Definitions

Ising model

The Ising model, named after the physicist Ernst Ising, is a mathematical model in statistical mechanics. It has since been used to model diverse phenomena in which bits of information, interacting in pairs, produce collective effects.

Definition

The Ising model is defined on a discrete collection of variables called spins, which can take on the value 1 or −1. The spins $S_i$ interact in pairs, with an energy which has one value when the two spins are the same, and a second value when the two spins are different.

Energy function

The energy of the Ising model is defined to be:

E = - sum_{ij} J_{ij} S_i S_j ,.

Where the sum counts each pairs of spins only once. Notice that the product of spins is either +1 if the two spins are the same, or aligned, and −1 if they are different, antialigned. J is half the difference in energy between the two possibilities. Magnetic interaction tries to align all the atoms in one direction, while thermal energy tries to break the order.

For each pair, if

$J_\left\{ij\right\} > 0$ the interaction is called ferromagnetic
$J_\left\{ij\right\} < 0$ the interaction is called antiferromagnetic
$J_\left\{ij\right\} = 0$ the spins are noninteracting

A ferromagnetic interaction tends to align spins, and an antiferromagnetic tends to antialign them.

The spins can be thought of as living on a graph, where each node has exactly one spin, and each edge connects two spins with a nonzero value of J. If all the Js are equal, it is convenient to measure energy in units of J. Then a model is completely specified by the graph and the sign of J.

Simple examples

The antiferromagnetic one-dimensional Ising model has the energy function:

E = sum_{i} S_{i} S_{i+1} , where i runs over all the integers. This links each pair of nearest neighbors.

The ferromagnetic two-dimensional Ising model on a square lattice is a collection of spins $S_\left\{i,j\right\}$ on each node (i,j) of a two dimensional square lattice and the Energy is:


E= - sum_{ij} (S_{i,j} S_{i,j+1} + S_{i,j} S_{i+1,j}) ,

Notice that the sum links every site to its right-neighbor and its down-neighbor. In this way, every edge is only counted once.

The mean field Ising model is the Ising model on a complete graph, where all the nodes are connected to all the other nodes:


E = - sum_{i

Magnetic field

The energy of the Ising model may be modified to bias the entire system. Normally, an Ising model is completely symmetric under interchange of + and −. A magnetic field $h_i$ may be added to the energy, and it breaks the symmetry, The full energy function is:

E= - frac{1}{2} sum_{langle i,jrangle} J_{ij} S_i S_j - sum_i h_i S_i ,

where the brackets indicate that i and j index neighboring positions on the graph.

Statistics

The model is a statistical model, so the energy is really the logarithm of the probability. The probability of each configuration of spins is the Boltzmann distribution with inverse temperature β.

P(S) propto e^{-beta E} ,.

To actually generate configurations using this probability distribution is conceptually easiest using the Metropolis Algorithm:

1. Pick a spin at random and calculate the contribution to the energy involving this spin.
2. flip the value of the spin and calculate the new contribution.
3. if the new energy is less, keep the flipped value.
4. if the new energy is more, only keep with probability $e^\left\{-beta Delta E\right\}$
5. Repeat

The change in energy $Delta E$ only depends on the value of the spin and its nearest graph neighbors. So if the graph is not too connected, the algorithm is fast. This process will eventually produce a pick from the distribution.

Questions

The interesting statistical questions to ask are all in the limit of large numbers of spins:

1. In a typical configuration, are most of the spins +1 or −1, or are they split equally?
2. If a spin at any given position is 1, what is the probability that the spin at position j is also 1?
3. If β is changed, is there a phase transition?
4. If J is random, how many different configurations are there at any given inverse temperature?
5. On a lattice, what is the fractal dimension of the shape of a large cluster of +1 spins?

General discussion

In his 1925 PhD thesis, Ising solved the model for the 1D case. In one dimension, the solution admits no phase transition. On the basis of this result, he incorrectly concluded that his model does not exhibit phase behaviour in any dimension.

Most numerical solutions use the Metropolis-Hastings algorithm run inside a Monte Carlo loop. Depending on the complexity only adjacent vertices can be taken into account or for long-range models other vertices can be included.

The Ising model undergoes a phase transition between an ordered and a disordered phase in 2 dimensions or more. In 2 dimensions, the Ising model has a strong/weak duality (between high temperatures and low ones) called the Kramers-Wannier duality. The fixed point of this duality is at the second-order phase transition temperature.

While the Ising model is a simplified microscopic description of ferromagnetism, it is still extremely important because of the universality of the continuum limit. Universality means that the fluctuations near the phase transition are described by a continuum field with a free energy or Lagrangian which is a function of the field values. Just as there are many ways to discretize a differential equation, all of which give the same answer when the lattice spacing is small, there are many different discrete models that have the exact same critical behavior, because they have the same continuum limit.

The experimentally observed critical fluctuations of ferromagnets near the Curie point and of fluids at the vapor/liquid critical point are described exactly by the critical fluctuations of the Ising model. The same is true for the simplest statistical models in three dimensions whose fluctuations can be described by a single scalar field, the local magnetization in a near-critical magnet or the local density in a near-critical fluid. All these systems have fluctuating clusters whose fractal scaling laws and long distance correlation functions are quantitatively predicted by the model.

Apart from the continuum limit, many discrete systems can be mapped exactly or approximately to the Ising system. The grand canonical ensemble formulation of the lattice gas model, for example, can be mapped exactly to the canonical ensemble formulation of the Ising model. The mapping allows one to exploit simulation and analytical results of the Ising model to answer questions about the related models.

