Ising model

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The Ising model (/ˈsɪŋ/; German: [ˈiːzɪŋ]), named after the physicist Ernst Ising, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic spins that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a lattice, allowing each spin to interact with its neighbors. The model allows the identification of phase transitions, as a simplified model of reality. The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition.[1]

The Ising model was invented by the physicist Wilhelm Lenz (1920), who gave it as a problem to his student Ernst Ising. The one-dimensional Ising model has no phase transition and was solved by Ising (1925) himself in his 1924 thesis.[2] The two-dimensional square lattice Ising model is much harder, and was given an analytic description much later, by Lars Onsager (1944). It is usually solved by a transfer-matrix method, although there exist different approaches, more related to quantum field theory.



Consider a set of lattice sites Λ, each with a set of adjacent sites (e.g. a graph) forming a d-dimensional lattice. For each lattice site j ∈ Λ there is a discrete variable σj such that σj  ∈ {+1, −1}. A spin configuration, σ = (σj)j∈Λ is an assignment of spin value to each lattice site.

For any two adjacent sites ij ∈ Λ one has an interaction Jij, and a site i ∈ Λ has an external magnetic field hi. The energy of a configuration σ is given by the Hamiltonian Function

H(\sigma) = - \sum_{<i~j>} J_{ij} \sigma_i \sigma_j -\mu \sum_{j} h_j\sigma_j

where the first sum is over pairs of adjacent spins (every pair is counted once). <ij> indicates that sites i and j are nearest neighbors. The magnetic moment is given by µ. Note that the sign in the second term of the Hamiltonian above should actually be positive because the electron's magnetic moment is antiparallel to its spin, but the negative term is used conventionally.[3] The configuration probability is given by the Boltzmann distribution with inverse temperature β ≥ 0:

P_\beta(\sigma) ={e^{-\beta H(\sigma)} \over Z_\beta},

where β = (kBT)−1

and the normalization constant

 Z_\beta = \sum_\sigma e^{-\beta H(\sigma)}

is the partition function. For a function f of the spins ("observable"), one denotes by

 \langle f \rangle_\beta = \sum_\sigma f(\sigma) P_\beta(\sigma) \,

the expectation (mean value) of f.

The configuration probabilities Pβ(σ) represent the probability of being in a state with configuration σ in equilibrium.


The minus sign on each term of the Hamiltonian function H(σ) is conventional. Using this sign convention, the Ising models can be classified according to the sign of the interaction: if, for all pairs i,j

Jij>0, the interaction is called ferromagnetic
Jij<0, the interaction is called antiferromagnetic
Jij=0, the spins are noninteracting

otherwise the system is called nonferromagnetic.

In a ferromagnetic Ising model, spins desire to be aligned: the configurations in which adjacent spins are of the same sign have higher probability. In an antiferromagnetic model, adjacent spins tend to have opposite signs.

The sign convention of H(σ) also explains how a spin site j interacts with the external field. Namely, the spin site wants to line up with the external field. If:

hj>0, the spin site j desires to line up in the positive direction
hj<0, the spin site j desires to line up in the negative direction
hj=0, there is no external influence on the spin site.

Ising models are often examined without an external field interacting with the lattice:


Another common simplification is to assume that all of the nearest neighbors <ij> have the same interaction strength. Then we can set Jij = J for all j in Λ such that:

H(\sigma) = -J\sum_{<i~j>}\sigma_i \sigma_j.

When the external field is everywhere zero, h = 0, the Ising model is symmetric under switching the value of the spin in all the lattice sites; a non zero field breaks this symmetry.


  2. If a spin at any given position i is 1, what is the probability that the spin at position j is also 1?

Basic properties and history[edit]

Visualization of the translation-invariant probability measure of the one-dimensional Ising model.

The most studied case of the Ising model is the translation-invariant ferromagnetic zero-field model on a d-dimensional lattice, namely, Λ = Zd, Jij = 1, h = 0.

