{"history":[],"article":{"id":"68089a33-c7e8-4f41-9660-7eaab6e1b148","title":"Ising model","slug":"ising-model","content":":::summary\n:::\n\n# \nThe **Ising model** (or Lenz–**Ising model**), named after the physicists [Ernst Ising](https://bliptext.com/articles/ernst-ising) and [Wilhelm Lenz](https://bliptext.com/articles/wilhelm-lenz), is a [mathematical model](https://bliptext.com/articles/mathematical-models-in-physics) of [ferromagnetism](https://bliptext.com/articles/ferromagnetism) in [statistical mechanics](https://bliptext.com/articles/statistical-mechanics).\nThe model consists of [discrete variables](https://bliptext.com/articles/discrete-variables) that represent [magnetic dipole moments of atomic \"spins\"](https://bliptext.com/articles/nuclear-magnetic-moment) that can be in one of two states (+1 or −1).\nThe spins are arranged in a graph, usually a [lattice](https://bliptext.com/articles/lattice-group) (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors.\nNeighboring 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.\nThe model allows the identification of [phase transitions](https://bliptext.com/articles/phase-transition) as a simplified model of reality.\nThe two-dimensional [square-lattice Ising model](https://bliptext.com/articles/square-lattice-ising-model) is one of the simplest statistical models to show a [phase transition](https://bliptext.com/articles/phase-transition).\nThe Ising model was invented by the physicist, who gave it as a problem to his student Ernst Ising.\nThe one-dimensional Ising model was solved by alone in his 1924 thesis; it has no phase transition.\nThe two-dimensional square-lattice Ising model is much harder and was only given an analytic description much later, by.\nIt is usually solved by a [transfer-matrix method](https://bliptext.com/articles/transfer-matrix-method-statistical-mechanics), although there exist different approaches, more related to [quantum field theory](https://bliptext.com/articles/quantum-field-theory).\nIn dimensions greater than four, the phase transition of the Ising model is described by [mean-field theory](https://bliptext.com/articles/mean-field-theory).\nThe 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.\nThe 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.\nThe Ising problem without an external field can be equivalently formulated as a [graph](https://bliptext.com/articles/graph-discrete-mathematics) [maximum cut](https://bliptext.com/articles/maximum-cut) (Max-Cut) problem that can be solved via [combinatorial optimization](https://bliptext.com/articles/combinatorial-optimization).\n\n# Definition\nConsider a set \\Lambda of lattice sites, each with a set of adjacent sites (e.g. a [graph](https://bliptext.com/articles/graph-discrete-mathematics)) forming a d-dimensional lattice.\nFor each lattice site k\\in\\Lambda there is a discrete variable \\sigma_k such that, representing the site's spin.\nA spin configuration, is an assignment of spin value to each lattice site.\nFor any two adjacent sites there is an interaction J_{ij}.\nAlso a site j\\in\\Lambda has an external magnetic field h_j interacting with it.\nThe energy of a configuration {\\sigma} is given by the [Hamiltonian function](https://bliptext.com/articles/hamiltonian-function)\nwhere the first sum is over pairs of adjacent spins (every pair is counted once).\nThe notation indicates that sites i and j are nearest neighbors.\nThe [magnetic moment](https://bliptext.com/articles/magnetic-moment) is given by \\mu.\nNote 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.\nThe configuration probability is given by the [Boltzmann distribution](https://bliptext.com/articles/boltzmann-distribution) with [inverse temperature](https://bliptext.com/articles/inverse-temperature) \\beta\\geq0:\nwhere, and the normalization constant\nis the [partition function](https://bliptext.com/articles/partition-function-statistical-mechanics).\nFor a function f of the spins (\"observable\"), one denotes by\nthe expectation (mean) value of f.\nThe configuration probabilities represent the probability that (in equilibrium) the system is in a state with configuration \\sigma.\n\n## Discussion\nThe minus sign on each term of the Hamiltonian function H(\\sigma) is conventional.\nUsing this sign convention, Ising models can be classified according to the sign of the interaction: if, for a pair i, j\nJ_{ij} > 0, the interaction is called ferromagnetic,\nJ_{ij} < 0, the interaction is called antiferromagnetic,\nJ_{ij} = 0, the spins are noninteracting.\nThe system is called ferromagnetic or antiferromagnetic if all interactions are ferromagnetic or all are antiferromagnetic.\nThe original Ising models were ferromagnetic, and it is still often assumed that \"Ising model\" means a ferromagnetic Ising model.\nIn a ferromagnetic Ising model, spins desire to be aligned: the configurations in which adjacent spins are of the same sign have higher probability.\nIn an antiferromagnetic model, adjacent spins tend to have opposite signs.\nThe sign convention of H(σ) also explains how a spin site j interacts with the external field.\nNamely, the spin site wants to line up with the external field.\nIf:\nh_j > 0, the spin site j desires to line up in the positive direction,\nh_j < 0, the spin site j desires to line up in the negative direction,\n\n## Simplifications\nIsing models are often examined without an external field interacting with the lattice, that is, h = 0 for all j in the lattice Λ.\nUsing this simplification, the Hamiltonian becomes\nWhen 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.\nAnother common simplification is to assume that all of the nearest neighbors ⟨ij⟩ have the same interaction strength.\nThen we can set Jij = J for all pairs i, j in Λ.\nIn this case the Hamiltonian is further simplified to\n\n## Connection to graph maximum cut\nA subset S of the [vertex](https://bliptext.com/articles/vertex-graph-theory) set V(G) of a weighted undirected graph G determines a cut of the graph G into S and its [complementary](https://bliptext.com/articles/complement-graph) subset G\\S.\nThe size of the cut is the sum of the weights of the edges between S and G\\S.\nA [maximum cut](https://bliptext.com/articles/maximum-cut) size is at least the size of any other cut, varying S.\nFor the Ising model without an external field on a graph G, the Hamiltonian becomes the following sum over the graph edges E(G)\n.\nHere each vertex i of the graph is a spin site that takes a spin value.\nA given spin configuration \\sigma partitions the set of vertices V(G) into two \\sigma-depended subsets, those with spin up V^+ and those with spin down V^-.\nWe denote by \\delta(V^+) the \\sigma-depended set of edges that connects the two complementary vertex subsets V^+ and V^-.\nThe size of the cut \\delta(V^+) to [bipartite](https://bliptext.com/articles/bipartite-graph) the weighted undirected graph G can be defined as\nwhere W_{ij} denotes a weight of the edge ij and the scaling 1/2 is introduced to compensate for double counting the same weights.\nThe identities\nwhere the total sum in the first term does not depend on \\sigma, imply that minimizing H(\\sigma) in \\sigma is equivalent to minimizing.\nDefining the edge weight thus turns the Ising problem without an external field into a graph Max-Cut problem\nmaximizing the cut size, which is related to the Ising Hamiltonian as follows,\n\n## Questions\nA significant number of statistical questions to ask about this model are in the limit of large numbers of spins:\n\n# Basic properties and history\nThe most studied case of the Ising model is the translation-invariant ferromagnetic zero-field model on a d-dimensional lattice, namely, Λ = **Z**d, Jij = 1, h = 0.\n# [Visualization of the translation-invariant probability measure of the one-dimensional Ising model | upload.wikimedia.org/wikipedia/commons/7/7d/Ising-tartan.png]\n\n## No phase transition in one dimension\nIn 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.\nIn one dimension, the solution admits no [phase transition](https://bliptext.com/articles/phase-transition).\nNamely, for any positive β, the correlations ⟨σiσj⟩ decay exponentially in |i − j|:\nand the system is disordered.\nOn the basis of this result, he incorrectly concluded that this model does not exhibit phase behaviour in any dimension.\n\n## Phase transition and exact solution in two dimensions\nThe Ising model undergoes a [phase transition](https://bliptext.com/articles/phase-transition) between an [ordered](https://bliptext.com/articles/ordered-phase) and a dis[ordered](https://bliptext.com/articles/ordered-phase) phase in 2 dimensions or more.\nNamely, the system is disordered for small β, whereas for large β the system exhibits ferromagnetic order:\nThis was first proven by [Rudolf Peierls](https://bliptext.com/articles/rudolf-peierls) in 1936, using what is now called a **Peierls argument**.\nThe Ising model on a two-dimensional square lattice with no magnetic field was analytically solved by.\nOnsager showed that the [correlation functions](https://bliptext.com/articles/correlation-function) and [free energy](https://bliptext.com/articles/thermodynamic-free-energy) of the Ising model are determined by a noninteracting lattice fermion.\nOnsager announced the formula for the [spontaneous magnetization](https://bliptext.com/articles/spontaneous-magnetization) for the 2-dimensional model in 1949 but did not give a derivation.\ngave the first published proof of this formula, using a [limit formula](https://bliptext.com/articles/szeg-limit-theorems) for [Fredholm determinants](https://bliptext.com/articles/fredholm-determinant), proved in 1951 by [Szegő](https://bliptext.com/articles/g-bor-szeg) in direct response to Onsager's work.\n\n## Correlation inequalities\nA number of [correlation inequalities](https://bliptext.com/articles/correlation-inequality) 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.\n\n### Griffiths inequality\nGiven any subset of spins \\sigma_A and \\sigma_B on the lattice, the following inequality holds,\nwhere.\nWith B = \\empty, the special case results.\nThis means that spins are positively correlated on the Ising ferromagnet.\nAn immediate application of this is that the magnetization of any set of spins is increasing with respect to any set of coupling constants J_B.\n\n### Simon-Lieb inequality\nThe Simon-Lieb inequality states that for any set S disconnecting x from y (e.g. the boundary of a box with x being inside the box and y being outside),\nThis inequality can be used to establish the sharpness of phase transition for the Ising model.\n\n### FKG inequality\nThis inequality is proven first for a type of [positively-correlated percolation model](https://bliptext.com/articles/random-cluster-model), of which includes a representation of the Ising model.\nIt is used to determine the critical temperatures of planar [Potts model](https://bliptext.com/articles/potts-model) using percolation arguments (which includes the Ising model as a special case).\n\n# Historical significance\nOne of [Democritus](https://bliptext.com/articles/democritus)' arguments in support of [atomism](https://bliptext.com/articles/atomism) was that atoms naturally explain the sharp phase boundaries observed in materials, as when ice melts to water or water turns to steam.\nHis idea was that small changes in atomic-scale properties would lead to big changes in the aggregate behavior.\nOthers believed that matter is inherently continuous, not atomic, and that the large-scale properties of matter are not reducible to basic atomic properties.\nWhile 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.\nAtomists, notably [James Clerk Maxwell](https://bliptext.com/articles/james-clerk-maxwell) and [Ludwig Boltzmann](https://bliptext.com/articles/ludwig-boltzmann), applied Hamilton's formulation of Newton's laws to large systems, and found that the [statistical behavior](https://bliptext.com/articles/statistical-mechanics) of the atoms correctly describes room temperature gases.\nBut classical statistical mechanics did not account for all of the properties of liquids and solids, nor of gases at low temperature.\nOnce modern [quantum mechanics](https://bliptext.com/articles/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.\n[Josiah Willard Gibbs](https://bliptext.com/articles/josiah-willard-gibbs) had given a complete formalism to reproduce the laws of thermodynamics from the laws of mechanics.\nBut many faulty arguments survived from the 19th century, when statistical mechanics was considered dubious.\nThe lapses in intuition mostly stemmed from the fact that the limit of an infinite statistical system has many [zero-one laws](https://bliptext.com/articles/zero-one-law) which are absent in finite systems: an infinitesimal change in a parameter can lead to big differences in the overall, aggregate behavior, as Democritus expected.\n\n## No phase transitions in finite volume\nIn the early part of the twentieth century, some believed that the [partition function](https://bliptext.com/articles/partition-function-statistical-mechanics) could never describe a phase transition, based on the following argument:\nThis 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.\nFor systems which are in the thermodynamic limit (that is, for infinite systems) the infinite sum can lead to singularities.\nThe 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.\nThis was first established by [Rudolf Peierls](https://bliptext.com/articles/rudolf-peierls) in the Ising model.\n\n## Peierls droplets\nShortly after Lenz and Ising constructed the Ising model, Peierls was able to explicitly show that a phase transition occurs in two dimensions.\nTo do this, he compared the high-temperature and low-temperature limits.\nAt infinite temperature (β = 0) all configurations have equal probability.\nEach 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](https://bliptext.com/articles/noise-video).\nFor 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.\nA quantitative measure of the excess is the **magnetization**, which is the average value of the spin:\nA bogus argument analogous to the argument in the last section now establishes that the magnetization in the Ising model is always zero.\nAs before, this only proves that the average magnetization is zero at any finite volume.\nFor an infinite system, fluctuations might not be able to push the system from a mostly plus state to a mostly minus with a nonzero probability.\nFor very high temperatures, the magnetization is zero, as it is at infinite temperature.\nTo see this, note that if spin A has only a small correlation ε with spin B, and B is only weakly correlated with C, but C is otherwise independent of A, the amount of correlation of A and C goes like ε2.\nFor two spins separated by distance L, the amount of correlation goes as εL, but if there is more than one path by which the correlations can travel, this amount is enhanced by the number of paths.\nThe number of paths of length L on a square lattice in d dimensions is\nsince there are 2d choices for where to go at each step.\nA bound on the total correlation is given by the contribution to the correlation by summing over all paths linking two points, which is bounded above by the sum over all paths of length L divided by\nwhich goes to zero when ε is small.\nAt low temperatures (β ≫ 1) the configurations are near the lowest-energy configuration, the one where all the spins are plus or all the spins are minus.\nPeierls asked whether it is statistically possible at low temperature, starting with all the spins minus, to fluctuate to a state where most of the spins are plus.\nFor this to happen, droplets of plus spin must be able to congeal to make the plus state.\nThe energy of a droplet of plus spins in a minus background is proportional to the perimeter of the droplet L, where plus spins and minus spins neighbor each other.\nFor a droplet with perimeter L, the area is somewhere between (L − 2)/2 (the straight line) and (L/4)2 (the square box).\nThe probability cost for introducing a droplet has the factor e−βL, but this contributes to the partition function multiplied by the total number of droplets with perimeter L, which is less than the total number of paths of length L:\nSo that the total spin contribution from droplets, even overcounting by allowing each site to have a separate droplet, is bounded above by\nwhich goes to zero at large β.