The Ising model on a two dimensional square lattice with no magnetic field was analytically solved in 1944 by Lars Onsager. Onsager showed that the correlation functions and free energy of the Ising model are determined by a noninteracting lattice fermion. The 3D Ising model does not have a representation in terms of free fields.

In 2000, Sorin Istrail established that computing the free energy of the Ising model on an arbitrary sublattice of a three dimensional square lattice is computationally intractable. While this means that it is impossible to efficiently compute all possible thermodynamic quantities with arbitrary external fields, it does not mean that the critical exponents or spin-spin correlations cannot be computed near criticality. In particular, Istrail's proof is valid in four or higher dimensions, where the model is also exactly solvable near criticality, since its correlation functions at long distances are those of a free scalar field.

The Ising model can be thought of as a Markov random field on a square grid, where the maximal graph cliques are edges (i.e. pairs of neighboring verticies).

Historical Significance

One of Democritus' arguments in support of atomism was that atoms naturally explain the sharp phase boundaries observed in materials, like when ice melts to water or water turns to steam. His idea was that small changes in the atomic scale properties would lead to big changes in their aggregate statistical behavior. Others believed that matter is inherently continuous, not atomic, and that the large-scale properties of matter are not reducible to basic atomic properties.

While the laws of chemical binding made it clear to nineteenth century chemists that atoms were real, among physicists the debate continued well into the early twentieth century. Atomists, notably James Clerk Maxwell and Ludwig Boltzmann, applied Hamilton's formulation of Newton's laws to large systems, and found that the statistical behavior of the atoms correctly describe room temperature gasses. But classical statistical mechanics did not account for all of the properties of liquids and solids, nor of gasses at low temperature.

Once modern quantum mechanics was formulated, atomism was no longer in conflict with experiment, but this did not lead to a universal acceptance of statistical mechanics, which went beyond atomism. Josiah Willard Gibbs had given a complete formalism to reproduce the laws of thermodynamics from the laws of mechanics. But from the 19th century, when statistical mechanics was considered dubious, many faulty arguments survived. The lapses in intuition mostly stemmed from the fact that the limit of an infinite statistical system has many zero-one laws which are absent in finite systems: an infinitesimal change in a parameter can lead to big differences in the statistical behavior, as Democritus expected.

No Phase Transitions In Finite Volume

In the early part of the twentieth century, some believed that the partition function could never describe a phase transition, based on the following argument:

1. The partition function is a sum of $scriptstyle e^\left\{-beta E\right\}$ over all configurations.
2. the exponential function is everywhere analytic as a function $beta$.
3. the sum of analytic things is analytic.

But the logarithm of the partition function is not analytic as a function of the temperature near a phase transition, so the theory doesn't work.

This argument works for a finite sum of exponentials, and correctly establishes that there are no singularities in the free energy of a system of a finite size. For systems which in the thermodynamic limit, for infinite systems, the infinite sum can lead to singularities. The convergence to the thermodynamic limit is fast, so that the phase behavior is apparent already on a relatively small lattice, even though the singularities are smoothed out by the system's finite size.

This was first established by Rudolph Peierls in the Ising model.

Peierls Droplets

Shortly after Lenz and Ising constructed the Ising model, Peierls was able to explicitly show that a phase transition occurs in two dimensions.

To do this, he compared the high-temperature and low temperature limits. At infinite temperature, zero beta, all configurations have equal probability. Each spin is completely independent of any other, and if typical configurations at infinite temperature are plotted so that plus/minus are represented by black and white, they look like television snow. For high, but not infinite temperature, there are small correlations between neighboring positions, the snow tends to clump a little bit, but the screen stays random looking, and there is no net excess of black or white.

A quantitative measure of the excess is the magnetization, which is the average value of the spin:


M= {1over N} sum_{i=0}^{N-1} S_i

A bogus argument analogous to the argument in the last section now establishes that the magnetization in the Ising model is always zero.

1. Every configurations of spin has equal energy to the configuration with all spins flipped.
2. So for every configuration with magnetization M there is a configuration with magnetization -M with equal probability
3. So the magnetization is zero.

As before, this only proves that the magnetization is zero at any finite volume. For an infinite system, fluctuations might not be able to push the system from a mostly-plus state to a mostly minus with any nonzero probability.

For very high temperatures, the magnetization is zero, as it is at infinite temperature. To see this, note that if spin A has only a small correlation $epsilon$ 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 $epsilon^2$. For two spins separated by distance L, the amount of correlation goes as $epsilon^L$ but if there is more than one path by which the correlations can travel, this amount is enhanced by the number of paths.

The number of paths of length L on a square lattice in d dimensions:


N(L) = (2d)^L , since there are 2d choices for where to go at each step.

A 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 lengths paths of length L divided by the :


sum_L (2d)^L (epsilon)^L , which goes to zero when $epsilon$ is small.

At low temperatures, infinite beta, the configurations are near the lowest energy configuration, the one where all the spins are plus or all the spins are minus. Peierls 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. For this to happen, droplets of plus spin must be able to congeal to make the plus state.

The 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. For a droplet with perimeter L, the area is somewhere between $\left(L-2\right)/2$ (the straight line) and $\left(L/4\right)^2$(the square box). The probability cost for introducing a droplet is the factor:


e^{-beta L} , 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:

N(L)< 4^{2L} , So that the total spin contribution from droplets, even overcounting by allowing each site to have a separate droplet, is bounded above by:


sum_L L^2 4^{-2L} e^{-4beta L} ,

which goes to zero at large $beta$. For $beta$ sufficiently large, this exponentially suppresses long loops, so that they cannot occur, and the magnetization never fluctuates too far from -1.