In his 1924 PhD thesis, Ising solved the model for the 1D case, which can be thought of as a linear horizontal lattice where each site only interacts with its left and right neighbor. In one dimension, the solution admits no phase transition.[4] Namely, for any positive β, the correlations <σiσj> decay exponentially in |ij|:

 \langle \sigma_i \sigma_j \rangle_\beta \leq C \exp(-c(\beta) |i-j|),\,

and the system is disordered. On the basis of this result, he incorrectly concluded that this model does not exhibit phase behaviour in any dimension.

The Ising model undergoes a phase transition between an ordered and a disordered phase in 2 dimensions or more. Namely, the system is disordered for small β, whereas for large β the system exhibits ferromagnetic order:

 \langle \sigma_i \sigma_j \rangle_\beta \geq c(\beta) > 0.\,

This was first proven by Rudolf Peierls in 1936,[5] using what is now called a Peierls argument.

The Ising model on a two-dimensional square lattice with no magnetic field was analytically solved by Lars Onsager (1944). Onsager showed that the correlation functions and free energy of the Ising model are determined by a noninteracting lattice fermion. Onsager announced the formula for the spontaneous magnetization for the 2-dimensional model in 1949 but did not give a derivation. Yang (1952) gave the first published proof of this formula, using a limit formula for Fredholm determinants, proved in 1951 by Szegő in direct response to Onsager's work.[6]

Historical significance[edit]

One of Democritus' arguments in support of atomism was that atoms naturally explain the sharp phase boundaries observed in materials, as when ice melts to water or water turns to steam. His idea was that small changes in atomic-scale properties would lead to big changes in the aggregate 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.



No phase transitions in finite volume[edit]


  1. The partition function is a sum of e−βE over all configurations.
  2. the exponential function is everywhere analytic as a function β.
  3. the sum of analytic things is analytic.


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

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

Peierls droplets[edit]




M= {1\over N} \sum_{i=1}^{N} S_i.


  2. So for every configuration with magnetization M there is a configuration with magnetization −M with equal probability
  3. So the magnetization is zero.


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 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. For 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.

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

N(L) = (2d)^L



\sum_L (2d)^L (\varepsilon)^L

which goes to zero when ε is small.


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 (L − 2)/2 (the straight line) and (L/4)2 (the square box). The 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:

N(L)< 4^{2L}.


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

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


Kramers–Wannier duality[edit]

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[edit]


Monte Carlo Methods for Numerical Simulation[edit]


The Ising model can often be difficult to evaluate numerically if there are many states in the system. Consider an Ising Model with L lattice sites. Let:

  • L = |Λ|: the total number of sites on the lattice,
  • σj ∈ {−1, +1}: an individual spin site on the lattice, j = 1, ..., L,
  • S ∈ {−1, +1}L: state of the system.

Since every spin site has ±1 spin, there are 2L different states that are possible.[7] This motivates the reason for the Ising Model to be simulated using Monte Carlo Methods.[7]

The Hamiltonian Function that is commonly used representing the energy of the model when using Monte Carlo Methods is:

H(\sigma) = - J\sum_{<ij>}\sigma_i \sigma_j -h\sum_{j}\sigma_j.

Furthermore, 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. This leads us to the following energy equation for state σ:


Two such examples of estimates of interest are the specific heat or the magnetization of the magnet at a given temperature.[7]

The Metropolis Algorithm[edit]

Overview of Algorithm[edit]

The Metropolis–Hastings algorithm is the most commonly used Monte Carlo algorithm to calculate Ising Model estimations.[7] The 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 we are in state μ. It then uses acceptance probabilities A(μ, ν) so that detailed balance is satisfied. If the new state ν is accepted, then we move to that state and repeat with selecting a new state and deciding to accept it. If ν is not accepted then we stay in μ. This process is repeated until some stopping criteria is met, which for the Ising model is often when the lattice becomes ferromagnetic, meaning all of the sites point in the same direction.[7]

When implementing the algorithm, we must ensure that g(μ, ν) is selected such that ergodicity is met. In thermal equilibrium a system's energy only fluctuates within a small range.[7] This 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.[7] Furthermore, by using single- spin-flip dynamics, we can get from any state to any other state by flipping each site that differs between the two states one at a time.

