Ising model
Encyclopedia
The Ising model (ˈaɪsɪŋ, iːzɪŋ) is a mathematical model
of ferromagnetism
in statistical mechanics
. The model consists of discrete variables called spins that can be in one of two states (+1 or −1). The spins are arranged in a graph (usually, a lattice
), and each spin interacts with its nearest neighbors. The goal is to find phase transition
s in the Ising model, as a simplified model of phase transitions in real substances. The two-dimensional square lattice Ising model is one of the simplest statistical models to show a phase transition.
The Ising model was invented by the physicist who gave it as a problem to his student Ernst Ising
after whom it is named. The one-dimensional Ising model has no phase transition and was solved by himself. The two-dimensional square lattice Ising model is much harder, and was given an analytic description much later, by . It is usually solved by a transfer-matrix method
, although there exist different approaches, more related to quantum field theory
.
In dimensions greater than four, the phase transition of the Ising model is described by mean field theory
.
showed that computing the free energy
of an arbitrary subgraph of an Ising model on a lattice of dimension three or more is computationally intractable so that no method of approximation will allow computation of the thermodynamic properties of arbitrary subgraphs in dimensions higher than two.
For any two adjacent sites i, j ∈Λ one has an interaction Jij, and for any i ∈ Λ one has an external field hi. The energy of a configuration σ is given by
where the first sum is over pairs of adjacent spins (every pair is counted once). The configuration probability is given by the Boltzmann distribution
with inverse temperature β ≥0:
where the normalization constant
is the partition function
. For a function f of the spins ("observable"), one denotes by
the expectation (mean value) of f.
otherwise the system is called nonferromagnetic.
In a ferromagnetic Ising model, spins tend 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.
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.
The interesting statistical questions to ask are all in the limit of large numbers of spins:
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. In one dimension, the solution admits no phase transition
. Namely, for any positive β, the correlations <σiσj> decay exponentially in |i−j|:
and the system is disordered. On the basis of this result, he incorrectly concluded that this model does not exhibit phase behaviour in any dimension.
The Ising model undergoes a phase transition
between an ordered and a disordered phase in 2 dimensions or more. Namely, the system is disordered for small β, whereas for large β the system exhibits ferromagnetic order:
This was first proved by Rudolph Peierls in 1933, using what is now called a Peierls argument.
The Ising model on a two dimensional square lattice with no magnetic field was analytically solved by . Onsager showed that the correlation function
s 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. gave the first published proof of this formula, using a limit formula
for Fredholm determinant
s, proved in 1951 by Szegő
in direct response to Onsager's work.
' 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.
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 describes room temperature gases. But classical statistical mechanics did not account for all of the properties of liquids and solids, nor of gases at low temperature.
Once modern quantum mechanics
was formulated, atomism was no longer in conflict with experiment, but this did not lead to a universal acceptance of statistical mechanics, which went beyond atomism. Josiah Willard Gibbs
had given a complete formalism to reproduce the laws of thermodynamics from the laws of mechanics. But many faulty arguments survived from the 19th century, when statistical mechanics was considered dubious. The lapses in intuition mostly stemmed from the fact that the limit of an infinite statistical system has many zero-one law
s which are absent in finite systems: an infinitesimal change in a parameter can lead to big differences in the overall, aggregate behavior, as Democritus expected.
could never describe a phase transition, based on the following argument:
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 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.
To do this, he compared the high-temperature and low temperature limits. At infinite temperature,
, 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:
Mathematical models in physics
Mathematical models are of great importance in physics. Physical theories are almost invariably expressed using mathematical models, and the mathematics involved is generally more complicated than in the other sciences. Different mathematical models use different geometries that are not necessarily...
of ferromagnetism
Ferromagnetism
Ferromagnetism is the basic mechanism by which certain materials form permanent magnets, or are attracted to magnets. In physics, several different types of magnetism are distinguished...
in statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
. The model consists of discrete variables called spins that can be in one of two states (+1 or −1). The spins are arranged in a graph (usually, a lattice
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
), and each spin interacts with its nearest neighbors. The goal is to find phase transition
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
s in the Ising model, as a simplified model of phase transitions in real substances. The two-dimensional square lattice Ising model is one of the simplest statistical models to show a phase transition.