So Peierls established that the magnetization in the Ising model eventually defines superselection sectors, separated domains which are not linked by finite fluctuations.

Kramers-Wannier Duality

Kramers 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. This allowed the phase transition point in the two-dimensional model to be determined exactly (under the assumption that there is a unique critical point).

Yang Lee Zeros

After Onsager's solution, Yang and Lee investigated the way in which the partition function becomes singular as the temperature approaches the critical temperature.

Applications

Magnetism

The original motivation for the model was the phenomenon of ferromagnetism. Iron is magnetic; once it is magnetized it stays magnetized for a long time compared to any atomic time.

Since the 19th century, it was clear that magnetic fields are due to currents in matter, and Ampère postulated that permanent magnets are caused by permanent atomic currents. The motion of classical charged particles could not explain permanent currents though, as shown by Larmor. In order to have ferromagnetism, the atoms must have permanent magnetic moments which are not due to the motion of classical charges.

Once the electron's spin was discovered, it was clear that the magnetism should be due to a large number of electrons spinning in the same direction. It was natural to ask how the electrons all know which direction to spin, because the electrons on one side of a magnet don't directly interact with the electrons on the other side. They can only influence their neighbors. The Ising model was designed to investigate whether a large fraction of the electrons could be made to spin in the same direction using only local forces.

Lattice gas

The Ising model can be reinterpreted as a statistical model for the motion of atoms. Since the kinetic energy doesn't depend on the position only on the momentum, 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.

A 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 $B_i$, where each bit is either 0 or 1 depending on whether the position is occupied or not. An attractive interaction reduces the energy of two nearby atoms. If the attraction is only between nearest neighbors, the energy is reduced by $- 4J B_i B_j$ for each occupied neighboring pair.

The density of the atoms can be controlled by adding a chemical potential, which is a multiplicative probability cost for adding one more atom. A multiplicative factor in probability can be reinterpreted as an additive term in the logarithm – the energy. The extra energy of a configuration with N atoms is changed by $mu N$. The probability cost of one more atom is a factor of $exp\left\{\left(-betamu\right)\right\}$.

So the energy of the lattice gas is:


E = - frac{1}{2} sum_{langle i,j rangle} 4 J B_i B_j + sum_i mu B_i ,

Rewriting the bits in terms of spins, $B_i = \left(S_i + 1\right)/2$.


E = - frac{1}{2} sum_{langle i,j rangle} J S_i S_j - frac{1}{2} sum_i (4 J - mu) S_i ,

For lattices where every site has an equal number of neighbors, this is the Ising model with a magnetic field $h = \left(z J - mu\right)/2$, where $z$ is the number of neighbors.

Pairwise correlated bits

The activity of neurons in the brain can be modelled statistically. Each neuron at any time is either active + or inactive −. The active neurons are those that send an action potential down the axon in any given time window, and the inactive ones are those that do not. Because the neural activity at any one time is modelled by independent bits, Hopfield suggested that a dynamical Ising model would provide a first approximation to a neural network which is capable of learning.

Following the general approach of Jaynes , a recent interpretation of Schneidman, Berry, Segev and Bialek, is 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. Given a collection of neurons, a statistical model which can reproduce the average firing rate for each neuron introduces a Lagrange multiplier for each neuron:


E = - sum_i h_i S_i But the activity of each neuron in this model is statistically independent. To allow for pair correlations, when one neuron tends to fire (or not to fire) along with another, introduce pair-wise lagrange multipliers:

E= - frac{1}{2} sum_{ij} J_{ij} S_i S_j - sum_i h_i S_i This energy function only introduces probability biases for a spin having a value and for a pair of spins having the same value. Higher order correlations are unconstrained by the multipliers. An 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. This 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.

Critical Behavior

One dimension – independent spin flips

The energy of the one dimensional ferromagnetic Ising model is:


- sum_i S_i S_{i+1} ,.

Where $i$ runs from $0$ to $L$, where $L$ is the length of the line. The energy of the lowest state is $-L$, when all the spins are the same. For any other configuration, the extra energy is equal to the number of sign changes as you scan the configuration from left to right.

If we call the number of sign changes in a configuration $k$, the difference in energy from the lowest energy state is $2k$. Since the energy is additive in the number of flips, the probability $p$ of having a spin-flip at each position is independent. The ratio of the probability of finding a flip to the probability of not finding one is the Boltzmann factor:


{p over 1-p} = e^{-2beta} ,.

The problem is reduced to independent biased coin tosses. This essentially completes the mathematical description.

From the description in terms of independent tosses, the statistics of the model for long lines can be understood. The line splits into domains. Each domain is of average length $exp\left(2beta\right)$. The length of a domain is distributed exponentially, since there is a constant probability at any step of encountering a flip. The domains never become infinite, so a long system is never magnetized. Each step reduces the correlation between a spin and its neighbor by an amount proportional to $p$, so the correlations fall off exponentially.


langle S_i S_j rangle ,propto, e^{-p|i-j|} ,.

The partition function is the volume of configurations, each configuration weighted by its Boltzmann weight. Since each configuration is described by a the sign-changes, the Partition function factorizes:


Z = sum_{mathrm{configs}} e^{sum_k S_i} = prod_k (1 + p ) = (1+p)^L ,.

The logarithm divided by $L$ is the free energy density:


beta f = log(1+p) = log(1 + {e^{-2beta}over 1+e^{-2beta}} ) ,.

which is analytic away from $beta=infty$. A sign of a phase transition is a non-analytic free energy, so the one dimensional model does not have a phase transition.