The 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.[7] Thus, in a 1D-Ising Model, where each site has 2 neighbors (left, and right), the maximum difference in energy would be 4J.

Let c represent the lattice coordination number; the number of nearest neighbors that any lattice site has. We assume that all sites have the same number of neighbors due to periodic boundary conditions.[7]

Algorithm Specification[edit]

Specifically for the Ising model and using single-spin-flip dynamics, we can establish the following.

Since 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. The algorithm assumes that the selection probabilities are equal to the L states: g(μ, ν) = 1/L. Detailed balance tells us that the following equation must hold:

 \frac{P(\mu,\nu)}{P(\nu,\mu)} = \frac{g(\mu,\nu)A(\mu,\nu)}{g(\nu,\mu)A(\nu,\mu)} = \frac{A(\mu,\nu)}{A(\nu,\mu)} = \frac{P_\beta(\nu)}{P_\beta(\mu)} = \frac{\frac{1}{Z}e^{-\beta(H_\nu)}}{\frac{1}{Z}e^{-\beta(H_\mu)}} = e^{-\beta(H_\nu-H_\mu)}.

Thus, we want to select the acceptance probability for our algorithm to satisfy:


If Hν > Hμ then A(ν, μ) > A(μ, ν) Metropolis sets the larger of A(μ, ν) or A(ν, μ) to be 1. By this reasoning the acceptance algorithm is:[7]

 e^{-\beta(H_\nu-H_\mu)}, & \text{if }H_\nu-H_\mu>0 \\
 1, & \text{otherwise}.

The basic form of the algorithm is as follows:

  1. Pick a spin site using selection probability g(μ, ν) 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^{-\beta(H_\nu-H_\mu)}.
  5. Repeat.


Viewing the Ising Model as a Markov Chain[edit]

It is easy to view the Ising Model as a Markov Chain, as the immediate future state ν transition probability: Pβ(ν) only depends on the present state μ. The Metropolis Algorithm is actually a version of 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 dipole. [8] Furthermore, since the energy equation Hσ change only depending on the nearest neighbors interaction influence 'J', we can view the Ising model as a form of Voter Model

One dimension[edit]

The thermodynamic limit exists as soon as the interaction decay is  J_{ij} \sim |i-j|^{-\alpha} with α > 1.[9]

  • In the case of ferromagnetic interaction  J_{ij} \sim |i-j|^{-\alpha} with 1< α < 2 Dyson proved, by comparison with the hierarchical case, that there is phase transition at small enough temperature.[10]
  • In the case of ferromagnetic interaction  J_{ij} \sim |i-j|^{-2} , Fröhlich and Spencer proved that there is phase transition at small enough temperature (in contrast with the hierarchical case).[11]
  • In the case of interaction  J_{ij} \sim |i-j|^{-\alpha} with α > 2 (that includes the case of finite range interactions) there is no phase transition at any positive temperature (i.e. finite β) since the free energy is analytic in the thermodynamic parameters.[9]
  • In the case of nearest neighbor interactions, E. Ising provided an exact solution of the model. At any positive temperature (i.e. finite β) the free energy is analytic in the thermodynamics parameters and the truncated two-point spin correlation decays exponentially fast. At zero temperature, (i.e. infinite β), there is a second order phase transition: the free energy is infinite and the truncated two point spin correlation does not decay (remains constant). Therefore T = 0 is the critical temperature of this case. Scaling formulas are satisfied.[12]

Ising's exact solution[edit]

In the nearest neighbor case (with periodic or free boundary conditions) an exact solution is available. The energy of the one-dimensional Ising model on a lattice of L sites with periodic boundary conditions is

H(\sigma)=-J\sum_{i=1,\ldots,L} \sigma_i \sigma_{i+1} - h \sum_i \sigma_i

where 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. Then the free energy is

f(\beta, h)=-\lim_{L\to \infty} \frac{1}{\beta L} \ln (Z(\beta))=-\frac{1}{\beta} \ln\left(e^{\beta J} \cosh \beta h+\sqrt{e^{2\beta J}(\sinh\beta h)^2+e^{-2\beta J}}\right)

and the spin-spin correlation is

\langle \sigma_i \sigma_j\rangle-\langle \sigma_i \rangle\langle\sigma_j\rangle=C(\beta)e^{-c(\beta)|i-j|}

where C(β) and c(β) are positive functions for T > 0. For T → 0, though, the inverse correlation length, c(β), vanishes.