\nFor β sufficiently large, this exponentially suppresses long loops, so that they cannot occur, and the magnetization never fluctuates too far from −1.\nSo Peierls established that the magnetization in the Ising model eventually defines [superselection sectors](https://bliptext.com/articles/superselection-sector), separated domains not linked by finite fluctuations.\n\n## Kramers–Wannier duality\nKramers and Wannier were able to show that the high-temperature expansion and the low-temperature expansion of the model are equal up to an overall rescaling of the free energy.\nThis allowed the phase-transition point in the two-dimensional model to be determined exactly (under the assumption that there is a unique critical point).\n\n## Yang–Lee zeros\nAfter Onsager's solution, Yang and Lee investigated the way in which the partition function becomes singular as the temperature approaches the critical temperature.\n\n# Applications\n\n## Magnetism\nThe original motivation for the model was the phenomenon of [ferromagnetism](https://bliptext.com/articles/ferromagnetism).\nIron is magnetic; once it is magnetized it stays magnetized for a long time compared to any atomic time.\nIn the 19th century, it was thought that magnetic fields are due to currents in matter, and [Ampère](https://bliptext.com/articles/andr-marie-amp-re) postulated that permanent magnets are caused by permanent atomic currents.\nThe motion of classical charged particles could not explain permanent currents though, as shown by [Larmor](https://bliptext.com/articles/joseph-larmor).\nIn order to have ferromagnetism, the atoms must have permanent [magnetic moments](https://bliptext.com/articles/magnetic-moment) which are not due to the motion of classical charges.\nOnce the electron's spin was discovered, it was clear that the magnetism should be due to a large number of electron spins all oriented in the same direction.\nIt was natural to ask how the electrons' spins all know which direction to point in, because the electrons on one side of a magnet don't directly interact with the electrons on the other side.\nThey can only influence their neighbors.\nThe Ising model was designed to investigate whether a large fraction of the electron spins could be oriented in the same direction using only local forces.\n\n## Lattice gas\nThe Ising model can be reinterpreted as a statistical model for the motion of atoms.\nSince the kinetic energy depends only on momentum and not on position, while the statistics of the positions only depends on the potential energy, the thermodynamics of the gas only depends on the potential energy for each configuration of atoms.\nA coarse model is to make space-time a lattice and imagine that each position either contains an atom or it doesn't. The space of configuration is that of independent bits Bi, where each bit is either 0 or 1 depending on whether the position is occupied or not.\nAn attractive interaction reduces the energy of two nearby atoms.\nIf the attraction is only between nearest neighbors, the energy is reduced by −4JBiBj for each occupied neighboring pair.\nThe density of the atoms can be controlled by adding a [chemical potential](https://bliptext.com/articles/chemical-potential), which is a multiplicative probability cost for adding one more atom.\nA multiplicative factor in probability can be reinterpreted as an additive term in the logarithm – the energy.\nThe extra energy of a configuration with N atoms is changed by μN.\nThe probability cost of one more atom is a factor of exp(−βμ).\nSo the energy of the lattice gas is:\nRewriting the bits in terms of spins,\nFor lattices where every site has an equal number of neighbors, this is the Ising model with a magnetic field h = (zJ − μ)/2, where z is the number of neighbors.\nIn biological systems, modified versions of the lattice gas model have been used to understand a range of binding behaviors.\nThese include the binding of ligands to receptors in the cell surface, the binding of [chemotaxis](https://bliptext.com/articles/chemotaxis) proteins to the flagellar motor, and the condensation of DNA.\n\n## Neuroscience\nThe activity of [neurons](https://bliptext.com/articles/neuron) in the brain can be modelled statistically.\nEach neuron at any time is either active + or inactive −.\nThe active neurons are those that send an [action potential](https://bliptext.com/articles/action-potential) down the axon in any given time window, and the inactive ones are those that do not.\nFollowing the general approach of Jaynes, a later interpretation of Schneidman, Berry, Segev and Bialek,\nis that the Ising model is useful for any model of neural function, because a statistical model for neural activity should be chosen using the [principle of maximum entropy](https://bliptext.com/articles/principle-of-maximum-entropy).\nGiven a collection of neurons, a statistical model which can reproduce the average firing rate for each neuron introduces a [Lagrange multiplier](https://bliptext.com/articles/lagrange-multiplier) for each neuron:\nBut the activity of each neuron in this model is statistically independent.\nTo allow for pair correlations, when one neuron tends to fire (or not to fire) along with another, introduce pair-wise lagrange multipliers:\nwhere J_{ij} are not restricted to neighbors.\nNote that this generalization of Ising model is sometimes called the quadratic exponential binary distribution in statistics.\nThis energy function only introduces probability biases for a spin having a value and for a pair of spins having the same value.\nHigher order correlations are unconstrained by the multipliers.\nAn activity pattern sampled from this distribution requires the largest number of bits to store in a computer, in the most efficient coding scheme imaginable, as compared with any other distribution with the same average activity and pairwise correlations.\nThis means that Ising models are relevant to any system which is described by bits which are as random as possible, with constraints on the pairwise correlations and the average number of 1s, which frequently occurs in both the physical and social sciences.\n\n## Spin glasses\nWith the Ising model the so-called [spin glasses](https://bliptext.com/articles/spin-glasses) can also be described, by the usual Hamiltonian where the S-variables describe the Ising spins, while the Ji,k are taken from a random distribution.\nFor spin glasses a typical distribution chooses antiferromagnetic bonds with probability p and ferromagnetic bonds with probability 1 − p (also known as the random-bond Ising model).\nThese bonds stay fixed or \"quenched\" even in the presence of thermal fluctuations.\nWhen p = 0 we have the original Ising model.\nThis system deserves interest in its own; particularly one has \"non-ergodic\" properties leading to strange relaxation behaviour.\nMuch attention has been also attracted by the related bond and site dilute Ising model, especially in two dimensions, leading to intriguing critical behavior.\n\n## Artificial neural network\nIsing model was instrumental in the development of the [Hopfield network](https://bliptext.com/articles/hopfield-network).\nThe original Ising model is a model for equilibrium.\n[Roy J. Glauber](https://bliptext.com/articles/roy-j-glauber) in 1963 studied the Ising model evolving in time, as a process towards thermal equilibrium ([Glauber dynamics](https://bliptext.com/articles/glauber-dynamics)), adding in the component of time.\n(Kaoru Nakano, 1971) and ([Shun'ichi Amari](https://bliptext.com/articles/shun-ichi-amari), 1972), proposed to modify the weights of an Ising model by [Hebbian learning](https://bliptext.com/articles/hebbian-theory) rule as a model of associative memory.\nThe same idea was published by (William A. Little (physicist), 1974), who was cited by Hopfield in his 1982 paper.\nThe [Sherrington–Kirkpatrick model](https://bliptext.com/articles/spin-glass) of spin glass, published in 1975, is the Hopfield network with random initialization.\nSherrington and Kirkpatrick found that it is highly likely for the energy function of the SK model to have many local minima.\nIn the 1982 paper, Hopfield applied this recently developed theory to study the Hopfield network with binary activation functions.\nIn a 1984 paper he extended this to continuous activation functions.\nIt became a standard model for the study of neural networks through statistical mechanics.\n\n## Sea ice\nThe [melt pond](https://bliptext.com/articles/melt-pond) can be modelled by the Ising model; sea ice topography data bears rather heavily on the results.\nThe state variable is binary for a simple 2D approximation, being either water or ice.\n\n## Cayley tree topologies and large neural networks\nIn order to investigate an Ising model with potential relevance for large (e.g. with 10^4 or 10^5 interactions per node) neural nets, at the suggestion of Krizan in 1979, obtained the exact analytical expression for the free energy of the Ising model on the closed Cayley tree (with an arbitrarily large branching ratio) for a zero-external magnetic field (in the thermodynamic limit) by applying the methodologies of and\nwhere \\gamma is an arbitrary branching ratio (greater than or equal to 2),, , (with \\epsilon representing the nearest-neighbor interaction energy) and there are k (→ ∞ in the thermodynamic limit) generations in each of the tree branches (forming the closed tree architecture as shown in the given closed Cayley tree diagram.) The sum in the last term can be shown to converge uniformly and rapidly (i.e. for z → ∞, it remains finite) yielding a continuous and monotonous function, establishing that, for \\gamma greater than or equal to 2, the free energy is a continuous function of temperature T. Further analysis of the free energy indicates that it exhibits an unusual discontinuous first derivative at the critical temperature (, .)\nThe spin-spin correlation between sites (in general, m and n) on the tree was found to have a transition point when considered at the vertices (e.g. A and Ā, its reflection), their respective neighboring sites (such as B and its reflection), and between sites adjacent to the top and bottom extreme vertices of the two trees (e.g. A and B), as may be determined from\nwhere N_b is equal to the number of bonds, is the number of graphs counted for odd vertices with even intermediate sites (see cited methodologies and references for detailed calculations), 2^N is the multiplicity resulting from two-valued spin possibilities and the partition function {Z_N} is derived from.\n(Note: s_i is consistent with the referenced literature in this section and is equivalent to S_i or \\sigma_i utilized above and in earlier sections; it is valued at \\pm 1.) The critical temperature T_C is given by\nThe critical temperature for this model is only determined by the branching ratio \\gamma and the site-to-site interaction energy \\epsilon, a fact which may have direct implications associated with neural structure vs. its function (in that it relates the energies of interaction and branching ratio to its transitional behavior.) For example, a relationship between the transition behavior of activities of neural networks between sleeping and wakeful states (which may correlate with a spin-spin type of phase transition) in terms of changes in neural interconnectivity (\\gamma) and/or neighbor-to-neighbor interactions (\\epsilon), over time, is just one possible avenue suggested for further experimental investigation into such a phenomenon.\nIn any case, for this Ising model it was established, that “the stability of the long-range correlation increases with increasing \\gamma or increasing \\epsilon.”\nFor this topology, the spin-spin correlation was found to be zero between the extreme vertices and the central sites at which the two trees (or branches) are joined (i.e. between A and individually C, D, or E.) This behavior is explained to be due to the fact that, as k increases, the number of links increases exponentially (between the extreme vertices) and so even though the contribution to spin correlations decrease exponentially, the correlation between sites such as the extreme vertex (A) in one tree and the extreme vertex in the joined tree (Ā) remains finite (above the critical temperature.) In addition, A and B also exhibit a non-vanishing correlation (as do their reflections) thus lending itself to, for B level sites (with A level), being considered “clusters” which tend to exhibit synchronization of firing.\nBased upon a review of other classical network models as a comparison, the Ising model on a closed Cayley tree was determined to be the first classical statistical mechanical model to demonstrate both local and long-range sites with non-vanishing spin-spin correlations, while at the same time exhibiting intermediate sites with zero correlation, which indeed was a relevant matter for large neural networks at the time of its consideration.\nThe model's behavior is also of relevance for any other divergent-convergent tree physical (or biological) system exhibiting a closed Cayley tree topology with an Ising-type of interaction.\nThis topology should not be ignored since its behavior for Ising models has been solved exactly, and presumably nature will have found a way of taking advantage of such simple symmetries at many levels of its designs.\nearly on noted the possibility of interrelationships between (1) the classical large neural network model (with similar coupled divergent-convergent topologies) with (2) an underlying statistical quantum mechanical model (independent of topology and with persistence in fundamental quantum states):\n\"The most significant result obtained from the closed Cayley tree model involves the occurrence of long-range correlation in the absence of intermediate-range correlation. This result has not been demonstrated by other classical models. The failure of the classical view of impulse transmission to account for this phenomenon has been cited by numerous investigators (Ricciiardi and Umezawa, 1967, Hokkyo 1972, Stuart, Takahashi and Umezawa 1978, 1979) as significant enough to warrant radically new assumptions on a very fundamental level and have suggested the existence of quantum cooperative modes within the brain…In addition, it is interesting to note that the (modeling) of…Goldstone particles or bosons (as per Umezawa, et al)…within the brain, demonstrates the long-range correlation of quantum numbers preserved in the ground state…In the closed Cayley tree model ground states of pairs of sites, as well as the state variable of individual sites, (can) exhibit long-range correlation.\"\nIt was a natural and common belief among early neurophysicists (e.g. Umezawa, Krizan, Barth, etc.) that classical neural models (including those with statistical mechanical aspects) will one day have to be integrated with quantum physics (with quantum statistical aspects), similar perhaps to how the domain of chemistry has historically integrated itself into quantum physics via quantum chemistry.\nSeveral additional statistical mechanical problems of interest remain to be solved for the closed Cayley tree, including the time-dependent case and the external field situation, as well as theoretical efforts aimed at understanding interrelationships with underlying quantum constituents and their physics.\n# [Ising model | upload.wikimedia.org/wikipedia/commons/d/d1/Cayley///Tree///Branch///with///Branching///Ratio///%3D///2.jpg]\n# [Ising model | upload.wikimedia.org/wikipedia/commons/f/fc/Closed///Cayley///Tree///with///Branching///Ratio///%3D///4.jpg]\n\n# Numerical simulation\nThe Ising model can often be difficult to evaluate numerically if there are many states in the system.\nConsider an Ising model with\nSince every spin site has ±1 spin, there are 2L different states that are possible.\nThis motivates the reason for the Ising model to be simulated using [Monte Carlo methods](https://bliptext.com/articles/monte-carlo-methods).\nThe [Hamiltonian](https://bliptext.com/articles/hamiltonian-mechanics) that is commonly used to represent the energy of the model when using Monte Carlo methods is:\nFurthermore, the Hamiltonian is further simplified by assuming zero external field h, since many questions that are posed to be solved using the model can be answered in absence of an external field.