Infinite dimensions – mean field

The behavior of an Ising model on a fully connected graph may be completely understood by mean field theory. This type of description is appropriate to very high dimensional square lattices, because then each site has a very large number of neighbors.

The idea is that if each spin is connected to a large number of spins, only the average number of + spins to − spins is important, since the fluctuations about this mean will be small. The mean field H is the average fraction of spins which are +. The energy cost of flipping a single spin in the mean field H is 2JNH. It is convenient to redefine J to absorb the factor N, so that the limit $scriptstyle Nrightarrow infty$ is smooth. In terms of the new J, the energy cost for flipping a spin is $2JH$.

This energy cost gives the ratio of probability p that the spin is + to the probability 1-p that the spin is −. This ratio is the Boltzmann factor.


{pover 1-p} = e^{-2beta JH} ,

so that


p = {1 over 1 + e^{-2beta JH} } ,

The 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. But this average is the same for all spins, and is therefore equal to H.


H = 2p - 1 = { 1 - e^{-2beta JH} over 1 + e^{-2beta JH}} = tanh (beta JH) ,

The solutions to this equation are the possible consistent mean fields. For $beta J <1$ there is only the one solution at H=0. For bigger values of β there are three solutions, and the solution at H=0 is unstable.

The 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. So a mean field which fluctuates above zero will produce an even greater mean field, and will eventually settle at the stable solution. This means that for temperatures below the critical value $beta J=1$ the mean field Ising model undergoes a phase transition in the limit of large N.

Above the critical temperature, fluctuations in H are damped because the mean field restores the fluctuation to zero field. Below 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.

For $beta J = 1+epsilon$, just below the critical temperature, the value of H can be calculated from the Taylor expansion of the Hyperbolic tangent:


H = tanh(beta J H) = (1+epsilon)H - {(1+epsilon)^3H^3over 3} ,

dividing by H to discard the unstable solution at H=0, the stable solutions are:


H = sqrt{3epsilon} ,

The spontaneous magnetization H grows near the critical point as the square root of the change in temperature. This 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 to suspect that all Ising type phase transitions in all dimensions should follow this law.

The mean field exponent is universal because changes in the character of solutions of analytic equations are always described by catastrophies in the Taylor series, which is a polynomial equation. By symmetry, the equation for H must only have odd powers of H on the right hand side. Changing β should only smoothly change the coefficients. The transition happens when the coefficient of H on the right hand side is 1. Near the transition:


H = {partial (beta F) over partial h} = (1+Aepsilon) H + B H^3 + ... ,

Whatever A and B are, so long as neither of them is tuned to zero, the sponetaneous magnetization will grow as the square root of ε. This argument can only fail if the free energy $beta F$ is either non-analytic or non-generic at the exact β where the transition occurs.

But the spontaneous magnetization in magnetic systems and the density in gasses near the critical point are measured very accuratedly. The 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:


H propto epsilon^{0.308} ,

The exponent is also universal, 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. This was a great surprise.

This is also true in two dimensions, where


Hpropto epsilon^{0.125} ,

But there it was not a surprise, because it was predicted by Onsager.

Two dimensions – Onsager's solution

The partition function of the Ising model in two dimensions on a square lattice can be mapped to a two dimensional free fermion. This allows the specific heat to be calculated exactly.

Transfer matrix

Start with an analogy with quantum mechanics. The Ising model on a long periodic lattice has a partition function


sum_S expbiggl(sum_{ij} S_{i,j} S_{i,j+1} + S_{i,j} S_{i+1,j}biggr) ,

Think of the i direction as space, and the j direction as time. This is an independent sum over all the values that the spins can take at each time slice. This is a type of path integral, it is the sum over all spin histories.

A path integral can be rewritten as a Hamiltonian evolution. The Hamiltonian steps through time by performing a unitary rotation between time $t$ and time $t+Delta t$:

$U = e^\left\{i H Delta t\right\}$
,

The product of the U matrices, one after the other, is the total time evolution operator, which is the path integral we started with.

$U^N = \left(e^\left\{i H Delta t\right\}\right)^N = int DX e^\left\{iL\right\}$
,

Where N is the number of time slices. The sum over all paths is given by a product of matrices, each matrix element is the transition probability from one slice to the next.

Similarly, one can divide the sum over all partition function configurations into slices, where each slice is the one-dimensional configuration at time 1. This defines the transfer matrix:


T_{C_1 C_2} ,

The configurations in each slice is a one dimensional collection of spins. At each time slice, T has matrix elements between two configurations of spins, one in the immediate future and one in the immediate past. These two configurations are $C_1$ and $C_2$, and they are all one dimensional spin configurations. We can think of the vector space that T acts on as all complex linear combinations of these. Using quantum mechanical notation:


|Arangle = sum_S A(S) |Srangle ,

Where each basis vector $|Srangle$ is a spin configuration of a one dimensional Ising model.

Like the Hamiltonian, the transfer matrix acts on all linear combinations of states. The partition function is a matrix function of T, which is defined by the sum over all histories which come back to the original configuration after N steps:


Z= mathrm{tr}(T^N) ,

Since this is a matrix equation, it can be evaluated in any basis. So if we can diagonalize the matrix T, we can find Z.

T in terms of Pauli matrices

The contribution to the partition function for each past/future pair of configurations on a slice is the sum of two terms. There is the number of spin flips in the past slice and there is the number of spin flips between the past and future slice. Define an operator on configurations which flips the spin at site i:

sigma^x_i ,

In 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.