The proof of this result is a simple computation.

If h = 0, it is very easy to obtain the free energy in the case of free boundary condition, i.e. when


Then the model factorizes under the change of variables

\sigma'_j=\sigma_j\sigma_{j-1} \qquad j\ge 2.

That gives

 Z(\beta) =\sum_{\sigma_1,\ldots, \sigma_L} e^{\beta J\sigma_1\sigma_2}\; e^{\beta J\sigma_2\sigma_3}\; \cdots e^{\beta J\sigma_{L-1}\sigma_L}= 2\prod_{j=2}^L \sum_{\sigma'_j}  e^{\beta J\sigma'_j} =2\left[ e^{\beta J}+e^{-\beta J}\right]^{L-1}.

Therefore the free energy is

f(\beta,0)=-\frac{1}{\beta } \ln\left[e^{\beta J}+ e^{-\beta J}\right].

With the same change of variables

\langle \sigma_{j}\sigma_{j+N}\rangle=\left[\frac{e^{\beta J}- e^{-\beta J}}{e^{\beta J}+ e^{-\beta J}}\right]^N

hence it decays exponentially as soon as T ≠ 0; but for T = 0, i.e. in the limit β → ∞ there is no decay.

If h ≠ 0 we need the transfer matrix method. For the periodic boundary conditions case is the following. The partition function is

Z(\beta)=\sum_{\sigma_1,\ldots, \sigma_L} e^{\beta h \sigma_1}e^{\beta J\sigma_1\sigma_2}\; e^{\beta h \sigma_2}e^{\beta J\sigma_2\sigma_3}\; \cdots e^{\beta h \sigma_L}e^{\beta J\sigma_L\sigma_1}=  \sum_{\sigma_1,\ldots, \sigma_L} V_{\sigma_1,\sigma_2}V_{\sigma_2,\sigma_3}\cdots V_{\sigma_L,\sigma_1}.

The coefficients V_{\sigma, \sigma'}'s can be seen as the entries of a matrix. There are different possible choices: a convenient one (because the matrix is symmetric) is

 V_{\sigma, \sigma'} = e^{\frac{\beta h}{2} \sigma} e^{\beta J\sigma\sigma'} e^{\frac{\beta h}{2} \sigma'}


V=\begin{bmatrix} e^{\beta(h+J)}&e^{-\beta J}\\ e^{-\beta J}&e^{-\beta(h-J)} \end{bmatrix}.

In matrix formalism

 Z(\beta)={\rm Tr} V^L= \lambda_1^L + \lambda_2^L= \lambda_1^L\left[1+ \left(\frac{\lambda_2}{\lambda_1}\right)^L\right]

where λ1 is the highest eigenvalue of V, while λ2 is the other eigenvalue:


and |λ2| < λ1. This gives the formula of the free energy.


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.


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



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


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

The logarithm divided by L is the free energy density:

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

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

Two dimensions[edit]

  • In the ferromagnetic case there is a phase transition: at low temperature, Peierls argument proves positive magnetization for the nearest neighbor case and then, by Griffiths inequality, also when longer range interactions are added; while, at high temperature, cluster expansion gives analyticity of the thermodynamic functions.
  • In the nearest-neighbor case, the free energy has been exactly computed by Onsager, through the equivalence of the model with free fermions on lattice. The spin-spin correlation functions has been computed by McCoy and Wu.