\nThis leads us to the following energy equation for state σ:\nGiven this Hamiltonian, quantities of interest such as the specific heat or the magnetization of the magnet at a given temperature can be calculated.\n\n## Metropolis algorithm\nThe [Metropolis–Hastings algorithm](https://bliptext.com/articles/metropolis-hastings-algorithm) is the most commonly used Monte Carlo algorithm to calculate Ising model estimations.\nThe algorithm first chooses selection probabilities g(μ, ν), which represent the probability that state ν is selected by the algorithm out of all states, given that one is in state μ.\nIt then uses acceptance probabilities A(μ, ν) so that [detailed balance](https://bliptext.com/articles/detailed-balance) is satisfied.\nIf the new state ν is accepted, then we move to that state and repeat with selecting a new state and deciding to accept it.\nIf ν is not accepted then we stay in μ.\nThis process is repeated until some stopping criterion is met, which for the Ising model is often when the lattice becomes [ferromagnetic](https://bliptext.com/articles/ferromagnetic), meaning all of the sites point in the same direction.\nWhen implementing the algorithm, one must ensure that g(μ, ν) is selected such that [ergodicity](https://bliptext.com/articles/ergodicity) is met.\nIn [thermal equilibrium](https://bliptext.com/articles/thermal-equilibrium) a system's energy only fluctuates within a small range.\nThis is the motivation behind the concept of **single-spin-flip dynamics**, which states that in each transition, we will only change one of the spin sites on the lattice.\nFurthermore, by using single- spin-flip dynamics, one can get from any state to any other state by flipping each site that differs between the two states one at a time.\nThe maximum amount of change between the energy of the present state, Hμ and any possible new state's energy Hν (using single-spin-flip dynamics) is 2J between the spin we choose to \"flip\" to move to the new state and that spin's neighbor.\nThus, in a 1D Ising model, where each site has two neighbors (left and right), the maximum difference in energy would be 4J.\nLet c represent the lattice coordination number; the number of nearest neighbors that any lattice site has.\nWe assume that all sites have the same number of neighbors due to [periodic boundary conditions](https://bliptext.com/articles/periodic-boundary-conditions).\nIt is important to note that the Metropolis–Hastings algorithm does not perform well around the critical point due to critical slowing down.\nOther techniques such as multigrid methods, Niedermayer's algorithm, Swendsen–Wang algorithm, or the Wolff algorithm are required in order to resolve the model near the critical point; a requirement for determining the critical exponents of the system.\nSpecifically for the Ising model and using single-spin-flip dynamics, one can establish the following.\nSince there are L total sites on the lattice, using single-spin-flip as the only way we transition to another state, we can see that there are a total of L new states ν from our present state μ.\nThe algorithm assumes that the selection probabilities are equal to the L states: g(μ, ν) = 1/L.\n[Detailed balance](https://bliptext.com/articles/detailed-balance) tells us that the following equation must hold:\nThus, we want to select the acceptance probability for our algorithm to satisfy\nIf Hν > Hμ, then A(ν, μ) > A(μ, ν).\nMetropolis sets the larger of A(μ, ν) or A(ν, μ) to be 1.\nBy this reasoning the acceptance algorithm is:\nThe basic form of the algorithm is as follows:\nThe change in energy Hν − Hμ only depends on the value of the spin and its nearest graph neighbors.\nSo if the graph is not too connected, the algorithm is fast.\nThis process will eventually produce a pick from the distribution.\n\n## As a Markov chain\nIt is possible to view the Ising model as a [Markov chain](https://bliptext.com/articles/markov-chain), as the immediate probability Pβ(ν) of transitioning to a future state ν only depends on the present state μ.\nThe Metropolis algorithm is actually a version of a [Markov chain Monte Carlo](https://bliptext.com/articles/markov-chain-monte-carlo) simulation, and since we use single-spin-flip dynamics in the Metropolis algorithm, every state can be viewed as having links to exactly L other states, where each transition corresponds to flipping a single spin site to the opposite value.\nFurthermore, since the energy equation Hσ change only depends on the nearest-neighbor interaction strength J, the Ising model and its variants such the [Sznajd model](https://bliptext.com/articles/sznajd-model) can be seen as a form of a [voter model](https://bliptext.com/articles/contact-process-mathematics) for opinion dynamics.\n\n# Solutions\n\n## One dimension\nThe thermodynamic limit exists as long as the interaction decay is with α > 1.\n\n### Ising's exact solution\nIn the nearest neighbor case (with periodic or free boundary conditions) an exact solution is available.\nThe Hamiltonian of the one-dimensional Ising model on a lattice of L sites with free boundary conditions is\nwhere J and h can be any number, since in this simplified case J is a constant representing the interaction strength between the nearest neighbors and h is the constant external magnetic field applied to lattice sites.\nThen the\n[free energy](https://bliptext.com/articles/thermodynamic-free-energy) is\nand the spin-spin correlation (i.e. the covariance) is\nwhere C(β) and c(β) are positive functions for T > 0.\nFor T → 0, though, the inverse correlation length c(β) vanishes.\n\n#### Proof\nThe proof of this result is a simple computation.\nIf h = 0, it is very easy to obtain the free energy in the case of free boundary condition, i.e. when\nThen the model factorizes under the change of variables\nThis gives\nTherefore, the free energy is\nWith the same change of variables\nhence it decays exponentially as soon as T ≠ 0; but for T = 0, i.e. in the limit β → ∞ there is no decay.\nIf h ≠ 0 we need the transfer matrix method.\nFor the periodic boundary conditions case is the following.\nThe partition function is\nThe coefficients can be seen as the entries of a matrix.\nThere are different possible choices: a convenient one (because the matrix is symmetric) is\nor\nIn matrix formalism\nwhere λ1 is the highest eigenvalue of V, while λ2 is the other eigenvalue:\nand |λ2| < λ1.\nThis gives the formula of the free energy.\n\n#### Comments\nThe energy of the lowest state is −JL, when all the spins are the same.\nFor any other configuration, the extra energy is equal to 2J times the number of sign changes that are encountered when scanning the configuration from left to right.\nIf we designate the number of sign changes in a configuration as k, the difference in energy from the lowest energy state is 2k.\nSince the energy is additive in the number of flips, the probability p of having a spin-flip at each position is independent.\nThe ratio of the probability of finding a flip to the probability of not finding one is the Boltzmann factor:\nThe problem is reduced to independent biased [coin tosses](https://bliptext.com/articles/coin-toss).\nThis essentially completes the mathematical description.\nFrom the description in terms of independent tosses, the statistics of the model for long lines can be understood.\nThe line splits into domains.\nEach domain is of average length exp(2β).\nThe length of a domain is distributed exponentially, since there is a constant probability at any step of encountering a flip.\nThe domains never become infinite, so a long system is never magnetized.\nEach step reduces the correlation between a spin and its neighbor by an amount proportional to p, so the correlations fall off exponentially.\nThe [partition function](https://bliptext.com/articles/partition-function-statistical-mechanics) is the volume of configurations, each configuration weighted by its Boltzmann weight.\nSince each configuration is described by the sign-changes, the partition function factorizes:\nThe logarithm divided by L is the free energy density:\nwhich is [analytic](https://bliptext.com/articles/analytic-function) away from β = ∞.\nA sign of a [phase transition](https://bliptext.com/articles/phase-transition) is a non-analytic free energy, so the one-dimensional model does not have a [phase transition](https://bliptext.com/articles/phase-transition).\n\n### One-dimensional solution with transverse field\nTo express the Ising Hamiltonian using a quantum mechanical description of spins, we replace the spin variables with their respective [Pauli matrices](https://bliptext.com/articles/pauli-matrices).\nHowever, depending on the direction of the magnetic field, we can create a transverse-field or longitudinal-field Hamiltonian.\nThe [transverse-field](https://bliptext.com/articles/transverse-field-ising-model) Hamiltonian is given by\nThe transverse-field model experiences a phase transition between an ordered and disordered regime at J ~ h.\nThis can be shown by a mapping of Pauli matrices\nUpon rewriting the Hamiltonian in terms of this change-of-basis matrices, we obtain\nSince the roles of h and J are switched, the Hamiltonian undergoes a transition at J = h.\n\n### Renormalization\nWhen there is no external field, we can derive a functional equation that satisfies using renormalization.\nSpecifically, let be the partition function with N sites.\nNow we have:where.\nWe sum over each of, to obtainNow, since the cosh function is even, we can solve as.\nNow we have a self-similarity relation:Taking the limit, we obtainwhere.\nWhen \\beta is small, we have, so we can numerically evaluate f(\\beta) by iterating the functional equation until K is small.\n\n## Two dimensions\n\n### Onsager's exact solution\nobtained the following analytical expression for the free energy of the Ising model on the anisotropic square lattice when the magnetic field h=0 in the thermodynamic limit as a function of temperature and the horizontal and vertical interaction energies J_1 and J_2, respectively\nFrom this expression for the free energy, all thermodynamic functions of the model can be calculated by using an appropriate derivative.\nThe 2D Ising model was the first model to exhibit a continuous phase transition at a positive temperature.\nIt occurs at the temperature T_c which solves the equation\nIn the isotropic case when the horizontal and vertical interaction energies are equal J_1=J_2=J, the critical temperature T_c occurs at the following point\nWhen the interaction energies J_1, J_2 are both negative, the Ising model becomes an antiferromagnet.\nSince the square lattice is bi-partite, it is invariant under this change when the magnetic field h=0, so the free energy and critical temperature are the same for the antiferromagnetic case.\nFor the triangular lattice, which is not bi-partite, the ferromagnetic and antiferromagnetic Ising model behave notably differently.\nSpecifically, around a triangle, it is impossible to make all 3 spin-pairs antiparallel, so the antiferromagnetic Ising model cannot reach the minimal energy state.\nThis is an example of [geometric frustration](https://bliptext.com/articles/geometric-frustration).\n\n#### Transfer matrix\nStart with an analogy with quantum mechanics.\nThe Ising model on a long periodic lattice has a partition function\nThink of the i direction as space, and the j direction as time.\nThis is an independent sum over all the values that the spins can take at each time slice.\nThis is a type of [path integral](https://bliptext.com/articles/path-integral-formulation), it is the sum over all spin histories.\nA path integral can be rewritten as a Hamiltonian evolution.\nThe Hamiltonian steps through time by performing a unitary rotation between time t and time t + Δt:\nThe product of the U matrices, one after the other, is the total time evolution operator, which is the path integral we started with.\nwhere N is the number of time slices.\nThe sum over all paths is given by a product of matrices, each matrix element is the transition probability from one slice to the next.\nSimilarly, one can divide the sum over all partition function configurations into slices, where each slice is the one-dimensional configuration at time 1.\nThis defines the [transfer matrix](https://bliptext.com/articles/transfer-matrix-method-statistical-mechanics):\nThe configuration in each slice is a one-dimensional collection of spins.\nAt each time slice, T has matrix elements between two configurations of spins, one in the immediate future and one in the immediate past.\nThese two configurations are C1 and C2, and they are all one-dimensional spin configurations.\nWe can think of the vector space that T acts on as all complex linear combinations of these.\nUsing quantum mechanical notation:\nwhere each basis vector |S\\rangle is a spin configuration of a one-dimensional Ising model.\nLike the Hamiltonian, the transfer matrix acts on all linear combinations of states.\nThe partition function is a matrix function of T, which is defined by the [sum](https://bliptext.com/articles/trace-linear-algebra) over all histories which come back to the original configuration after N steps:\nSince this is a matrix equation, it can be evaluated in any basis.\nSo if we can diagonalize the matrix T, we can find Z.\n\n#### T in terms of Pauli matrices\nThe contribution to the partition function for each past/future pair of configurations on a slice is the sum of two terms.\nThere is the number of spin flips in the past slice and there is the number of spin flips between the past and future slice.\nDefine an operator on configurations which flips the spin at site i:\n\\sigma^x_i.\nIn the usual Ising basis, acting on any linear combination of past configurations, it produces the same linear combination but with the spin at position i of each basis vector flipped.\nDefine a second operator which multiplies the basis vector by +1 and −1 according to the spin at position i:\n\\sigma^z_i.\nT can be written in terms of these:\nwhere A and B are constants which are to be determined so as to reproduce the partition function.\nThe interpretation is that the statistical configuration at this slice contributes according to both the number of spin flips in the slice, and whether or not the spin at position i has flipped.\n\n#### Spin flip creation and annihilation operators\nJust as in the one-dimensional case, we will shift attention from the spins to the spin-flips.\nThe σz term in T counts the number of spin flips, which we can write in terms of spin-flip creation and annihilation operators:\nThe first term flips a spin, so depending on the basis state it either:\nWriting this out in terms of creation and annihilation operators:\nIgnore the constant coefficients, and focus attention on the form.\nThey are all quadratic.\nSince the coefficients are constant, this means that the T matrix can be diagonalized by Fourier transforms.\nCarrying out the diagonalization produces the Onsager free energy.\n\n#### Onsager's formula for spontaneous magnetization\nOnsager famously announced the following expression for the [spontaneous magnetization](https://bliptext.com/articles/spontaneous-magnetization) M of a two-dimensional Ising ferromagnet on the square lattice at two different conferences in 1948, though without proof\nwhere J_1 and J_2 are horizontal and vertical interaction energies.\nA complete derivation was only given in 1951 by using a limiting process of transfer matrix eigenvalues.\nThe proof was subsequently greatly simplified in 1963 by Montroll, Potts, and Ward using [Szegő](https://bliptext.com/articles/g-bor-szeg)'s [limit formula](https://bliptext.com/articles/szeg-limit-theorems) for [Toeplitz determinants](https://bliptext.com/articles/toeplitz-determinant) by treating the magnetization as the limit of correlation functions.\n\n### Minimal model\nAt the critical point, the two-dimensional Ising model is a [two-dimensional conformal field theory](https://bliptext.com/articles/two-dimensional-conformal-field-theory).\nThe spin and energy correlation functions are described by a [minimal model](https://bliptext.com/articles/minimal-model-physics), which has been exactly solved.