Define a second operator which multiplies the basis vector by +1 and −1 according to the spin at position i:


sigma^z_i ,

T can be written in terms of these:


sum_i A sigma^x_i + B sigma^z_i sigma^z_{i+1} ,

Where A and B are constants which are to be determined so as to reproduce the partition function. The 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.

Spin flip creation and annihilation operators

Just as in the one dimensional case, we will shift attention from the spins to the spin-flips. The $sigma_z$ term in T counts the number of spin flips, which we can write in terms of spin-flip creation and annihilation operators:
$sum C psi^dagger_i psi_i ,$

The first term flips a spin, so depending on the basis state it either:

1. moves a spin-flip one unit to the right
2. moves a spin-flip one unit to the left
3. produces two spin-flips on neighboring sites
4. destroys two spin-flips on neighboring sites.

Writing this out in terms of creation and annihilation operators:

$sigma^x_i = D \left\{psi^dagger\right\}_i psi_\left\{i+1\right\} + D^* \left\{psi^dagger\right\}_i psi_\left\{i-1\right\} + Cpsi_i psi_\left\{i+1\right\} + C^* \left\{psi^dagger\right\}_i \left\{psi^dagger\right\}_\left\{i+1\right\}$
,

Ignore the constant coefficients, and focus attention on the form. They are all quadratic. Since the coefficients are constant, this means that the T matrix can be diagonalized by Fourier transforms.

Carrying out the diagonalization produces the Onsager free energy.

Dimensions 5 and above – free field

In any dimension, the Ising model can be productively described by a locally varying mean field. The field is defined as the average spin value over a large region, but not so large so as to include the entire system. The field still has slow variations from point to point, as the averaging volume moves. These fluctuations in the field are described by a continuum field theory in the infinite system limit.

Local field

The field H is defined as the long wavelength Fourier components of the spin variable, in the limit that the wavelengths are long. There are many ways to take the long wavelength average, depending on the details of how high wavelengths are cut off. The details are not too important, since the goal is to find the statistics of H and not the spins. Once the correlations in H are known, the long-distance correlations between the spins will be proportional to the long-distance correlations in H.

For any value of the slowly varying field H, the free energy (log-probability) is a local analytic function of H and its gradients. The free energy F(H) is defined to be the sum over all Ising configurations which are consistent with the long wavelength field. Since 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.

Since the allowed range of values of the spin in any region only depend 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. So F is a sum over all regions of a local contribution, which only depends on H and its derivatives.

By symmetry in H, only even powers contribute. By reflection symmetry on a square lattice, only even powers of gradients contribute. Writing out the first few terms in the free energy:


beta F = int d^dx A H^2 + sum_{i=1}^{d} Z_i (partial_i H)^2 + lambda H^4 ... ,

On a square lattice, symmetries guarantee that the coefficients $Z_i$ of the derivative terms are all equal. But even for an anisotropic Ising model, where the Z's in different directions are different, the fluctuations in H are isotropic in a coordinate system where the different directions of space are rescaled.

On any lattice, the derivative term $scriptstyle Z_\left\{ij\right\} partial_i H partial_j H$ is a positive definite quadratic form, and can be used to define the metric for space. So any translationally invariant Ising model is rotationally invariant at long distances, in coordinates that make $Z_\left\{ij\right\}=delta_\left\{ij\right\}$. Rotational symmetry emerges spontaneously at large distances just because there aren't very many low order terms. At higher order multicritical points, this accidental symmetry is lost.

Since $beta F$ is a function of a slowly spatially varying field. The probability of any field configuration is:


P(H) propto e^{ - int d^dx AH^2 + Z |nabla H|^2 + lambda H^4 } ,

The statistical average of any product of H's is equal to:


langle H(x_1) H(x_2)....H(x_n) rangle = { int DH P(H) H(x_1) H(x_2) ... H(x_n) over int DH P(H) } ,

The denominator in this expression is called the partition function, and the integral over all possible values of H is a statistical path integral. It integrates $exp\left(beta F\right)$ over all values of H, over all the long wavelength fourier components of the spins. F is a Euclidean Lagrangian for the field H, the only difference between this and the quantum field theory of a scalar field is that all the derivative terms enter with a positive sign, and there is no overall factor of i.


Z = int DH e ^ { int d^dx A H^2 + Z |nabla H|^2 + lambda H^4} ,

Dimensional analysis

The form of F can be used to predict which terms are most important by dimensional analysis. Dimensional analysis is not completely straightforward, because the scaling of H needs to be determined.

In the generic case, choosing the scaling law for H is easy, the only term that contributes is the first one,


F = int d^dx A H^2 ,

This term is the most significant, but it gives trivial behavior. This form of the free energy is ultralocal, meaning that it is a sum of an independent contribution from each point. This is like the spin-flips in the one-dimensional Ising model. Every value of H at any point fluctuates completely independently of the value at any other point.

The 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. The 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.

To find the critical point, lower the temperature. As the temperature goes down, the fluctuations in H go up because the fluctuations are more correlated. This 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. This corresponds to decreasing A in the system of units where H does not absorb A. The 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. This is the location of the critical point:


F= int d^dx t H^2 + lambda H^4 + Z (nabla H)^2 ,

Where t is a parameter which goes through zero at the transition.

Since t is vanishing, fixing the scale of the field using this term makes the other terms blow up. Once t is small, the scale of the field can either be set to fix the coefficient of the $H^4$ term or the $scriptstyle \left(nabla H\right)^2$ term to 1.