Onsager's exact solution[edit]

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. Onsager obtained the following analytical expression for the magnetization as a function of temperature:

M = \left(1-\left[\sinh\left(\log(1+\sqrt{2})\frac{T_c}{T}\right)\right]^{-4}\right)^{\frac{1}{8}}


T_c = \frac{2J}{k_B\log(1+\sqrt{2})}.

Transfer matrix[edit]


\sum_S \exp\biggl(\sum_{ij} S_{i,j} S_{i,j+1} + S_{i,j} S_{i+1,j}\biggr).



 U = e^{i H \Delta t}


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



T_{C_1 C_2}.

The configuration 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 C1 and C2, 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:

|A\rangle = \sum_S A(S) |S\rangle

where each basis vector |S\rangle is a spin configuration of a one-dimensional Ising model.


Z= \mathrm{tr}(T^N).


T in terms of Pauli matrices[edit]






T can be written in terms of these:

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


Spin flip creation and annihilation operators[edit]

Just as in the one-dimensional case, we will shift attention from the spins to the spin-flips. The σz 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. \,


  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 {\psi^\dagger}_i \psi_{i+1} + D^* {\psi^\dagger}_i \psi_{i-1} + C\psi_i \psi_{i+1} + C^* {\psi^\dagger}_i {\psi^\dagger}_{i+1}.


Carrying out the diagonalization produces the Onsager free energy.

Onsager's formula for spontaneous magnetization[edit]

Onsager (1949) obtained the following formula for the spontaneous magnetization M of a two-dimensional Ising ferromagnet

M = \left(1 - \left[\sinh 2\beta E_1 \sinh 2\beta E_2\right]^{-2}\right)^{\frac{1}{8}}.

A complete derivation was later given by Yang (1952), using Szegő's limit formula for Toeplitz determinants, proved in 1951 in response to Onsager's work.[13][6] In this formula the total energy of Onsager's lattice model is given by

E=-E_1 \sum_{j,k} S_{j,k}S_{j,k+1} - E_2\sum_{j,k} S_{j,k} S_{j+1,k}

and β−1 = kT where k is Boltzmann's constant and T is the absolute temperature.

Three and four dimensions[edit]

In three dimensions, the Ising model was shown to have a representation in terms of non-interacting Fermionic lattice strings by Alexander Polyakov. In dimensions near four, the critical behavior of the model is understood to correspond to the renormalization behavior of the scalar phi-4 theory (see Kenneth Wilson).

More than four dimensions[edit]


Local field[edit]





\beta F  =  \int d^dx \left[ A H^2 + \sum_{i=1}^{d} Z_i (\partial_i H)^2 + \lambda H^4 ... \right].

On a square lattice, symmetries guarantee that the coefficients Zi 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

Z_{ij} \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 Zij = δij. 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 βF is a function of a slowly spatially varying field. The probability of any field configuration is:

P(H) \propto e^{ - \int d^dx \left[ AH^2 + Z |\nabla H|^2 + \lambda H^4 \right]}.

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

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


Z = \int DH e^{ \int d^dx \left[ A H^2 + Z  |\nabla H|^2 + \lambda H^4 \right]}

Dimensional analysis[edit]



F = \int d^dx A H^2.




F= \int d^dx  \left[ t H^2 + \lambda H^4  + Z (\nabla H)^2 \right],

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 H4 term or the (∇H)2 term to 1.


To find the magnetization, fix the scaling of H so that λ is one. Now the field H has dimension −d/4, so that H4ddx is dimensionless, and Z has dimension 2−d/2. In this scaling, the gradient term is only important at long distances for d ≤ 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 H4 split in the following way:

H(x)^4 = -\langle H(x)^2\rangle^2  + 2\langle H(x)^2\rangle H(x)^2 + \left(H(x)^2 - \langle H(x)^2\rangle\right)^2



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




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 H2 term is 2, while the scale dimension of the H4 term is 4−d. For d < 4, the H4 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 H4 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.