\n\n## Three dimensions\nIn three as in two dimensions, the most studied case of the Ising model is the translation-invariant model on a cubic lattice with nearest-neighbor coupling in the zero magnetic field.\nMany** **theoreticians** **searched** **for** **an** **analytical** **three-dimensional** **solution** **for** **many** **decades,** **which** **would** **be** **analogous** **to** **Onsager's** **solution** **in** **the** **two-dimensional** **case.** **** **Such** **a** **solution** **has** **not** **been** **found** **until** **now,** **although** **there** **is** **no** **proof** **that** **it** **may** **not** **exist.\nIn three dimensions, the Ising model was shown to have a representation in terms of non-interacting fermionic strings by [Alexander Polyakov](https://bliptext.com/articles/alexander-markovich-polyakov) and [Vladimir Dotsenko](https://bliptext.com/articles/vladimir-dotsenko).\nThis construction has been carried on the lattice, and the [continuum limit](https://bliptext.com/articles/continuum-limit), conjecturally describing the critical point, is unknown.\nIn three as in two dimensions, Peierls' argument shows that there is a phase transition.\nThis phase transition is rigorously known to be continuous (in the sense that correlation length diverges and the magnetization goes to zero), and is called the [critical point](https://bliptext.com/articles/critical-point-thermodynamics).\nIt is believed that the critical point can be described by a renormalization group fixed point of the Wilson-Kadanoff renormalization group transformation.\nIt is also believed that the phase transition can be described by a three-dimensional unitary conformal field theory, as evidenced by [Monte Carlo](https://bliptext.com/articles/metropolis-hastings-algorithm) simulations, exact diagonalization results in quantum models, and quantum field theoretical arguments.\nAlthough it is an open problem to establish rigorously the renormalization group picture or the conformal field theory picture, theoretical physicists have used these two methods to compute the [critical exponents](https://bliptext.com/articles/critical-exponents) of the phase transition, which agree with the experiments and with the Monte Carlo simulations.\nThis conformal field theory describing the three-dimensional Ising critical point is under active investigation using the method of the [conformal bootstrap](https://bliptext.com/articles/conformal-bootstrap).\nThis method currently yields the most precise information about the structure of the critical theory (see [Ising critical exponents](https://bliptext.com/articles/ising-critical-exponents)).\nIn 2000, [Sorin Istrail](https://bliptext.com/articles/sorin-istrail) of [Sandia National Laboratories](https://bliptext.com/articles/sandia-national-laboratories) proved that the spin glass Ising model on a [nonplanar](https://bliptext.com/articles/nonplanar) lattice is [NP-complete](https://bliptext.com/articles/np-completeness).\nThat is, assuming **P** ≠ N**P**, the general spin glass Ising model is exactly solvable only in [planar](https://bliptext.com/articles/planar-graph) cases, so solutions for dimensions higher than two are also intractable.\nIstrail's result only concerns the spin glass model with spatially varying couplings, and tells nothing about Ising's original ferromagnetic model with equal couplings.\n\n## Four dimensions and above\nIn any dimension, the Ising model can be productively described by a locally varying mean field.\nThe field is defined as the average spin value over a large region, but not so large so as to include the entire system.\nThe field still has slow variations from point to point, as the averaging volume moves.\nThese fluctuations in the field are described by a continuum field theory in the infinite system limit.\n\n### Local field\nThe field H is defined as the long wavelength Fourier components of the spin variable, in the limit that the wavelengths are long.\nThere are many ways to take the long wavelength average, depending on the details of how high wavelengths are cut off.\nThe details are not too important, since the goal is to find the statistics of H and not the spins.\nOnce the correlations in H are known, the long-distance correlations between the spins will be proportional to the long-distance correlations in H.\nFor any value of the slowly varying field H, the free energy (log-probability) is a local analytic function of H and its gradients.\nThe free energy F(H) is defined to be the sum over all Ising configurations which are consistent with the long wavelength field.\nSince H is a coarse description, there are many Ising configurations consistent with each value of H, so long as not too much exactness is required for the match.\nSince the allowed range of values of the spin in any region only depends on the values of H within one averaging volume from that region, the free energy contribution from each region only depends on the value of H there and in the neighboring regions.\nSo F is a sum over all regions of a local contribution, which only depends on H and its derivatives.\nBy symmetry in H, only even powers contribute.\nBy reflection symmetry on a square lattice, only even powers of gradients contribute.\nWriting out the first few terms in the free energy:\nOn a square lattice, symmetries guarantee that the coefficients Zi of the derivative terms are all equal.\nBut even for an anisotropic Ising model, where the Zi's in different directions are different, the fluctuations in H are isotropic in a coordinate system where the different directions of space are rescaled.\nOn any lattice, the derivative term\nis a positive definite [quadratic form](https://bliptext.com/articles/quadratic-form), and can be used to define the metric for space.\nSo any translationally invariant Ising model is rotationally invariant at long distances, in coordinates that make Zij = δij.\nRotational symmetry emerges spontaneously at large distances just because there aren't very many low order terms.\nAt higher order multicritical points, this [accidental symmetry](https://bliptext.com/articles/accidental-symmetry) is lost.\nSince βF is a function of a slowly spatially varying field, the probability of any field configuration is (omitting higher-order terms):\nThe statistical average of any product of H terms is equal to:\nThe denominator in this expression is called the partition function:and the integral over all possible values of H is a statistical path integral.\nIt integrates exp(βF) over all values of H, over all the long wavelength fourier components of the spins.\nF is a \"Euclidean\" Lagrangian for the field H.\nIt is similar to the Lagrangian in of a scalar field in [quantum field theory](https://bliptext.com/articles/quantum-field-theory), the difference being that all the derivative terms enter with a positive sign, and there is no overall factor of i (thus \"Euclidean\").\n\n### Dimensional analysis\nThe form of F can be used to predict which terms are most important by dimensional analysis.\nDimensional analysis is not completely straightforward, because the scaling of H needs to be determined.\nIn the generic case, choosing the scaling law for H is easy, since the only term that contributes is the first one,\nThis term is the most significant, but it gives trivial behavior.\nThis form of the free energy is ultralocal, meaning that it is a sum of an independent contribution from each point.\nThis is like the spin-flips in the one-dimensional Ising model.\nEvery value of H at any point fluctuates completely independently of the value at any other point.\nThe scale of the field can be redefined to absorb the coefficient A, and then it is clear that A only determines the overall scale of fluctuations.\nThe ultralocal model describes the long wavelength high temperature behavior of the Ising model, since in this limit the fluctuation averages are independent from point to point.\nTo find the critical point, lower the temperature.\nAs the temperature goes down, the fluctuations in H go up because the fluctuations are more correlated.\nThis means that the average of a large number of spins does not become small as quickly as if they were uncorrelated, because they tend to be the same.\nThis corresponds to decreasing A in the system of units where H does not absorb A.\nThe phase transition can only happen when the subleading terms in F can contribute, but since the first term dominates at long distances, the coefficient A must be tuned to zero.\nThis is the location of the critical point:\nwhere t is a parameter which goes through zero at the transition.\nSince t is vanishing, fixing the scale of the field using this term makes the other terms blow up.\nOnce t is small, the scale of the field can either be set to fix the coefficient of the H4 term or the (∇H)2 term to 1.\n\n### Magnetization\nTo find the magnetization, fix the scaling of H so that λ is one.\nNow the field H has dimension −d/4, so that H4ddx is dimensionless, and Z has dimension 2 − d/2.\nIn this scaling, the gradient term is only important at long distances for d ≤ 4.\nAbove four dimensions, at long wavelengths, the overall magnetization is only affected by the ultralocal terms.\nThere is one subtle point.\nThe field H is fluctuating statistically, and the fluctuations can shift the zero point of t.\nTo see how, consider H4 split in the following way:\nThe first term is a constant contribution to the free energy, and can be ignored.\nThe second term is a [finite shift](https://bliptext.com/articles/subshift-of-finite-type) in t.\nThe third term is a quantity that scales to zero at long distances.\nThis means that when analyzing the scaling of t by dimensional analysis, it is the shifted t that is important.\nThis was historically very confusing, because the shift in t at any finite λ is finite, but near the transition t is very small.\nThe fractional change in t is very large, and in units where t is fixed the shift looks infinite.\nThe magnetization is at the minimum of the free energy, and this is an analytic equation.\nIn terms of the shifted t,\nFor t < 0, the minima are at H proportional to the square root of t.\nSo Landau's [catastrophe](https://bliptext.com/articles/catastrophe-theory) argument is correct in dimensions larger than 5.\nThe magnetization exponent in dimensions higher than 5 is equal to the mean-field value.\nWhen t is negative, the fluctuations about the new minimum are described by a new positive quadratic coefficient.\nSince this term always dominates, at temperatures below the transition the fluctuations again become ultralocal at long distances.\n\n### Fluctuations\nTo find the behavior of fluctuations, rescale the field to fix the gradient term.\nThen the length scaling dimension of the field is 1 − d/2.\nNow the field has constant quadratic spatial fluctuations at all temperatures.\nThe scale dimension of the H2 term is 2, while the scale dimension of the H4 term is 4 − d.\nFor d < 4, the H4 term has positive scale dimension.\nIn dimensions higher than 4 it has negative scale dimensions.\nThis is an essential difference.\nIn dimensions higher than 4, fixing the scale of the gradient term means that the coefficient of the H4 term is less and less important at longer and longer wavelengths.\nThe dimension at which nonquadratic contributions begin to contribute is known as the critical dimension.\nIn the Ising model, the critical dimension is 4.\nIn dimensions above 4, the critical fluctuations are described by a purely quadratic free energy at long wavelengths.\nThis means that the correlation functions are all computable from as [Gaussian](https://bliptext.com/articles/gaussian-distribution) averages:\nvalid when x − y is large.\nThe function G(x − y) is the analytic continuation to imaginary time of the [Feynman propagator](https://bliptext.com/articles/propagator), since the free energy is the analytic continuation of the quantum field action for a free scalar field.\nFor dimensions 5 and higher, all the other correlation functions at long distances are then determined by [Wick's theorem](https://bliptext.com/articles/s-matrix).\nAll the odd moments are zero, by ± symmetry.\nThe even moments are the sum over all partition into pairs of the product of G(x − y) for each pair.\nwhere C is the proportionality constant.\nSo knowing G is enough.\nIt determines all the multipoint correlations of the field.\n\n### The critical two-point function\nTo determine the form of G, consider that the fields in a path integral obey the classical equations of motion derived by varying the free energy:\nThis is valid at noncoincident points only, since the correlations of H are singular when points collide.\nH obeys classical equations of motion for the same reason that quantum mechanical operators obey them—its fluctuations are defined by a path integral.\nAt the critical point t = 0, this is [Laplace's equation](https://bliptext.com/articles/laplace-s-equation), which can be solved by [Gauss's method](https://bliptext.com/articles/gaussian-surface) from electrostatics.\nDefine an electric field analog by\nAway from the origin:\nsince G is spherically symmetric in d dimensions, and E is the radial gradient of G.\nIntegrating over a large d − 1 dimensional sphere,\nThis gives:\nand G can be found by integrating with respect to r.\nThe constant C fixes the overall normalization of the field.\n\n### G(r) away from the critical point\nWhen t does not equal zero, so that H is fluctuating at a temperature slightly away from critical, the two point function decays at long distances.\nThe equation it obeys is altered:\nFor r small compared with \\sqrt{t}, the solution diverges exactly the same way as in the critical case, but the long distance behavior is modified.\nTo see how, it is convenient to represent the two point function as an integral, introduced by Schwinger in the quantum field theory context:\nThis is G, since the Fourier transform of this integral is easy.\nEach fixed τ contribution is a Gaussian in x, whose Fourier transform is another Gaussian of reciprocal width in k.\nThis is the inverse of the operator ∇2 − t in k-space, acting on the unit function in k-space, which is the Fourier transform of a delta function source localized at the origin.\nSo it satisfies the same equation as G with the same boundary conditions that determine the strength of the divergence at 0.\nThe interpretation of the integral representation over the proper time τ is that the two point function is the sum over all random walk paths that link position 0 to position x over time τ.\nThe density of these paths at time τ at position x is Gaussian, but the random walkers disappear at a steady rate proportional to t so that the Gaussian at time τ is diminished in height by a factor that decreases steadily exponentially.\nIn the quantum field theory context, these are the paths of relativistically localized quanta in a formalism that follows the paths of individual particles.\nIn the pure statistical context, these paths still appear by the mathematical correspondence with quantum fields, but their interpretation is less directly physical.\nThe integral representation immediately shows that G(r) is positive, since it is represented as a weighted sum of positive Gaussians.\nIt also gives the rate of decay at large r, since the proper time for a random walk to reach position τ is r2 and in this time, the Gaussian height has decayed by.\nThe decay factor appropriate for position r is therefore.\nA heuristic approximation for G(r) is:\nThis is not an exact form, except in three dimensions, where interactions between paths become important.\nThe exact forms in high dimensions are variants of [Bessel functions](https://bliptext.com/articles/bessel-functions).\n\n### Symanzik polymer interpretation\nThe interpretation of the correlations as fixed size quanta travelling along random walks gives a way of understanding why the critical dimension of the H4 interaction is 4.\nThe term H4 can be thought of as the square of the density of the random walkers at any point.\nIn order for such a term to alter the finite order correlation functions, which only introduce a few new random walks into the fluctuating environment, the new paths must intersect.\nOtherwise, the square of the density is just proportional to the density and only shifts the H2 coefficient by a constant.