Magnetization

To find the magnetization, fix the scaling of H so that λ is one. Now the field H has dimension −d/4, so that $H^4 d^dx$ is dimensionless, and Z has dimension 2−d/2. In this scaling, the gradient term is only important at long distances for $scriptstyle dle 4$. Above four dimensions, at long wavelengths, the overall magnetization is only affected by the ultralocal terms.

There is one subtle point. The field H is fluctuating statistically, and the fluctuations can shift the zero point of t. To see how, consider $H^4$ split in the following way:


H(x)^4 = langle H(x)^2rangle^2 + 2langle H(x)^2rangle H(x)^2 + (H(x)^2-langle H(x)^2rangle)^2 ,

The first term is a constant contribution to the free energy, and can be ignored. The second term is a finite shift in t. The third term is a quantity that scales to zero at long distances. This means that when analyzing the scaling of t by dimensional analysis, it is the shifted t that is important. This was historically very confusing, because the shift in t at any finite λ is finite, but near the transition t is very small. The fractional change in t is very large, and in units where t is fixed the shift looks infinite. The magnetization is at the minimum of the free energy, and this is an analytic equation. In terms of the shifted t,


{partial over partial H } (t H^2 + lambda H^4 ) = 2t H + 4lambda H^3 = 0 ,

For t<0, the minima are at H proportional to the square root of t. So Landau's catastrophe argument is correct in dimensions larger than 5. The magnetization exponent in dimensions higher than 5 is equal to the mean field value.

When t is negative, the fluctuations about the new minimum are described by a new positive quadratic coefficient. Since this term always dominates, at temperatures below the transition the flucuations again become ultralocal at long distances.

Fluctuations

To find the behavior of fluctuations, rescale the field to fix the gradient term. Then the length scaling dimension of the field is 1−d/2. Now the field has constant quadratic spatial fluctuations at all temperatures. The scale dimension of the $H^2$ term is 2, while the scale dimension of the $H^4$ term is 4−d. For d<4, the $H^4$ term has positive scale dimension. In dimensions higher than 4 it has negative scale dimensions.

This is an essential difference. In dimensions higher than 4, fixing the scale of the gradient term means that the coefficient of the $H^4$ term is less and less important at longer and longer wavelengths. The dimension at which nonquadratic contributions begin to contribute is known as the critical dimension. In the Ising model, the critical dimension is 4.

In dimensions above 4, the critical fluctuations are described by a purely quadratic free energy at long wavelengths. This means that the correlation functions are all computable from as Gaussian averages:


langle S(x)S(y)rangle propto langle H(x)H(y)rangle = G(x-y) = int {dk over (2pi)^d} { e^{ik(x-y)}over k^2 + t } ,

valid when x-y is large. The function G(x-y) is the analytic continuation to imaginary time of the Feynman propagator, since the free energy is the analytic continuation of the quantum field action for a free scalar field. For dimensions 5 and higher, all the other correlation functions at long distances are then determined by Wick's theorem. All the odd moments are zero, by +/- symmetry. The even moments are the sum over all partition into pairs of the product of G(x-y) for each pair.


langle S(x_1) S(x_2) ... S(x_{2n})rangle = C^n sum G(x_{i1},x_{j1}) G(x_{i2},X_{j2}) ldots G(x_{in},x_{jn}) ,

Where C is the proportionality constant. So knowing G is enough. It determines all the multipoint correlations of the field.

The critical two point function

To 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:

(-nabla_x^2 + t) langle H(x)H(y) rangle = 0 rightarrow nabla^2 G(x) +t G(x) = 0 , This is valid at noncoindent points only, since the correlations of H are singular when points collide. H obeys classical equations of motion for the same reason that quantum mechanical operators obey them – its fluctuations are defined by a path integral.

At the critical point t=0, this is Laplace's equation, which can be solved by Gauss's method from electrostatics. Define an electric field analog by


E = nabla G ,

away from the origin:


nabla cdot E = 0 ,

since G is spherically symmetric in d dimensions, E is the radial gradient of G. Integrating over a large d-1 dimensional sphere,


int d^{d-1}S E_r = mathrm{constant} ,

This gives:


E = {C over r^{d-1} } ,

and G can be found by integrating with respect to r.


G(r) = {C over r^{d-2} } ,

The constant C fixes the overall normalization of the field.

G(r) away from the critical point

When 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. The equation it obeys is altered:

nabla^2 G + t G = 0 rightarrow {1over r^{d-1}} {dover dr} (r^{d-1} {dGover dr}) + t G(r) =0 ,

For r small compared with $sqrt\left\{t\right\}$, the solution diverges exactly the same way as in the critical case, but the long distance behavior is modified.

To see how, it is convenient to represent the two point function as an integral, introduced by Schwinger in the quantum field theory context:


G(x) = int dtau {1over (sqrt{2pitau})^d} e^{- {x^2over 4tau} -t tau} ,

This is G, since the Fourier transform of this integral is easy. Each fixed τ contribution is a Gaussian in x, whose Fourier transform is another Gaussian of reciprocal width in k.


G(k) = int dtau e^{- (k^2 - t)tau} = {1over k^2 - t} ,

This is the inverse of the operator $scriptstyle nabla^2 -t$ 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. So it satisfies the same equation as G with the same boundary conditions that determine the strength of the divergence at 0.

The 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. The 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. In the quantum field theory context, these are the paths of relativistically localized quanta in a formalism that follows the paths of individual particles. In the pure statistical context, these paths still appear by the mathematical correspondence with quantum fields, but their interpretation is less directly physical.

