation invariant ferromagnetic zero field model on a d dimensional lattice namely λ z d j ij 1 h 0 no phase transition in one dimension edit 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 6 namely for any positive β the correlations σ i σ j decay exponentially in i j σ i σ j β c exp c β i j displaystyle langle sigma _ i sigma _ j rangle _ beta leq c exp left c beta i j right and the system is disordered on the basis of this result he incorrectly concluded citation needed that this model does not exhibit phase behaviour in any dimension phase transition and exact solution in two dimensions edit 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 σ i σ j β c β 0 displaystyle langle sigma _ i sigma _ j rangle _ beta geq c beta 0 this was first proven by rudolf peierls in 1936 7 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 lars onsager 1944 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 yang 1952 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 8 correlation inequalities edit 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 edit main article griffiths inequality given any subset of spins σ a displaystyle sigma _ a and σ b displaystyle sigma _ b on the lattice the following inequality holds σ a σ b σ a σ b displaystyle langle sigma _ a sigma _ b rangle geq langle sigma _ a rangle langle sigma _ b rangle where σ a j a σ j displaystyle langle sigma _ a rangle langle prod _ j in a sigma _ j rangle with b displaystyle b emptyset the special case σ a 0 displaystyle langle sigma _ a rangle geq 0 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 σ a displaystyle langle sigma _ a rangle is increasing with respect to any set of coupling constants j b displaystyle j_ b simon lieb inequality edit the simon lieb inequality 9 states that for any set s displaystyle s disconnecting x displaystyle x from y displaystyle y e g the boundary of a box with x displaystyle x being inside the box and y displaystyle y being outside σ x σ y z s σ x σ z σ z σ y displaystyle langle sigma _ x sigma _ y rangle leq sum _ z in s langle sigma _ x sigma _ z rangle langle sigma _ z sigma _ y rangle this inequality can be used to establish the sharpness of phase transition for the ising model 10 fkg inequality edit main article 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 11 historical significance edit 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 edit 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 β e 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 this was first established by rudolf peierls in the ising model peierls droplets edit shortly after lenz and ising constructed the ising model peierls was able to explicitly show that a phase transition occurs in two dimensions to do this he compared the high temperature and low temperature limits at infinite temperature β 0 all configurations have equal probability each spin is completely independent of any other and if typical configurations at infinite temperature are plotted so that plus minus are represented by black and white they look like television snow for high but not infinite temperature there are small correlations between neighboring positions the snow tends to clump a little bit but the screen stays randomly looking and there is no net excess of black or white a quantitative measure of the excess is the magnetization which is the average value of the spin m 1 n i 1 n σ i displaystyle m frac 1 n sum _ i 1 n sigma _ i a bogus argument analogous to the argument in the last section now establis...
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