The Ising model was invented by the physicist who gave it as a problem to his student Ernst Ising
Ernst Ising
Ernst Ising was a German physicist, who is best remembered for the development of the Ising model. He was a professor of physics at Bradley University until his retirement in 1976.-Life:Ernst Ising was born in Cologne in 1900...
after whom it is named. The one-dimensional Ising model has no phase transition and was solved by himself. The two-dimensional square lattice Ising model is much harder, and was given an analytic description much later, by . It is usually solved by a transfer-matrix method
Transfer-matrix method
In physics and mathematics, the transfer-matrix method is a general technique for solving problems in statistical mechanics.The basic idea is to write the partition function in the form...
, although there exist different approaches, more related to quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
.
In dimensions greater than four, the phase transition of the Ising model is described by mean field theory
Mean field theory
Mean field theory is a method to analyse physical systems with multiple bodies. A many-body system with interactions is generally very difficult to solve exactly, except for extremely simple cases . The n-body system is replaced by a 1-body problem with a chosen good external field...
.
showed that computing the free energy
Thermodynamic free energy
The thermodynamic free energy is the amount of work that a thermodynamic system can perform. The concept is useful in the thermodynamics of chemical or thermal processes in engineering and science. The free energy is the internal energy of a system less the amount of energy that cannot be used to...
of an arbitrary subgraph of an Ising model on a lattice of dimension three or more is computationally intractable so that no method of approximation will allow computation of the thermodynamic properties of arbitrary subgraphs in dimensions higher than two.
Definition
Given a graph Λ (for example, a d-dimensional lattice), per each lattice site j ∈ Λ there is a discrete variable σj that can be +1 or −1. A spin configuration, σ = (σj)j∈Λ is an assignment of spin value to each lattice site.For any two adjacent sites i, j ∈Λ one has an interaction Jij, and for any i ∈ Λ one has an external field hi. The energy of a configuration σ is given by
where the first sum is over pairs of adjacent spins (every pair is counted once). The configuration probability is given by the Boltzmann distribution
Boltzmann distribution
In chemistry, physics, and mathematics, the Boltzmann distribution is a certain distribution function or probability measure for the distribution of the states of a system. It underpins the concept of the canonical ensemble, providing its underlying distribution...
with inverse temperature β ≥0:
where the normalization constant
is the partition function
Partition function (statistical mechanics)
Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...
. For a function f of the spins ("observable"), one denotes by
the expectation (mean value) of f.
Discussion
Ising models can be classified according to the signum 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 tend 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.
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.
The interesting statistical questions to ask are all in the limit of large numbers of spins:
- In a typical configuration, are most of the spins +1 or −1, or are they split equally?
- If a spin at any given position i is 1, what is the probability that the spin at position j is also 1?
- If β is changed, is there a phase transition?
- On a lattice, what is the fractal dimension of the shape of a large cluster of +1 spins?
Basic properties and history
In his 1924 PhD thesis, Ising solved the model for the 1D case. In one dimension, the solution admits no phase transition
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
. Namely, for any positive β, the correlations <σiσj> decay exponentially in |i−j|:
and the system is disordered. On the basis of this result, he incorrectly concluded that this model does not exhibit phase behaviour in any dimension.
The Ising model undergoes a phase transition
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
between an ordered and a disordered phase in 2 dimensions or more. Namely, the system is disordered for small β, whereas for large β the system exhibits ferromagnetic order:
This was first proved by Rudolph Peierls in 1933, using what is now called a Peierls argument.
The Ising model on a two dimensional square lattice with no magnetic field was analytically solved by . Onsager showed that the correlation function
Correlation function
A correlation function is the correlation between random variables at two different points in space or time, usually as a function of the spatial or temporal distance between the points...
s and free energy
Thermodynamic free energy
The thermodynamic free energy is the amount of work that a thermodynamic system can perform. The concept is useful in the thermodynamics of chemical or thermal processes in engineering and science. The free energy is the internal energy of a system less the amount of energy that cannot be used to...
of the Ising model are determined by a noninteracting lattice fermion. Onsager announced the formula for the spontaneous magnetization
Spontaneous magnetization
Spontaneous magnetization is the term used to describe the appearance of an ordered spin state at zero applied magnetic field in a ferromagnetic or ferrimagnetic material below a critical point called the Curie temperature or .-Overview:...