The integral representation immediately shows that G(r) is positive, since it is represented as a weighted sum of positive Gaussians. It 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 $e^\left\{-ttau\right\}=e^\left\{-tr^2\right\}$. The decay factor appropriate for position r is therefore $e^\left\{-sqrt t r\right\}$.

A heuristic approximation for G(r) is:


G(r) approx { e^{-sqrt t r} over r^{d-2} } ,

This is not an exact form, except in three dimensions, where interactions between paths become important. The exact forms in high dimensions are variants of Bessel functions.

Symanzik polymer interpretation

The interpretation of the correlations as fixed size quanta travelling along random walks gives a way of understanding why the critical dimension of the $H^4$ interaction is 4. The term H4 can be thought of as the square of the density of the random walkers at any point. In 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. Otherwise, the square of the density is just proportional to the density and only shifts the H2 coefficient by a constant. But the intersection probability of random walks depends on the dimension, and random walks in dimension higher than 4 don't intersect.

The fractal dimension of an ordinary random walk is 2. The number of balls of size ε required to cover the path increase as $1/epsilon^2$. Two 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. Kurt Symanzik argued that this implies that the critical Ising fluctuations in dimensions higher than 4 should be described by a free field. This argument eventually became a mathematical proof.

4-ε dimensions – renormalization group

The Ising model in four dimensions is described by a fluctuating field, but now the fluctuations are interacting. In the polymer representation, intersections of random walks are marginally possible. In the quantum field continuation, the quanta interact.

The negative logarithm of the probability of any field configuration H is the free energy function


F= int d^4 x {Z over 2} |nabla H|^2 + {tover 2} H^2 + {lambda over 4!} H^4 ,

The numerical factors are there to simplify the equations of motion. The goal is to understand the statistical fluctuations. Like any other non-quadratic path integral, The correlation functions have a Feynman expansion as particles travelling along random walks, splitting and rejoining at vertices. The interaction strength is parametrized by the classically dimensionless quantity λ.

Although dimensional analysis shows that both λ and Z dimensionless, this is misleading. The long wavelength statistical fluctuations are not exactly scale invariant, and only become scale invariant when the interaction strength vanishes.

The reason is that there is a cutoff used to define H, and the cutoff defines the shortest wavelength. Fluctuations of H at wavelengths near the cutoff can affect the longer-wavelength fluctuations. If 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. If the system is rescaled in such a way that the short wavelength cutoff remains fixed, the long-wavelength fluctuations are modified.

Wilson renormalization

A quick heuristic way of studying the scaling is to cut off the H wavenumbers at a point λ. Fourier modes of H with wavenumbers larger than λ are not allowed to fluctuate. A rescaling of length that make the whole system smaller increases all wavenumbers, and moves some fluctuations above the cutoff.

To restore the old cutoff, perform a partial integration over all the wavenumbers which used to be forbidden, but are now fluctuating. In 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.

Under 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. The change in t for infinitesimal b is 2bt. The other two coefficients are dimensionless and don't change at all.

The lowest order effect of integrating out can be calculated from the equations of motion:


nabla^2 H + t H = - {lambda over 6} H^3 ,

This equation is an identity inside any correlation function away from other insertions. After integrating out the modes with $Lambda< k < \left(1+b\right) Lambda$, it will be a slightly different identity.

Since the form of the equation will be preserved, to find the change in coefficients it is sufficient to analyze the change in the $H^3$ term. In a Feynman diagram expansion, the $H^3$ term in a correlation function inside a correlation has three dangling lines. Joining two of them at large wavenumber k gives a change $H^3$ with one dangling line, so proportional to H:


delta H^3 = 3H int_{Lambda<|k|<(1+b)Lambda} {d^4k over (2pi)^4} {1over (k^2 + t)}

The factor of 3 comes from the fact that the loop can be closed in three different ways.

The integral should be split into two parts:


int dk {1over k^2} - t int dk { 1over k^2(k^2 + t)} = ALambda^2 b + B b t ,

the first part is not proportional to t, and in the equation of motion it can be absorbed by a constant shift in t. It is caused by the fact that the $H^3$ term has a linear part. part is independent of the value of t. Only the second term, which varies from t to t, contributes to the critical scaling.

This new linear term adds to the first term on the left hand side, changing t by an amount proportional to t. The total change in t is the sum of the term from dimensional analysis and this second term from operator products:


delta t = Bigl(2 - {Blambda over 2} Bigl)b t So t is rescaled, but its dimension is anomalous, it is changed by an amount proportional to the value of λ.

But λ also changes. The change in lambda requires considering the lines splitting and then quickly rejoining. The lowest order process is one where one of the three lines from $H^3$ splits into three, which quickly joins with one of the other lines from the same vertex. The correction to the vertex is


delta lambda = - {3 lambda^2 over 2} int_k dk {1 over (k^2 + t)^2} = -{3lambda^2 over 2} b , The numerical factor is three times bigger because there is an extra factor of three in choosing which of the three new lines to contract.

So


delta lambda = - 3 B lambda^2 b ,

These two equations together define the renormalization group equations in four dimensions:


{dt over t} = Bigl(2 - {Blambda over 2}Bigr) b ,
$\left\{dlambda over lambda\right\} = \left\{-3 B lambda over 2\right\} b$
,

The coefficient B is determined by the formula


B b = int_{Lambda<|k|<(1+b)Lambda} {d^4kover (2pi)^4} {1 over k^4} , And is proportional to the area of a three dimensional sphere of radius λ, times the width of the integration region $bLambda$ divided by $Lambda^4$

B= (2 pi^2 Lambda^3) {1over (2pi)^4} { b Lambda} {1 over bLambda^4} = {1over 8pi^2} ,

In other dimensions, the constant B changes, but the same constant appears both in the t flow and in the coupling flow. The reason is that the derivative with respect to t of the closed loop with a single vertex is a closed loop with two vertices. This means that the only difference between the scaling of the coupling and the t is the combinatorial factors from joining and splitting.