\nBut the intersection probability of random walks depends on the dimension, and random walks in dimension higher than 4 do not intersect.\nThe [fractal dimension](https://bliptext.com/articles/fractal-dimension) of an ordinary random walk is 2.\nThe number of balls of size ε required to cover the path increase as ε−2.\nTwo objects of fractal dimension 2 will intersect with reasonable probability only in a space of dimension 4 or less, the same condition as for a generic pair of planes.\n[Kurt Symanzik](https://bliptext.com/articles/kurt-symanzik) argued that this implies that the critical Ising fluctuations in dimensions higher than 4 should be described by a free field.\nThis argument eventually became a mathematical proof.\n\n### 4 − ε dimensions – renormalization group\nThe Ising model in four dimensions is described by a fluctuating field, but now the fluctuations are interacting.\nIn the polymer representation, intersections of random walks are marginally possible.\nIn the quantum field continuation, the quanta interact.\nThe negative logarithm of the probability of any field configuration H is the [free energy](https://bliptext.com/articles/thermodynamic-free-energy) function\nThe numerical factors are there to simplify the equations of motion.\nThe goal is to understand the statistical fluctuations.\nLike any other non-quadratic path integral, the correlation functions have a [Feynman expansion](https://bliptext.com/articles/feynman-diagram) as particles travelling along random walks, splitting and rejoining at vertices.\nThe interaction strength is parametrized by the classically dimensionless quantity λ.\nAlthough dimensional analysis shows that both λ and Z are dimensionless, this is misleading.\nThe long wavelength statistical fluctuations are not exactly scale invariant, and only become scale invariant when the interaction strength vanishes.\nThe reason is that there is a cutoff used to define H, and the cutoff defines the shortest wavelength.\nFluctuations of H at wavelengths near the cutoff can affect the longer-wavelength fluctuations.\nIf the system is scaled along with the cutoff, the parameters will scale by dimensional analysis, but then comparing parameters doesn't compare behavior because the rescaled system has more modes.\nIf the system is rescaled in such a way that the short wavelength cutoff remains fixed, the long-wavelength fluctuations are modified.\n\n#### Wilson renormalization\nA quick heuristic way of studying the scaling is to cut off the H wavenumbers at a point λ.\nFourier modes of H with wavenumbers larger than λ are not allowed to fluctuate.\nA rescaling of length that make the whole system smaller increases all wavenumbers, and moves some fluctuations above the cutoff.\nTo restore the old cutoff, perform a partial integration over all the wavenumbers which used to be forbidden, but are now fluctuating.\nIn Feynman diagrams, integrating over a fluctuating mode at wavenumber k links up lines carrying momentum k in a correlation function in pairs, with a factor of the inverse propagator.\nUnder rescaling, when the system is shrunk by a factor of (1+b), the t coefficient scales up by a factor (1+b)2 by dimensional analysis.\nThe change in t for infinitesimal b is 2bt.\nThe other two coefficients are dimensionless and do not change at all.\nThe lowest order effect of integrating out can be calculated from the equations of motion:\nThis equation is an identity inside any correlation function away from other insertions.\nAfter integrating out the modes with Λ < k < (1+b)Λ, it will be a slightly different identity.\nSince the form of the equation will be preserved, to find the change in coefficients it is sufficient to analyze the change in the H3 term.\nIn a Feynman diagram expansion, the H3 term in a correlation function inside a correlation has three dangling lines.\nJoining two of them at large wavenumber k gives a change H3 with one dangling line, so proportional to H:\nThe factor of 3 comes from the fact that the loop can be closed in three different ways.\nThe integral should be split into two parts:\nThe first part is not proportional to t, and in the equation of motion it can be absorbed by a constant shift in t.\nIt is caused by the fact that the H3 term has a linear part.\nOnly the second term, which varies from t to t, contributes to the critical scaling.\nThis new linear term adds to the first term on the left hand side, changing t by an amount proportional to t.\nThe total change in t is the sum of the term from dimensional analysis and this second term from [operator products](https://bliptext.com/articles/operator-product-expansion):\nSo t is rescaled, but its dimension is [anomalous](https://bliptext.com/articles/anomalous-dimension), it is changed by an amount proportional to the value of λ.\nBut λ also changes.\nThe change in λ requires considering the lines splitting and then quickly rejoining.\nThe lowest order process is one where one of the three lines from H3 splits into three, which quickly joins with one of the other lines from the same vertex.\nThe correction to the vertex is\nThe numerical factor is three times bigger because there is an extra factor of three in choosing which of the three new lines to contract.\nSo\nThese two equations together define the renormalization group equations in four dimensions:\nThe coefficient B is determined by the formula\nand is proportional to the area of a three-dimensional sphere of radius λ, times the width of the integration region bΛ divided by Λ4:\nIn other dimensions, the constant B changes, but the same constant appears both in the t flow and in the coupling flow.\nThe reason is that the derivative with respect to t of the closed loop with a single vertex is a closed loop with two vertices.\nThis means that the only difference between the scaling of the coupling and the t is the combinatorial factors from joining and splitting.\n\n#### Wilson–Fisher fixed point\nTo investigate three dimensions starting from the four-dimensional theory should be possible, because the intersection probabilities of random walks depend continuously on the dimensionality of the space.\nIn the language of Feynman graphs, the coupling does not change very much when the dimension is changed.\nThe process of continuing away from dimension 4 is not completely well defined without a prescription for how to do it.\nThe prescription is only well defined on diagrams.\nIt replaces the Schwinger representation in dimension 4 with the Schwinger representation in dimension 4 − ε defined by:\nIn dimension 4 − ε, the coupling λ has positive scale dimension ε, and this must be added to the flow.\nThe coefficient B is dimension dependent, but it will cancel.\nThe fixed point for λ is no longer zero, but at:\nwhere the scale dimensions of t is altered by an amount λB = ε/3.\nThe magnetization exponent is altered proportionately to:\nwhich is .333 in 3 dimensions (ε = 1) and .166 in 2 dimensions (ε = 2).\nThis is not so far off from the measured exponent .308 and the Onsager two dimensional exponent .125.\n\n### Infinite dimensions – mean field\nThe behavior of an Ising model on a fully connected graph may be completely understood by [mean-field theory](https://bliptext.com/articles/mean-field-theory).\nThis type of description is appropriate to very-high-dimensional square lattices, because then each site has a very large number of neighbors.\nThe idea is that if each spin is connected to a large number of spins, only the average ratio of + spins to − spins is important, since the fluctuations about this mean will be small.\nThe [mean field](https://bliptext.com/articles/mean-field) H is the average fraction of spins which are + minus the average fraction of spins which are −.\nThe energy cost of flipping a single spin in the mean field H is ±2JNH.\nIt is convenient to redefine J to absorb the factor N, so that the limit N → ∞ is smooth.\nIn terms of the new J, the energy cost for flipping a spin is ±2JH.\nThis energy cost gives the ratio of probability p that the spin is + to the probability 1−p that the spin is −.\nThis ratio is the Boltzmann factor:\nso that\nThe mean value of the spin is given by averaging 1 and −1 with the weights p and 1 − p, so the mean value is 2p − 1.\nBut this average is the same for all spins, and is therefore equal to H.\nThe solutions to this equation are the possible consistent mean fields.\nFor βJ < 1 there is only the one solution at H = 0.\nFor bigger values of β there are three solutions, and the solution at H = 0 is unstable.\nThe instability means that increasing the mean field above zero a little bit produces a statistical fraction of spins which are + which is bigger than the value of the mean field.\nSo a mean field which fluctuates above zero will produce an even greater mean field, and will eventually settle at the stable solution.\nThis means that for temperatures below the critical value βJ = 1 the mean-field Ising model undergoes a phase transition in the limit of large N.\nAbove the critical temperature, fluctuations in H are damped because the mean field restores the fluctuation to zero field.\nBelow the critical temperature, the mean field is driven to a new equilibrium value, which is either the positive H or negative H solution to the equation.\nFor βJ = 1 + ε, just below the critical temperature, the value of H can be calculated from the Taylor expansion of the hyperbolic tangent:\nDividing by H to discard the unstable solution at H = 0, the stable solutions are:\nThe spontaneous magnetization H grows near the critical point as the square root of the change in temperature.\nThis is true whenever H can be calculated from the solution of an analytic equation which is symmetric between positive and negative values, which led [Landau](https://bliptext.com/articles/lev-landau) to suspect that all Ising type phase transitions in all dimensions should follow this law.\nThe mean-field exponent is [universal](https://bliptext.com/articles/universality-dynamical-systems) because changes in the character of solutions of analytic equations are always described by [catastrophes](https://bliptext.com/articles/catastrophe-theory) in the [Taylor series](https://bliptext.com/articles/taylor-series), which is a polynomial equation.\nBy symmetry, the equation for H must only have odd powers of H on the right hand side.\nChanging β should only smoothly change the coefficients.\nThe transition happens when the coefficient of H on the right hand side is 1.\nNear the transition:\nWhatever A and B are, so long as neither of them is tuned to zero, the spontaneous magnetization will grow as the square root of ε.\nThis argument can only fail if the free energy βF is either non-analytic or non-generic at the exact β where the transition occurs.\nBut the spontaneous magnetization in magnetic systems and the density in gasses near the critical point are measured very accurately.\nThe density and the magnetization in three dimensions have the same power-law dependence on the temperature near the critical point, but the behavior from experiments is:\nThe exponent is also universal, since it is the same in the Ising model as in the experimental magnet and gas, but it is not equal to the mean-field value.\nThis was a great surprise.\nThis is also true in two dimensions, where\nBut there it was not a surprise, because it was predicted by [Onsager](https://bliptext.com/articles/lars-onsager).\n\n### Low dimensions – block spins\nIn three dimensions, the perturbative series from the field theory is an expansion in a coupling constant λ which is not particularly small.\nThe effective size of the coupling at the fixed point is one over the branching factor of the particle paths, so the expansion parameter is about 1/3.\nIn two dimensions, the perturbative expansion parameter is 2/3.\nBut renormalization can also be productively applied to the spins directly, without passing to an average field.\nHistorically, this approach is due to [Leo Kadanoff](https://bliptext.com/articles/leo-kadanoff) and predated the perturbative ε expansion.\nThe idea is to integrate out lattice spins iteratively, generating a flow in couplings.\nBut now the couplings are lattice energy coefficients.\nThe fact that a continuum description exists guarantees that this iteration will converge to a fixed point when the temperature is tuned to criticality.\n\n#### Migdal–Kadanoff renormalization\nWrite the two-dimensional Ising model with an infinite number of possible higher order interactions.\nTo keep spin reflection symmetry, only even powers contribute:\nBy translation invariance, Jij is only a function of i-j.\nBy the accidental rotational symmetry, at large i and j its size only depends on the magnitude of the two-dimensional vector i − j.\nThe higher order coefficients are also similarly restricted.\nThe renormalization iteration divides the lattice into two parts – even spins and odd spins.\nThe odd spins live on the odd-checkerboard lattice positions, and the even ones on the even-checkerboard.\nWhen the spins are indexed by the position (i,j), the odd sites are those with i + j odd and the even sites those with i + j even, and even sites are only connected to odd sites.\nThe two possible values of the odd spins will be integrated out, by summing over both possible values.\nThis will produce a new free energy function for the remaining even spins, with new adjusted couplings.\nThe even spins are again in a lattice, with axes tilted at 45 degrees to the old ones.\nUnrotating the system restores the old configuration, but with new parameters.\nThese parameters describe the interaction between spins at distances \\sqrt{2} larger.\nStarting from the Ising model and repeating this iteration eventually changes all the couplings.\nWhen the temperature is higher than the critical temperature, the couplings will converge to zero, since the spins at large distances are uncorrelated.\nBut when the temperature is critical, there will be nonzero coefficients linking spins at all orders.\nThe flow can be approximated by only considering the first few terms.\nThis truncated flow will produce better and better approximations to the critical exponents when more terms are included.\nThe simplest approximation is to keep only the usual J term, and discard everything else.\nThis will generate a flow in J, analogous to the flow in t at the fixed point of λ in the ε expansion.\nTo find the change in J, consider the four neighbors of an odd site.\nThese are the only spins which interact with it.\nThe multiplicative contribution to the partition function from the sum over the two values of the spin at the odd site is:\nwhere N± is the number of neighbors which are ±.\nIgnoring the factor of 2, the free energy contribution from this odd site is:\nThis includes nearest neighbor and next-nearest neighbor interactions, as expected, but also a four-spin interaction which is to be discarded.\nTo truncate to nearest neighbor interactions, consider that the difference in energy between all spins the same and equal numbers + and – is:\nFrom nearest neighbor couplings, the difference in energy between all spins equal and staggered spins is 8J.\nThe difference in energy between all spins equal and nonstaggered but net zero spin is 4J.\nIgnoring four-spin interactions, a reasonable truncation is the average of these two energies or 6J.\nSince each link will contribute to two odd spins, the right value to compare with the previous one is half that:\nFor small J, this quickly flows to zero coupling.\nLarge J's flow to large couplings.\nThe magnetization exponent is determined from the slope of the equation at the fixed point.\nVariants of this method produce good numerical approximations for the critical exponents when many terms are included, in both two and three dimensions.\n\n# Footnotes","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:03:11.918Z","updated_at":"2025-01-25T19:03:11.918Z","search_vector":"'ise':1 'model':2","search_content":" \n \n\n# \nThe **Ising model** (or Lenz–**Ising model**), named after the physicists [Ernst Ising](https://bliptext.com/articles/ernst-ising) and [Wilhelm Lenz](https://bliptext.com/articles/wilhelm-lenz), is a [mathematical model](https://bliptext.com/articles/mathematical-models-in-physics) of [ferromagnetism](https://bliptext.com/articles/ferromagnetism) in [statistical mechanics](https://bliptext.com/articles/statistical-mechanics).