for the 2-dimensional model in 1949 but did not give a derivation. gave the first published proof of this formula, using a limit formula
Szegő limit theorems
In mathematical analysis, the Szegő limit theorems describe the asymptotic behaviour of the determinants of large Toeplitz matrices. They were first proved by Gábor Szegő.-Notation:...
for Fredholm determinant
Fredholm determinant
In mathematics, the Fredholm determinant is a complex-valued function which generalizes the determinant of a matrix. It is defined for bounded operators on a Hilbert space which differ from the identity operator by a trace-class operator...
s, proved in 1951 by Szegő
Gábor Szego
Gábor Szegő was a Hungarian mathematician. He was one of the foremost analysts of his generation and made fundamental contributions to the theory of Toeplitz matrices and orthogonal polynomials.-Life:...
in direct response to Onsager's work.
Historical significance
One of DemocritusDemocritus
Democritus was an Ancient Greek philosopher born in Abdera, Thrace, Greece. He was an influential pre-Socratic philosopher and pupil of Leucippus, who formulated an atomic theory for the cosmos....
' arguments in support of atomism
Atomism
Atomism is a natural philosophy that developed in several ancient traditions. The atomists theorized that the natural world consists of two fundamental parts: indivisible atoms and empty void.According to Aristotle, atoms are indestructible and immutable and there are an infinite variety of shapes...
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.
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
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
and Ludwig Boltzmann
Ludwig Boltzmann
Ludwig Eduard Boltzmann was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics...
, applied Hamilton's formulation of Newton's laws to large systems, and found that the statistical behavior
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
of the atoms correctly describes room temperature gases. But classical statistical mechanics did not account for all of the properties of liquids and solids, nor of gases at low temperature.
Once modern quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
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
Josiah Willard Gibbs
Josiah Willard Gibbs was an American theoretical physicist, chemist, and mathematician. He devised much of the theoretical foundation for chemical thermodynamics as well as physical chemistry. As a mathematician, he invented vector analysis . Yale University awarded Gibbs the first American Ph.D...
had given a complete formalism to reproduce the laws of thermodynamics from the laws of mechanics. But many faulty arguments survived from the 19th century, when statistical mechanics was considered dubious. The lapses in intuition mostly stemmed from the fact that the limit of an infinite statistical system has many zero-one law
Zero-one law
In probability theory, a zero-one law is a result that states that an event must have probability 0 or 1 and no intermediate value. Sometimes, the statement is that the limit of certain probabilities must be 0 or 1.It may refer to:...
s which are absent in finite systems: an infinitesimal change in a parameter can lead to big differences in the overall, aggregate behavior, as Democritus expected.
No phase transitions in finite volume
In the early part of the twentieth century, some believed that the partition functionPartition function (statistical mechanics)
Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...
could never describe a phase transition, based on the following argument:
- The partition function is a sum of
over all configurations. - the exponential function is everywhere analyticAnalytic functionIn mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
as a function
. - 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 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
Rudolf Peierls
Sir Rudolf Ernst Peierls, CBE was a German-born British physicist. Rudolf Peierls had a major role in Britain's nuclear program, but he also had a role in many modern sciences...
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,
, 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 snowNoise (video)
Noise, in analog video and television, is a random dot pattern of static displayed when no transmission signal is obtained by the antenna receiver of television set and other display devices...
. 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:
-
A bogus argument analogous to the argument in the last section now establishes that the magnetization in the Ising model is always zero.- Every configurations of spin has equal energy to the configuration with all spins flipped.
- So for every configuration with magnetization M there is a configuration with magnetization -M with equal probability
- 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
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 
. For two spins separated by distance L, the amount of correlation goes as 
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:-
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 :-
which goes to zero when
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
(the straight line) and 
(the square box). The probability cost for introducing a droplet is the factor:
-
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:-
So that the total spin contribution from droplets, even overcounting by allowing each site to have a separate droplet, is bounded above by:
-
which goes to zero at large
. For 
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 sectorSuperselection sectorIn Quantum mechanics, superselection extends the concept of selection rules.Superselection rules are postulated rules forbidding the preparation of quantum states that exhibit coherence between eigenstates of certain observables....
s, 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.
Numerical simulations
To actually generate configurations using this probability distribution is conceptually
easiest using the Metropolis algorithm:- Pick a spin at random and calculate the contribution to the energy involving this spin.
- Flip the value of the spin and calculate the new contribution.