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

valid when xy is large. The function G(xy) 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(xy) 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})


The critical two-point function[edit]


                 &&\left(-\nabla_x^2 + t\right) \langle H(x)H(y) \rangle &= 0 \\
  \rightarrow {} &&                                \nabla^2 G(x) + tG(x) &= 0



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[edit]


\nabla^2 G  + t G = 0 \to {1 \over r^{d - 1}} {d \over dr} \left( r^{d-1} {dG \over dr} \right) + t G(r) = 0

For 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.


G(x) = \int d\tau  {1 \over \left(\sqrt{2\pi\tau}\right)^d} e^{-{x^2 \over 4\tau} - t\tau}


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

This is the inverse of the operator ∇2t 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 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^{-t\tau} = e^{-tr^2}. The decay factor appropriate for position r is therefore e^{-\sqrt t r}.

A heuristic approximation for G(r) is:

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


Symanzik polymer interpretation[edit]



4−ε dimensions – renormalization group[edit]



F= \int d^4 x  \left[ {Z \over 2} |\nabla H|^2 + {t\over 2} H^2  + {\lambda \over 4!} H^4 \right] \,




Wilson renormalization[edit]



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.


\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 Λ < k < (1+b)Λ, 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 H3 term. In a Feynman diagram expansion, the H3 term in a correlation function inside a correlation has three dangling lines. Joining two of them at large wavenumber k gives a change H3 with one dangling line, so proportional to H:

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


The integral should be split into two parts:

\int dk {1 \over k^2} - t \int dk { 1\over k^2(k^2 + t)} = A\Lambda^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 H3 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.


\delta t = \left(2 - {B\lambda \over 2} \right)b t


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 H3 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} = -{3\lambda^2 \over 2} b


\delta \lambda = - 3 B \lambda^2 b


              {dt \over t} &= \left(2  - {B\lambda \over 2}\right) b \\
  {d\lambda \over \lambda} &= {-3 B \lambda \over 2} b

The coefficient B is determined by the formula

B b  = \int_{\Lambda<|k|<(1+b)\Lambda} {d^4k\over (2\pi)^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 bΛ divided by Λ4

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


Wilson–Fisher point[edit]

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.


 G(x-y) = \int d\tau {1 \over t^{d\over 2}} e^{{x^2 \over 2\tau} + t \tau}

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

{d\lambda \over \lambda} = \varepsilon - 3 B \lambda
{dt \over t} = 2 - \lambda  B


\lambda = {\varepsilon \over 3B}

where the scale dimensions of t is altered by an amount λB = ε/3.

The magnetization exponent is altered proportionately to:

\tfrac{1}{2} \left( 1 - {\varepsilon \over 3}\right)

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

Infinite dimensions – mean field[edit]


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 + minus 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 N → ∞ 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.

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

so that

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

The solutions to this equation are the possible consistent mean fields. For βJ < 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.



For βJ = 1 + ε, 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+\varepsilon)H - {(1+\varepsilon)^3H^3\over 3}


H = \sqrt{3\varepsilon}



H = {\partial (\beta F) \over \partial h} = (1+A\varepsilon) H + B H^3 + \cdots

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 βF 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 accurately. 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 \varepsilon^{0.308}


This is also true in two dimensions, where

H \propto \varepsilon^{0.125}


Low dimensions – block spins[edit]




Migdal-Kadanoff renormalization[edit]

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 \ldots.

By translation invariance,Jij 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 ij. The higher order coefficients are also similarly restricted.






 e^{J (N_+ - N_-)} + e^{J (N_- - N_+)} = 2 \cosh(J[N_+ - N_-])

where N± is the number of neighbors which are ±. Ignoring the factor of 2, the free energy contribution from this odd site is:

 F = \log(\cosh[J(N_+ - N_-)]).


 \Delta F = \ln(\cosh[4J]).


3J' =  \ln(\cosh[4J]).








Lattice gas[edit]


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 Bi, 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 −4JBiBj 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 μN. The probability cost of one more atom is a factor of exp(−βμ).

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


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 = (zJ−μ)/2, where z is the number of neighbors.