\nThe model consists of [discrete variables](https://bliptext.com/articles/discrete-variables) that represent [magnetic dipole moments of atomic \"spins\"](https://bliptext.com/articles/nuclear-magnetic-moment) that can be in one of two states (+1 or −1).\nThe spins are arranged in a graph, usually a [lattice](https://bliptext.com/articles/lattice-group) (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors.\nNeighboring 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.\nThe model allows the identification of [phase transitions](https://bliptext.com/articles/phase-transition) as a simplified model of reality.\nThe two-dimensional [square-lattice Ising model](https://bliptext.com/articles/square-lattice-ising-model) is one of the simplest statistical models to show a [phase transition](https://bliptext.com/articles/phase-transition).\nThe Ising model was invented by the physicist, who gave it as a problem to his student Ernst Ising.\nThe one-dimensional Ising model was solved by alone in his 1924 thesis; it has no phase transition.\nThe two-dimensional square-lattice Ising model is much harder and was only given an analytic description much later, by.\nIt is usually solved by a [transfer-matrix method](https://bliptext.com/articles/transfer-matrix-method-statistical-mechanics), although there exist different approaches, more related to [quantum field theory](https://bliptext.com/articles/quantum-field-theory).\nIn dimensions greater than four, the phase transition of the Ising model is described by [mean-field theory](https://bliptext.com/articles/mean-field-theory).\nThe 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.\nThe 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.\nThe Ising problem without an external field can be equivalently formulated as a [graph](https://bliptext.com/articles/graph-discrete-mathematics) [maximum cut](https://bliptext.com/articles/maximum-cut) (Max-Cut) problem that can be solved via [combinatorial optimization](https://bliptext.com/articles/combinatorial-optimization).\n\n# Definition\nConsider a set \\Lambda of lattice sites, each with a set of adjacent sites (e.g. a [graph](https://bliptext.com/articles/graph-discrete-mathematics)) forming a d-dimensional lattice.\nFor each lattice site k\\in\\Lambda there is a discrete variable \\sigma_k such that, representing the site's spin.\nA spin configuration, is an assignment of spin value to each lattice site.\nFor any two adjacent sites there is an interaction J_{ij}.\nAlso a site j\\in\\Lambda has an external magnetic field h_j interacting with it.\nThe energy of a configuration {\\sigma} is given by the [Hamiltonian function](https://bliptext.com/articles/hamiltonian-function)\nwhere the first sum is over pairs of adjacent spins (every pair is counted once).\nThe notation indicates that sites i and j are nearest neighbors.\nThe [magnetic moment](https://bliptext.com/articles/magnetic-moment) is given by \\mu.\nNote 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.\nThe configuration probability is given by the [Boltzmann distribution](https://bliptext.com/articles/boltzmann-distribution) with [inverse temperature](https://bliptext.com/articles/inverse-temperature) \\beta\\geq0:\nwhere, and the normalization constant\nis the [partition function](https://bliptext.com/articles/partition-function-statistical-mechanics).\nFor a function f of the spins (\"observable\"), one denotes by\nthe expectation (mean) value of f.\nThe configuration probabilities represent the probability that (in equilibrium) the system is in a state with configuration \\sigma.\n\n## Discussion\nThe minus sign on each term of the Hamiltonian function H(\\sigma) is conventional.\nUsing this sign convention, Ising models can be classified according to the sign of the interaction: if, for a pair i, j\nJ_{ij} > 0, the interaction is called ferromagnetic,\nJ_{ij} 0, the spin site j desires to line up in the positive direction,\nh_j Hμ, then A(ν, μ) > A(μ, ν).\nMetropolis sets the larger of A(μ, ν) or A(ν, μ) to be 1.\nBy this reasoning the acceptance algorithm is:\nThe basic form of the algorithm is as follows:\nThe change in energy Hν − Hμ only depends on the value of the spin and its nearest graph neighbors.\nSo if the graph is not too connected, the algorithm is fast.\nThis process will eventually produce a pick from the distribution.\n\n## As a Markov chain\nIt is possible to view the Ising model as a [Markov chain](https://bliptext.com/articles/markov-chain), as the immediate probability Pβ(ν) of transitioning to a future state ν only depends on the present state μ.\nThe Metropolis algorithm is actually a version of a [Markov chain Monte Carlo](https://bliptext.com/articles/markov-chain-monte-carlo) simulation, and since we use single-spin-flip dynamics in the Metropolis algorithm, every state can be viewed as having links to exactly L other states, where each transition corresponds to flipping a single spin site to the opposite value.\nFurthermore, since the energy equation Hσ change only depends on the nearest-neighbor interaction strength J, the Ising model and its variants such the [Sznajd model](https://bliptext.com/articles/sznajd-model) can be seen as a form of a [voter model](https://bliptext.com/articles/contact-process-mathematics) for opinion dynamics.\n\n# Solutions\n\n## One dimension\nThe thermodynamic limit exists as long as the interaction decay is with α > 1.\n\n### Ising's exact solution\nIn the nearest neighbor case (with periodic or free boundary conditions) an exact solution is available.\nThe Hamiltonian of the one-dimensional Ising model on a lattice of L sites with free boundary conditions is\nwhere J and h can be any number, since in this simplified case J is a constant representing the interaction strength between the nearest neighbors and h is the constant external magnetic field applied to lattice sites.\nThen the\n[free energy](https://bliptext.com/articles/thermodynamic-free-energy) is\nand the spin-spin correlation (i.e. the covariance) is\nwhere C(β) and c(β) are positive functions for T > 0.\nFor T → 0, though, the inverse correlation length c(β) vanishes.\n\n#### Proof\nThe proof of this result is a simple computation.\nIf h = 0, it is very easy to obtain the free energy in the case of free boundary condition, i.e. when\nThen the model factorizes under the change of variables\nThis gives\nTherefore, the free energy is\nWith the same change of variables\nhence it decays exponentially as soon as T ≠ 0; but for T = 0, i.e. in the limit β → ∞ there is no decay.\nIf h ≠ 0 we need the transfer matrix method.\nFor the periodic boundary conditions case is the following.\nThe partition function is\nThe coefficients can be seen as the entries of a matrix.\nThere are different possible choices: a convenient one (because the matrix is symmetric) is\nor\nIn matrix formalism\nwhere λ1 is the highest eigenvalue of V, while λ2 is the other eigenvalue:\nand |λ2| < λ1.\nThis gives the formula of the free energy.\n\n#### Comments\nThe energy of the lowest state is −JL, when all the spins are the same.\nFor any other configuration, the extra energy is equal to 2J times the number of sign changes that are encountered when scanning the configuration from left to right.\nIf we designate the number of sign changes in a configuration as k, the difference in energy from the lowest energy state is 2k.\nSince the energy is additive in the number of flips, the probability p of having a spin-flip at each position is independent.\nThe ratio of the probability of finding a flip to the probability of not finding one is the Boltzmann factor:\nThe problem is reduced to independent biased [coin tosses](https://bliptext.com/articles/coin-toss).\nThis essentially completes the mathematical description.\nFrom the description in terms of independent tosses, the statistics of the model for long lines can be understood.\nThe line splits into domains.\nEach domain is of average length exp(2β).\nThe length of a domain is distributed exponentially, since there is a constant probability at any step of encountering a flip.\nThe domains never become infinite, so a long system is never magnetized.\nEach step reduces the correlation between a spin and its neighbor by an amount proportional to p, so the correlations fall off exponentially.\nThe [partition function](https://bliptext.com/articles/partition-function-statistical-mechanics) is the volume of configurations, each configuration weighted by its Boltzmann weight.\nSince each configuration is described by the sign-changes, the partition function factorizes:\nThe logarithm divided by L is the free energy density:\nwhich is [analytic](https://bliptext.com/articles/analytic-function) away from β = ∞.\nA sign of a [phase transition](https://bliptext.com/articles/phase-transition) is a non-analytic free energy, so the one-dimensional model does not have a [phase transition](https://bliptext.com/articles/phase-transition).\n\n### One-dimensional solution with transverse field\nTo express the Ising Hamiltonian using a quantum mechanical description of spins, we replace the spin variables with their respective [Pauli matrices](https://bliptext.com/articles/pauli-matrices).\nHowever, depending on the direction of the magnetic field, we can create a transverse-field or longitudinal-field Hamiltonian.\nThe [transverse-field](https://bliptext.com/articles/transverse-field-ising-model) Hamiltonian is given by\nThe transverse-field model experiences a phase transition between an ordered and disordered regime at J ~ h.\nThis can be shown by a mapping of Pauli matrices\nUpon rewriting the Hamiltonian in terms of this change-of-basis matrices, we obtain\nSince the roles of h and J are switched, the Hamiltonian undergoes a transition at J = h.\n\n### Renormalization\nWhen there is no external field, we can derive a functional equation that satisfies using renormalization.\nSpecifically, let be the partition function with N sites.\nNow we have:where.\nWe sum over each of, to obtainNow, since the cosh function is even, we can solve as.\nNow we have a self-similarity relation:Taking the limit, we obtainwhere.\nWhen \\beta is small, we have, so we can numerically evaluate f(\\beta) by iterating the functional equation until K is small.\n\n## Two dimensions\n\n### Onsager's exact solution\nobtained the following analytical expression for the free energy of the Ising model on the anisotropic square lattice when the magnetic field h=0 in the thermodynamic limit as a function of temperature and the horizontal and vertical interaction energies J_1 and J_2, respectively\nFrom this expression for the free energy, all thermodynamic functions of the model can be calculated by using an appropriate derivative.\nThe 2D Ising model was the first model to exhibit a continuous phase transition at a positive temperature.\nIt occurs at the temperature T_c which solves the equation\nIn the isotropic case when the horizontal and vertical interaction energies are equal J_1=J_2=J, the critical temperature T_c occurs at the following point\nWhen the interaction energies J_1, J_2 are both negative, the Ising model becomes an antiferromagnet.\nSince the square lattice is bi-partite, it is invariant under this change when the magnetic field h=0, so the free energy and critical temperature are the same for the antiferromagnetic case.\nFor the triangular lattice, which is not bi-partite, the ferromagnetic and antiferromagnetic Ising model behave notably differently.\nSpecifically, around a triangle, it is impossible to make all 3 spin-pairs antiparallel, so the antiferromagnetic Ising model cannot reach the minimal energy state.\nThis is an example of [geometric frustration](https://bliptext.com/articles/geometric-frustration).\n\n#### Transfer matrix\nStart with an analogy with quantum mechanics.\nThe Ising model on a long periodic lattice has a partition function\nThink of the i direction as space, and the j direction as time.\nThis is an independent sum over all the values that the spins can take at each time slice.\nThis is a type of [path integral](https://bliptext.com/articles/path-integral-formulation), it is the sum over all spin histories.\nA path integral can be rewritten as a Hamiltonian evolution.\nThe Hamiltonian steps through time by performing a unitary rotation between time t and time t + Δt:\nThe product of the U matrices, one after the other, is the total time evolution operator, which is the path integral we started with.\nwhere N is the number of time slices.\nThe sum over all paths is given by a product of matrices, each matrix element is the transition probability from one slice to the next.\nSimilarly, one can divide the sum over all partition function configurations into slices, where each slice is the one-dimensional configuration at time 1.\nThis defines the [transfer matrix](https://bliptext.com/articles/transfer-matrix-method-statistical-mechanics):\nThe configuration in each slice is a one-dimensional collection of spins.\nAt each time slice, T has matrix elements between two configurations of spins, one in the immediate future and one in the immediate past.\nThese two configurations are C1 and C2, and they are all one-dimensional spin configurations.\nWe can think of the vector space that T acts on as all complex linear combinations of these.\nUsing quantum mechanical notation:\nwhere each basis vector |S\\rangle is a spin configuration of a one-dimensional Ising model.\nLike the Hamiltonian, the transfer matrix acts on all linear combinations of states.\nThe partition function is a matrix function of T, which is defined by the [sum](https://bliptext.com/articles/trace-linear-algebra) over all histories which come back to the original configuration after N steps:\nSince this is a matrix equation, it can be evaluated in any basis.\nSo if we can diagonalize the matrix T, we can find Z.\n\n#### T in terms of Pauli matrices\nThe contribution to the partition function for each past/future pair of configurations on a slice is the sum of two terms.\nThere is the number of spin flips in the past slice and there is the number of spin flips between the past and future slice.\nDefine an operator on configurations which flips the spin at site i:\n\\sigma^x_i.\nIn the usual Ising basis, acting on any linear combination of past configurations, it produces the same linear combination but with the spin at position i of each basis vector flipped.\nDefine a second operator which multiplies the basis vector by +1 and −1 according to the spin at position i:\n\\sigma^z_i.\nT can be written in terms of these:\nwhere A and B are constants which are to be determined so as to reproduce the partition function.\nThe interpretation is that the statistical configuration at this slice contributes according to both the number of spin flips in the slice, and whether or not the spin at position i has flipped.\n\n#### Spin flip creation and annihilation operators\nJust as in the one-dimensional case, we will shift attention from the spins to the spin-flips.\nThe σz term in T counts the number of spin flips, which we can write in terms of spin-flip creation and annihilation operators:\nThe first term flips a spin, so depending on the basis state it either:\nWriting this out in terms of creation and annihilation operators:\nIgnore the constant coefficients, and focus attention on the form.\nThey are all quadratic.\nSince the coefficients are constant, this means that the T matrix can be diagonalized by Fourier transforms.\nCarrying out the diagonalization produces the Onsager free energy.\n\n#### Onsager's formula for spontaneous magnetization\nOnsager famously announced the following expression for the [spontaneous magnetization](https://bliptext.com/articles/spontaneous-magnetization) M of a two-dimensional Ising ferromagnet on the square lattice at two different conferences in 1948, though without proof\nwhere J_1 and J_2 are horizontal and vertical interaction energies.\nA complete derivation was only given in 1951 by using a limiting process of transfer matrix eigenvalues.\nThe proof was subsequently greatly simplified in 1963 by Montroll, Potts, and Ward using [Szegő](https://bliptext.com/articles/g-bor-szeg)'s [limit formula](https://bliptext.com/articles/szeg-limit-theorems) for [Toeplitz determinants](https://bliptext.com/articles/toeplitz-determinant) by treating the magnetization as the limit of correlation functions.