Wilson Fisher point

To 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. In the language of Feynman graphs, the coupling doesn't change very much when the dimension is changed.

The process of continuing away from dimension four is not completely well defined without a prescription for how to do it. The prescription is only well defined on diagrams. It replaces the Schwinger representation in dimension 4 with the Schwinger representation in dimension $4-epsilon$ defined by:


G(x-y) = int dtau {1 over t^{dover 2}} e^{{x^2 over 2tau} + t tau} ,

In dimension $4-epsilon$, the coupling λ has positive scale dimension ε, and this must be added to the flow.


{dlambda over lambda} = epsilon - 3 B lambda ,

{dt over t} = 2 - lambda B ,

The coefficient B is dimension dependent, but it will cancel. The fixed point for λ is no longer zero, but at:


lambda = {epsilon over 3B} where the scale dimensions of t is altered by an amount $lambda B = epsilon/3$.

The magnetization exponent is altered proportionately to:


{1over 2} Bigl(1 - {epsilon over 3}Bigr)

which is .333 in 3 dimensions ($epsilon=1$) and .166 in 2 dimensions ($epsilon=2$). This is not so far off from the measured exponent .308 and the Onsager two dimensional exponent .125.

Low dimensions – block spins

In three dimensions, the perturbative series from the field theory is an expansion in a coupling constant λ which is not particularly small. The 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. In two dimensions, the perturbative expansion parameter is 2/3.

But renormalization can also be productively applied to the spins directly, without passing to an average field. Historically, this approach is due to Leo Kadanoff and predated the perturbative ε expansion.

The idea is to integrate out lattice spins iteratively, generating a flow in couplings. But now the couplings are lattice energy coefficients. The fact that a continuum description exists guarantees that this iteration will converge to a fixed point when the temperature is tuned to criticality.

Write the two dimensional Ising model with an infinite number of possible higher order interactions. To keep spin reflection symmetry, only even powers contribute:

E = sum_{ij} J_{ij} S_i S_j + sum J_{ijkl} S_i S_j S_k S_l ... ,

By translation invariance, $J_\left\{ij\right\}$ is only a function of i-j. By the accidental rotational symmetry, at large i and j its size only depends on the magnitude of the two dimensional vector i-j. The higher order coefficients are also similarly restricted.

The renormalization iteration divides the lattice into two parts – even spins and odd spins. The odd spins live on the odd-checkerboard lattice positions, and the even ones on the even-checkerboard. When 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.

The two possible values of the odd spins will be integrated out, by summing over both possible values. This will produce a new free energy function for the remaining even spins, with new adjusted couplings. The even spins are again in a lattice, with axes tilted at 45 degrees to the old ones. Unrotating the system restores the old configuration, but with new parameters. These parameters describe the interaction between spins at distances $scriptstyle sqrt\left\{2\right\}$ larger.

Starting from the Ising model and repeating this iteration eventually changes all the couplings. When the temperature is higher than critical, the couplings will converge to zero, since the spins at large distances are uncorrelated. But when the temperature is critical, there will be nonzero coefficients linking spins at all orders. The flow can be approximated by only considering the first few terms. This truncated flow will produce better and better approximations to the critical exponents when more terms are included.

The simplest approximation is to keep only the usual J term, and discard everything else. This will generate a flow in J, analogous to the flow in t at the fixed point of λ in the ε expansion.

To find the change in J, consider the four neighbors of an odd site. These are the only spins which interact with it. The multiplicative contribution to the partition function from the sum over the two values of the spin at the odd site is:


e^{J (N_+ - N_-)} + e^{J (N_- - N_+)} = 2 cosh(J (N_+ - N_-)) ,

where $N_+,N_-$ are the number of neighbors which are + and −. Ignoring the factor of 2, the free energy contribution from this odd site is:


`F = log(cosh(J (N_+ - N_-)))`
,

This includes nearest neighbor and next-nearest neighbor interactions, as expected, but also a four-spin interaction which is to be discarded. To truncate to nearest neighbor interactions, consider that the difference in energy between all spins the same and equal numbers + and - is:


`Delta F = ln(cosh(4J))`
, Where D is the dimension of the lattice, D is three. From nearest neighbor couplings, the difference in energy between all spins equal and staggered spins is 8J. The difference in energy between all spins equal and nonstaggered but net zero spin is 4J. Ignoring four-spin interactions, a reasonable truncation is the average of these two energies or 6J. Since each link will contribute to two odd spins, the right value to compare with the previous one is half that:

3J' = ln(cosh(4J)) , For small J, this quickly flows to zero coupling. Large J's flow to large couplings. The magnetization exponent is determined from the slope of the equation at the fixed point.

Variants of this method produce good numerical approximations for the critical exponents when many terms are included, in two and three dimensions.

References

• Barry M. McCoy and Tai Tsun Wu (1973), The Two-Dimensional Ising Model. Harvard University Press, Cambridge Massachusetts, ISBN 0674914406
• Ross Kindermann and J. Laurie Snell (1980), . American Mathematical Society. ISBN 0-8218-3381-2.
• Stephen G. Brush (1967), History of the Lenz-Ising Model Reviews of Modern Physics (American Physical Society) vol. 39, pp 883–893. (DOI: 10.1103/RevModPhys.39.883)