Application to neuroscience[edit]


Following the general approach of Jaynes,[15][16] a recent interpretation of Schneidman, Berry, Segev and Bialek,[17] 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


E= - \tfrac{1}{2} \sum_{ij} J_{ij} S_i S_j - \sum_i h_i S_i

where J_{ij} are not restricted to neighbors. Note that this generalization of Ising model is sometimes called the quadratic exponential binary distribution in statistics. 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.

Spin Glasses[edit]

With the Ising model the so-called spin glasses can also be described, by the usual Hamiltonian \hat H=-\frac{1}{2}\,\sum J_{i,k}\,S_i\,S_k, where the S-variables describe the Ising spins, while the Ji,k are taken from a random distribution. For spin glasses a typical distribution chooses antiferromagnetic bonds with probability p and ferromagnetic bonds with probability 1−p. These bonds stay fixed or "quenched" even in the presence of thermal fluctuations. When p=0 we have the original Ising model. This system deserves interest in its own; particularly one has "non-ergodic" properties leading to strange relaxation behaviour.| Much attention has been also attracted by the related bond and site dilute Ising model, especially in two dimensions, leading to intriguing critical behavior.[18]

See also[edit]


  1. ^ See Gallavotti (1999), Chapters VI-VII.
  2. ^ Ernst Ising, Contribution to the Theory of Ferromagnetism
  3. ^ See Baierlein (1999), Chapter 16.
  4. ^
  5. ^
  6. ^ a b Montroll, Potts & Ward 1963, pp. 308–309
  7. ^ a b c d e f g h i j Newman MEJ, Barkema GT, "Monte Carlo Methods in Statistical Physics, Clarendon Press, 1999
  8. ^ Teif V.B. (2007). "General transfer matrix formalism to calculate DNA-protein-drug binding in gene regulation". Nucleic Acids Res. 35: e80. doi:10.1093/nar/gkm268. 
  9. ^ a b Ruelle (1969). Statistical Mechanics:Rigorous Results. New York: W.A. Benjamin Inc. 
  10. ^ Dyson, F.J. (1969). "Existence of a phase-transition in a one-dimensional Ising ferromagnet". Comm. Math. Phys. 12: 91–107. Bibcode:1969CMaPh..12...91D. doi:10.1007/BF01645907. 
  11. ^ Fröhlich, J.; Spencer, T. (1982). "The phase transition in the one-dimensional Ising model with 1/r 2 interaction energy.". Comm. Math. Phys. 84. 
  12. ^ Baxter, Rodney J. (1982), Exactly solved models in statistical mechanics*, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-083180-7, MR 690578 
  13. ^ McCoy & Wu 1973, p. 245
  14. ^ J. J. Hopfield (1982), "Neural networks and physical systems with emergent collective computational abilities", Proceedings of the National Academy of Sciences of the USA 79 (8): 2554–2558, Bibcode:1982PNAS...79.2554H, doi:10.1073/pnas.79.8.2554, PMC 346238, PMID 6953413. 
  15. ^ Jaynes, E. T. (1957), "Information Theory and Statistical Mechanics", Physical Review 106 (4): 620, Bibcode:1957PhRv..106..620J, doi:10.1103/PhysRev.106.620. 
  16. ^ Jaynes, Edwin T. (1957), "Information Theory and Statistical Mechanics II", Physical Review 108 (2): 71, Bibcode:1957PhRv..108..171J, doi:10.1103/PhysRev.108.171. 
  17. ^ Elad Schneidman, Michael J. Berry, Ronen Segev and William Bialek (2006), "Weak pairwise correlations imply strongly correlated network states in a neural population", Nature 440 (7087): 1007–1012, arXiv:q-bio/0512013, Bibcode:2006Natur.440.1007S, doi:10.1038/nature04701, PMC 1785327, PMID 16625187. 
  18. ^ J-S Wang, W Selke, VB Andreichenko, and VS Dotsenko (1990), "The critical behaviour of the two-dimensional dilute model", Physica A 164: 221–239, Bibcode:1990PhyA..164..221W, doi:10.1016/0378-4371(90)90196-Y 


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