\n\n### Minimal model\nAt the critical point, the two-dimensional Ising model is a [two-dimensional conformal field theory](https://bliptext.com/articles/two-dimensional-conformal-field-theory).\nThe spin and energy correlation functions are described by a [minimal model](https://bliptext.com/articles/minimal-model-physics), which has been exactly solved.\n\n## Three dimensions\nIn three as in two dimensions, the most studied case of the Ising model is the translation-invariant model on a cubic lattice with nearest-neighbor coupling in the zero magnetic field.\nMany** **theoreticians** **searched** **for** **an** **analytical** **three-dimensional** **solution** **for** **many** **decades,** **which** **would** **be** **analogous** **to** **Onsager's** **solution** **in** **the** **two-dimensional** **case.** **** **Such** **a** **solution** **has** **not** **been** **found** **until** **now,** **although** **there** **is** **no** **proof** **that** **it** **may** **not** **exist.\nIn three dimensions, the Ising model was shown to have a representation in terms of non-interacting fermionic strings by [Alexander Polyakov](https://bliptext.com/articles/alexander-markovich-polyakov) and [Vladimir Dotsenko](https://bliptext.com/articles/vladimir-dotsenko).\nThis construction has been carried on the lattice, and the [continuum limit](https://bliptext.com/articles/continuum-limit), conjecturally describing the critical point, is unknown.\nIn three as in two dimensions, Peierls' argument shows that there is a phase transition.\nThis phase transition is rigorously known to be continuous (in the sense that correlation length diverges and the magnetization goes to zero), and is called the [critical point](https://bliptext.com/articles/critical-point-thermodynamics).\nIt is believed that the critical point can be described by a renormalization group fixed point of the Wilson-Kadanoff renormalization group transformation.\nIt is also believed that the phase transition can be described by a three-dimensional unitary conformal field theory, as evidenced by [Monte Carlo](https://bliptext.com/articles/metropolis-hastings-algorithm) simulations, exact diagonalization results in quantum models, and quantum field theoretical arguments.\nAlthough it is an open problem to establish rigorously the renormalization group picture or the conformal field theory picture, theoretical physicists have used these two methods to compute the [critical exponents](https://bliptext.com/articles/critical-exponents) of the phase transition, which agree with the experiments and with the Monte Carlo simulations.\nThis conformal field theory describing the three-dimensional Ising critical point is under active investigation using the method of the [conformal bootstrap](https://bliptext.com/articles/conformal-bootstrap).\nThis method currently yields the most precise information about the structure of the critical theory (see [Ising critical exponents](https://bliptext.com/articles/ising-critical-exponents)).\nIn 2000, [Sorin Istrail](https://bliptext.com/articles/sorin-istrail) of [Sandia National Laboratories](https://bliptext.com/articles/sandia-national-laboratories) proved that the spin glass Ising model on a [nonplanar](https://bliptext.com/articles/nonplanar) lattice is [NP-complete](https://bliptext.com/articles/np-completeness).\nThat is, assuming **P** ≠ N**P**, the general spin glass Ising model is exactly solvable only in [planar](https://bliptext.com/articles/planar-graph) cases, so solutions for dimensions higher than two are also intractable.\nIstrail's result only concerns the spin glass model with spatially varying couplings, and tells nothing about Ising's original ferromagnetic model with equal couplings.\n\n## Four dimensions and above\nIn any dimension, the Ising model can be productively described by a locally varying mean field.\nThe field is defined as the average spin value over a large region, but not so large so as to include the entire system.\nThe field still has slow variations from point to point, as the averaging volume moves.\nThese fluctuations in the field are described by a continuum field theory in the infinite system limit.\n\n### Local field\nThe field H is defined as the long wavelength Fourier components of the spin variable, in the limit that the wavelengths are long.\nThere are many ways to take the long wavelength average, depending on the details of how high wavelengths are cut off.\nThe details are not too important, since the goal is to find the statistics of H and not the spins.\nOnce the correlations in H are known, the long-distance correlations between the spins will be proportional to the long-distance correlations in H.\nFor any value of the slowly varying field H, the free energy (log-probability) is a local analytic function of H and its gradients.\nThe free energy F(H) is defined to be the sum over all Ising configurations which are consistent with the long wavelength field.\nSince H is a coarse description, there are many Ising configurations consistent with each value of H, so long as not too much exactness is required for the match.\nSince the allowed range of values of the spin in any region only depends on the values of H within one averaging volume from that region, the free energy contribution from each region only depends on the value of H there and in the neighboring regions.\nSo F is a sum over all regions of a local contribution, which only depends on H and its derivatives.\nBy symmetry in H, only even powers contribute.\nBy reflection symmetry on a square lattice, only even powers of gradients contribute.\nWriting out the first few terms in the free energy:\nOn a square lattice, symmetries guarantee that the coefficients Zi of the derivative terms are all equal.\nBut even for an anisotropic Ising model, where the Zi's in different directions are different, the fluctuations in H are isotropic in a coordinate system where the different directions of space are rescaled.\nOn any lattice, the derivative term\nis a positive definite [quadratic form](https://bliptext.com/articles/quadratic-form), and can be used to define the metric for space.\nSo any translationally invariant Ising model is rotationally invariant at long distances, in coordinates that make Zij = δij.\nRotational symmetry emerges spontaneously at large distances just because there aren't very many low order terms.\nAt higher order multicritical points, this [accidental symmetry](https://bliptext.com/articles/accidental-symmetry) is lost.\nSince βF is a function of a slowly spatially varying field, the probability of any field configuration is (omitting higher-order terms):\nThe statistical average of any product of H terms is equal to:\nThe denominator in this expression is called the partition function:and the integral over all possible values of H is a statistical path integral.\nIt integrates exp(βF) over all values of H, over all the long wavelength fourier components of the spins.\nF is a \"Euclidean\" Lagrangian for the field H.\nIt is similar to the Lagrangian in of a scalar field in [quantum field theory](https://bliptext.com/articles/quantum-field-theory), the difference being that all the derivative terms enter with a positive sign, and there is no overall factor of i (thus \"Euclidean\").\n\n### Dimensional analysis\nThe form of F can be used to predict which terms are most important by dimensional analysis.\nDimensional analysis is not completely straightforward, because the scaling of H needs to be determined.\nIn the generic case, choosing the scaling law for H is easy, since the only term that contributes is the first one,\nThis term is the most significant, but it gives trivial behavior.\nThis form of the free energy is ultralocal, meaning that it is a sum of an independent contribution from each point.\nThis is like the spin-flips in the one-dimensional Ising model.\nEvery value of H at any point fluctuates completely independently of the value at any other point.\nThe scale of the field can be redefined to absorb the coefficient A, and then it is clear that A only determines the overall scale of fluctuations.\nThe ultralocal model describes the long wavelength high temperature behavior of the Ising model, since in this limit the fluctuation averages are independent from point to point.\nTo find the critical point, lower the temperature.\nAs the temperature goes down, the fluctuations in H go up because the fluctuations are more correlated.\nThis means that the average of a large number of spins does not become small as quickly as if they were uncorrelated, because they tend to be the same.\nThis corresponds to decreasing A in the system of units where H does not absorb A.\nThe phase transition can only happen when the subleading terms in F can contribute, but since the first term dominates at long distances, the coefficient A must be tuned to zero.\nThis is the location of the critical point:\nwhere t is a parameter which goes through zero at the transition.\nSince t is vanishing, fixing the scale of the field using this term makes the other terms blow up.\nOnce t is small, the scale of the field can either be set to fix the coefficient of the H4 term or the (∇H)2 term to 1.\n\n### Magnetization\nTo find the magnetization, fix the scaling of H so that λ is one.\nNow the field H has dimension −d/4, so that H4ddx is dimensionless, and Z has dimension 2 − d/2.\nIn this scaling, the gradient term is only important at long distances for d ≤ 4.\nAbove four dimensions, at long wavelengths, the overall magnetization is only affected by the ultralocal terms.\nThere is one subtle point.\nThe field H is fluctuating statistically, and the fluctuations can shift the zero point of t.\nTo see how, consider H4 split in the following way:\nThe first term is a constant contribution to the free energy, and can be ignored.\nThe second term is a [finite shift](https://bliptext.com/articles/subshift-of-finite-type) in t.\nThe third term is a quantity that scales to zero at long distances.\nThis means that when analyzing the scaling of t by dimensional analysis, it is the shifted t that is important.\nThis was historically very confusing, because the shift in t at any finite λ is finite, but near the transition t is very small.\nThe fractional change in t is very large, and in units where t is fixed the shift looks infinite.\nThe magnetization is at the minimum of the free energy, and this is an analytic equation.\nIn terms of the shifted t,\nFor t < 0, the minima are at H proportional to the square root of t.\nSo Landau's [catastrophe](https://bliptext.com/articles/catastrophe-theory) argument is correct in dimensions larger than 5.\nThe magnetization exponent in dimensions higher than 5 is equal to the mean-field value.\nWhen t is negative, the fluctuations about the new minimum are described by a new positive quadratic coefficient.\nSince this term always dominates, at temperatures below the transition the fluctuations again become ultralocal at long distances.\n\n### Fluctuations\nTo find the behavior of fluctuations, rescale the field to fix the gradient term.\nThen the length scaling dimension of the field is 1 − d/2.\nNow the field has constant quadratic spatial fluctuations at all temperatures.\nThe scale dimension of the H2 term is 2, while the scale dimension of the H4 term is 4 − d.\nFor d < 4, the H4 term has positive scale dimension.\nIn dimensions higher than 4 it has negative scale dimensions.\nThis is an essential difference.\nIn dimensions higher than 4, fixing the scale of the gradient term means that the coefficient of the H4 term is less and less important at longer and longer wavelengths.\nThe dimension at which nonquadratic contributions begin to contribute is known as the critical dimension.\nIn the Ising model, the critical dimension is 4.\nIn dimensions above 4, the critical fluctuations are described by a purely quadratic free energy at long wavelengths.\nThis means that the correlation functions are all computable from as [Gaussian](https://bliptext.com/articles/gaussian-distribution) averages:\nvalid when x − y is large.\nThe function G(x − y) is the analytic continuation to imaginary time of the [Feynman propagator](https://bliptext.com/articles/propagator), since the free energy is the analytic continuation of the quantum field action for a free scalar field.\nFor dimensions 5 and higher, all the other correlation functions at long distances are then determined by [Wick's theorem](https://bliptext.com/articles/s-matrix).\nAll the odd moments are zero, by ± symmetry.\nThe even moments are the sum over all partition into pairs of the product of G(x − y) for each pair.\nwhere C is the proportionality constant.\nSo knowing G is enough.\nIt determines all the multipoint correlations of the field.\n\n### The critical two-point function\nTo determine the form of G, consider that the fields in a path integral obey the classical equations of motion derived by varying the free energy:\nThis is valid at noncoincident points only, since the correlations of H are singular when points collide.\nH obeys classical equations of motion for the same reason that quantum mechanical operators obey them—its fluctuations are defined by a path integral.\nAt the critical point t = 0, this is [Laplace's equation](https://bliptext.com/articles/laplace-s-equation), which can be solved by [Gauss's method](https://bliptext.com/articles/gaussian-surface) from electrostatics.\nDefine an electric field analog by\nAway from the origin:\nsince G is spherically symmetric in d dimensions, and E is the radial gradient of G.\nIntegrating over a large d − 1 dimensional sphere,\nThis gives:\nand G can be found by integrating with respect to r.\nThe constant C fixes the overall normalization of the field.\n\n### G(r) away from the critical point\nWhen t does not equal zero, so that H is fluctuating at a temperature slightly away from critical, the two point function decays at long distances.\nThe equation it obeys is altered:\nFor r small compared with \\sqrt{t}, the solution diverges exactly the same way as in the critical case, but the long distance behavior is modified.\nTo see how, it is convenient to represent the two point function as an integral, introduced by Schwinger in the quantum field theory context:\nThis is G, since the Fourier transform of this integral is easy.\nEach fixed τ contribution is a Gaussian in x, whose Fourier transform is another Gaussian of reciprocal width in k.\nThis is the inverse of the operator ∇2 − t in k-space, acting on the unit function in k-space, which is the Fourier transform of a delta function source localized at the origin.\nSo it satisfies the same equation as G with the same boundary conditions that determine the strength of the divergence at 0.\nThe interpretation of the integral representation over the proper time τ is that the two point function is the sum over all random walk paths that link position 0 to position x over time τ.\nThe density of these paths at time τ at position x is Gaussian, but the random walkers disappear at a steady rate proportional to t so that the Gaussian at time τ is diminished in height by a factor that decreases steadily exponentially.\nIn the quantum field theory context, these are the paths of relativistically localized quanta in a formalism that follows the paths of individual particles.\nIn the pure statistical context, these paths still appear by the mathematical correspondence with quantum fields, but their interpretation is less directly physical.\nThe integral representation immediately shows that G(r) is positive, since it is represented as a weighted sum of positive Gaussians.\nIt also gives the rate of decay at large r, since the proper time for a random walk to reach position τ is r2 and in this time, the Gaussian height has decayed by.\nThe decay factor appropriate for position r is therefore.\nA heuristic approximation for G(r) is:\nThis is not an exact form, except in three dimensions, where interactions between paths become important.\nThe exact forms in high dimensions are variants of [Bessel functions](https://bliptext.com/articles/bessel-functions).\n\n### Symanzik polymer interpretation\nThe interpretation of the correlations as fixed size quanta travelling along random walks gives a way of understanding why the critical dimension of the H4 interaction is 4.\nThe term H4 can be thought of as the square of the density of the random walkers at any point.\nIn order for such a term to alter the finite order correlation functions, which only introduce a few new random walks into the fluctuating environment, the new paths must intersect.\nOtherwise, the square of the density is just proportional to the density and only shifts the H2 coefficient by a constant.\nBut the intersection probability of random walks depends on the dimension, and random walks in dimension higher than 4 do not intersect.\nThe [fractal dimension](https://bliptext.com/articles/fractal-dimension) of an ordinary random walk is 2.\nThe number of balls of size ε required to cover the path increase as ε−2.\nTwo objects of fractal dimension 2 will intersect with reasonable probability only in a space of dimension 4 or less, the same condition as for a generic pair of planes.\n[Kurt Symanzik](https://bliptext.com/articles/kurt-symanzik) argued that this implies that the critical Ising fluctuations in dimensions higher than 4 should be described by a free field.\nThis argument eventually became a mathematical proof.\n\n### 4 − ε dimensions – renormalization group\nThe Ising model in four dimensions is described by a fluctuating field, but now the fluctuations are interacting.\nIn the polymer representation, intersections of random walks are marginally possible.\nIn the quantum field continuation, the quanta interact.\nThe negative logarithm of the probability of any field configuration H is the [free energy](https://bliptext.com/articles/thermodynamic-free-energy) function\nThe numerical factors are there to simplify the equations of motion.\nThe goal is to understand the statistical fluctuations.\nLike any other non-quadratic path integral, the correlation functions have a [Feynman expansion](https://bliptext.com/articles/feynman-diagram) as particles travelling along random walks, splitting and rejoining at vertices.\nThe interaction strength is parametrized by the classically dimensionless quantity λ.\nAlthough dimensional analysis shows that both λ and Z are dimensionless, this is misleading.\nThe long wavelength statistical fluctuations are not exactly scale invariant, and only become scale invariant when the interaction strength vanishes.\nThe reason is that there is a cutoff used to define H, and the cutoff defines the shortest wavelength.\nFluctuations of H at wavelengths near the cutoff can affect the longer-wavelength fluctuations.\nIf the system is scaled along with the cutoff, the parameters will scale by dimensional analysis, but then comparing parameters doesn't compare behavior because the rescaled system has more modes.\nIf the system is rescaled in such a way that the short wavelength cutoff remains fixed, the long-wavelength fluctuations are modified.\n\n#### Wilson renormalization\nA quick heuristic way of studying the scaling is to cut off the H wavenumbers at a point λ.\nFourier modes of H with wavenumbers larger than λ are not allowed to fluctuate.\nA rescaling of length that make the whole system smaller increases all wavenumbers, and moves some fluctuations above the cutoff.\nTo restore the old cutoff, perform a partial integration over all the wavenumbers which used to be forbidden, but are now fluctuating.\nIn Feynman diagrams, integrating over a fluctuating mode at wavenumber k links up lines carrying momentum k in a correlation function in pairs, with a factor of the inverse propagator.\nUnder rescaling, when the system is shrunk by a factor of (1+b), the t coefficient scales up by a factor (1+b)2 by dimensional analysis.\nThe change in t for infinitesimal b is 2bt.\nThe other two coefficients are dimensionless and do not change at all.\nThe lowest order effect of integrating out can be calculated from the equations of motion:\nThis equation is an identity inside any correlation function away from other insertions.\nAfter integrating out the modes with Λ < k < (1+b)Λ, it will be a slightly different identity.\nSince the form of the equation will be preserved, to find the change in coefficients it is sufficient to analyze the change in the H3 term.\nIn a Feynman diagram expansion, the H3 term in a correlation function inside a correlation has three dangling lines.\nJoining two of them at large wavenumber k gives a change H3 with one dangling line, so proportional to H:\nThe factor of 3 comes from the fact that the loop can be closed in three different ways.\nThe integral should be split into two parts:\nThe first part is not proportional to t, and in the equation of motion it can be absorbed by a constant shift in t.\nIt is caused by the fact that the H3 term has a linear part.\nOnly the second term, which varies from t to t, contributes to the critical scaling.\nThis new linear term adds to the first term on the left hand side, changing t by an amount proportional to t.\nThe total change in t is the sum of the term from dimensional analysis and this second term from [operator products](https://bliptext.com/articles/operator-product-expansion):\nSo t is rescaled, but its dimension is [anomalous](https://bliptext.com/articles/anomalous-dimension), it is changed by an amount proportional to the value of λ.\nBut λ also changes.\nThe change in λ requires considering the lines splitting and then quickly rejoining.\nThe lowest order process is one where one of the three lines from H3 splits into three, which quickly joins with one of the other lines from the same vertex.\nThe correction to the vertex is\nThe numerical factor is three times bigger because there is an extra factor of three in choosing which of the three new lines to contract.\nSo\nThese two equations together define the renormalization group equations in four dimensions:\nThe coefficient B is determined by the formula\nand is proportional to the area of a three-dimensional sphere of radius λ, times the width of the integration region bΛ divided by Λ4:\nIn other dimensions, the constant B changes, but the same constant appears both in the t flow and in the coupling flow.\nThe reason is that the derivative with respect to t of the closed loop with a single vertex is a closed loop with two vertices.\nThis means that the only difference between the scaling of the coupling and the t is the combinatorial factors from joining and splitting.\n\n#### Wilson–Fisher fixed point\nTo investigate three dimensions starting from the four-dimensional theory should be possible, because the intersection probabilities of random walks depend continuously on the dimensionality of the space.\nIn the language of Feynman graphs, the coupling does not change very much when the dimension is changed.\nThe process of continuing away from dimension 4 is not completely well defined without a prescription for how to do it.\nThe prescription is only well defined on diagrams.\nIt replaces the Schwinger representation in dimension 4 with the Schwinger representation in dimension 4 − ε defined by:\nIn dimension 4 − ε, the coupling λ has positive scale dimension ε, and this must be added to the flow.\nThe coefficient B is dimension dependent, but it will cancel.\nThe fixed point for λ is no longer zero, but at:\nwhere the scale dimensions of t is altered by an amount λB = ε/3.\nThe magnetization exponent is altered proportionately to:\nwhich is .333 in 3 dimensions (ε = 1) and .166 in 2 dimensions (ε = 2).\nThis is not so far off from the measured exponent .308 and the Onsager two dimensional exponent .125.\n\n### Infinite dimensions – mean field\nThe behavior of an Ising model on a fully connected graph may be completely understood by [mean-field theory](https://bliptext.com/articles/mean-field-theory).\nThis type of description is appropriate to very-high-dimensional square lattices, because then each site has a very large number of neighbors.\nThe idea is that if each spin is connected to a large number of spins, only the average ratio of + spins to − spins is important, since the fluctuations about this mean will be small.\nThe [mean field](https://bliptext.com/articles/mean-field) H is the average fraction of spins which are + minus the average fraction of spins which are −.\nThe energy cost of flipping a single spin in the mean field H is ±2JNH.\nIt is convenient to redefine J to absorb the factor N, so that the limit N → ∞ is smooth.\nIn terms of the new J, the energy cost for flipping a spin is ±2JH.\nThis energy cost gives the ratio of probability p that the spin is + to the probability 1−p that the spin is −.\nThis ratio is the Boltzmann factor:\nso that\nThe mean value of the spin is given by averaging 1 and −1 with the weights p and 1 − p, so the mean value is 2p − 1.\nBut this average is the same for all spins, and is therefore equal to H.\nThe solutions to this equation are the possible consistent mean fields.\nFor βJ < 1 there is only the one solution at H = 0.\nFor bigger values of β there are three solutions, and the solution at H = 0 is unstable.\nThe instability means that increasing the mean field above zero a little bit produces a statistical fraction of spins which are + which is bigger than the value of the mean field.\nSo a mean field which fluctuates above zero will produce an even greater mean field, and will eventually settle at the stable solution.\nThis means that for temperatures below the critical value βJ = 1 the mean-field Ising model undergoes a phase transition in the limit of large N.\nAbove the critical temperature, fluctuations in H are damped because the mean field restores the fluctuation to zero field.\nBelow the critical temperature, the mean field is driven to a new equilibrium value, which is either the positive H or negative H solution to the equation.\nFor βJ = 1 + ε, just below the critical temperature, the value of H can be calculated from the Taylor expansion of the hyperbolic tangent:\nDividing by H to discard the unstable solution at H = 0, the stable solutions are:\nThe spontaneous magnetization H grows near the critical point as the square root of the change in temperature.\nThis is true whenever H can be calculated from the solution of an analytic equation which is symmetric between positive and negative values, which led [Landau](https://bliptext.com/articles/lev-landau) to suspect that all Ising type phase transitions in all dimensions should follow this law.\nThe mean-field exponent is [universal](https://bliptext.com/articles/universality-dynamical-systems) because changes in the character of solutions of analytic equations are always described by [catastrophes](https://bliptext.com/articles/catastrophe-theory) in the [Taylor series](https://bliptext.com/articles/taylor-series), which is a polynomial equation.\nBy symmetry, the equation for H must only have odd powers of H on the right hand side.\nChanging β should only smoothly change the coefficients.\nThe transition happens when the coefficient of H on the right hand side is 1.\nNear the transition:\nWhatever A and B are, so long as neither of them is tuned to zero, the spontaneous magnetization will grow as the square root of ε.\nThis argument can only fail if the free energy βF is either non-analytic or non-generic at the exact β where the transition occurs.\nBut the spontaneous magnetization in magnetic systems and the density in gasses near the critical point are measured very accurately.\nThe density and the magnetization in three dimensions have the same power-law dependence on the temperature near the critical point, but the behavior from experiments is:\nThe exponent is also universal, since it is the same in the Ising model as in the experimental magnet and gas, but it is not equal to the mean-field value.\nThis was a great surprise.\nThis is also true in two dimensions, where\nBut there it was not a surprise, because it was predicted by [Onsager](https://bliptext.com/articles/lars-onsager).\n\n### Low dimensions – block spins\nIn three dimensions, the perturbative series from the field theory is an expansion in a coupling constant λ which is not particularly small.\nThe effective size of the coupling at the fixed point is one over the branching factor of the particle paths, so the expansion parameter is about 1/3.\nIn two dimensions, the perturbative expansion parameter is 2/3.\nBut renormalization can also be productively applied to the spins directly, without passing to an average field.\nHistorically, this approach is due to [Leo Kadanoff](https://bliptext.com/articles/leo-kadanoff) and predated the perturbative ε expansion.\nThe idea is to integrate out lattice spins iteratively, generating a flow in couplings.\nBut now the couplings are lattice energy coefficients.\nThe fact that a continuum description exists guarantees that this iteration will converge to a fixed point when the temperature is tuned to criticality.\n\n#### Migdal–Kadanoff renormalization\nWrite the two-dimensional Ising model with an infinite number of possible higher order interactions.\nTo keep spin reflection symmetry, only even powers contribute:\nBy translation invariance, Jij is only a function of i-j.\nBy the accidental rotational symmetry, at large i and j its size only depends on the magnitude of the two-dimensional vector i − j.\nThe higher order coefficients are also similarly restricted.\nThe renormalization iteration divides the lattice into two parts – even spins and odd spins.\nThe odd spins live on the odd-checkerboard lattice positions, and the even ones on the even-checkerboard.\nWhen the spins are indexed by the position (i,j), the odd sites are those with i + j odd and the even sites those with i + j even, and even sites are only connected to odd sites.\nThe two possible values of the odd spins will be integrated out, by summing over both possible values.\nThis will produce a new free energy function for the remaining even spins, with new adjusted couplings.\nThe even spins are again in a lattice, with axes tilted at 45 degrees to the old ones.\nUnrotating the system restores the old configuration, but with new parameters.\nThese parameters describe the interaction between spins at distances \\sqrt{2} larger.\nStarting from the Ising model and repeating this iteration eventually changes all the couplings.\nWhen the temperature is higher than the critical temperature, the couplings will converge to zero, since the spins at large distances are uncorrelated.\nBut when the temperature is critical, there will be nonzero coefficients linking spins at all orders.\nThe flow can be approximated by only considering the first few terms.\nThis truncated flow will produce better and better approximations to the critical exponents when more terms are included.\nThe simplest approximation is to keep only the usual J term, and discard everything else.\nThis will generate a flow in J, analogous to the flow in t at the fixed point of λ in the ε expansion.\nTo find the change in J, consider the four neighbors of an odd site.\nThese are the only spins which interact with it.\nThe multiplicative contribution to the partition function from the sum over the two values of the spin at the odd site is:\nwhere N± is the number of neighbors which are ±.\nIgnoring the factor of 2, the free energy contribution from this odd site is:\nThis includes nearest neighbor and next-nearest neighbor interactions, as expected, but also a four-spin interaction which is to be discarded.\nTo truncate to nearest neighbor interactions, consider that the difference in energy between all spins the same and equal numbers + and – is:\nFrom nearest neighbor couplings, the difference in energy between all spins equal and staggered spins is 8J.\nThe difference in energy between all spins equal and nonstaggered but net zero spin is 4J.\nIgnoring four-spin interactions, a reasonable truncation is the average of these two energies or 6J.\nSince each link will contribute to two odd spins, the right value to compare with the previous one is half that:\nFor small J, this quickly flows to zero coupling.\nLarge J's flow to large couplings.\nThe magnetization exponent is determined from the slope of the equation at the fixed point.\nVariants of this method produce good numerical approximations for the critical exponents when many terms are included, in both two and three dimensions.\n\n# Footnotes","revisionCount":0},"hasMore":false,"nextCursor":null}

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