- If the new energy is less, keep the flipped value.
- If the new energy is more, only keep with probability
- Repeat.
The change in energy
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.
One dimension
The thermodynamic limit exists as soon as the interaction decay is
with 
.
- In the case of ferromagnetic interaction
with 
Dyson proved, by comparison with the hierarchical case, that there is phase transition at small enough temperature.
- In the case of ferromagnetic interaction
, Fröhlich and Spencer proved that there is phase transition at small enough temperature (in contrast with the hierarchical case).
- In the case of interaction
with 
(that includes the case of finite range interactions) there is no phase transition at any positive temperature (i.e. finite 
) since the free energyThermodynamic free energyThe thermodynamic free energy is the amount of work that a thermodynamic system can perform. The concept is useful in the thermodynamics of chemical or thermal processes in engineering and science. The free energy is the internal energy of a system less the amount of energy that cannot be used to...
is analytic in the thermodynamic parameters.
- In the case of nearest neighbor interactions, E. Ising provided an exact solution of the model. At any positive temperature (i.e. finite
) 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 
), 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 
is the critical temperature of this case. Scaling formulas are satisfied.
Ising's exact solution
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
sites with periodic boundary conditions is
where
and 
can be any number. Then the
free energyThermodynamic free energyThe thermodynamic free energy is the amount of work that a thermodynamic system can perform. The concept is useful in the thermodynamics of chemical or thermal processes in engineering and science. The free energy is the internal energy of a system less the amount of energy that cannot be used to...
is
and the spin-spin correlation is
where
and 
are positive functions for 
. For 
, though, the inverse correlation length, 
, vanishes.
Proof
The proof of this result is a simple computation.
If
, 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
That gives
Therefore the free energy is
With the same change of variables
hence it decays exponentially as soon as
; but for 
, i.e. in the limit 
there is no decay.
If
we need the transfer matrix method. For the periodic boundary conditions case is the following. The partition function is
The coefficients
's can be seen as the entries of a matrix. There are different possible choices: a convenient one (because the matrix is symmetric) is
In matrix formalism
where
is the highest eigenvalue of 
, while 
is the other eigenvalue:
and
.
This gives the formula of the free energy.
Comments
The energy of the lowest state is
, 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 designate the number of sign changes in a configuration as
, the difference in energy from the lowest energy state is 
. Since the energy is additive in the number of flips, the probability 
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:
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
. 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
, so the correlations fall off exponentially.
The partition functionPartition function (statistical mechanics)Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...
is the volume of configurations, each configuration weighted by its Boltzmann weight. Since each configuration is described by the sign-changes, the Partition function factorizes:
The logarithm divided by
is the free energy density:
which is analyticAnalytic functionIn mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
away from
. A sign of a phase transitionPhase transitionA phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
is a non-analytic free energy, so the one dimensional model does not have a phase transition.
Two dimensions
- 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 inequalityGriffiths inequalityIn statistical mechanics, the Griffiths inequality , named after Robert B. Griffiths, is a correlation inequality for ferromagnetic spin systems...
, also when longer range interactions are added; while, at high temperature, cluster expansionCluster expansionIn statistical mechanics, the cluster expansion is a power series expansion of the partition function of a statistical field theory around a model that is a union of non-interacting 0-dimensional field theories. Cluster expansions originated in the work of...
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
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:
where
Transfer matrix
Start with an analogy with quantum mechanics. The Ising model on a long periodic lattice has a partition function
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 integralPath integralPath integral may refer to:* Line integral, the integral of a function along a curve* Functional integration, the integral of a functional over a space of curves...
, 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
and time 
:
The product of the U matrices, one after the other, is the total time evolution operator, which is the path integral we started with.
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 matrixTransfer matrixThe transfer matrix is a formulation in terms of a block-Toeplitz matrix of the two-scale equation, which characterizes refinable functions. Refinable functions play an important role in wavelet theory and finite element theory....
:
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 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:
where each basis vector
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:
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:
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:
T can be written in terms of these:
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
term in T counts the number of spin flips, which we can write in terms of spin-flip creation and annihilation operators:
The first term flips a spin, so depending on the basis state it either:- moves a spin-flip one unit to the right
- moves a spin-flip one unit to the left
- produces two spin-flips on neighboring sites
- destroys two spin-flips on neighboring sites.
Writing this out in terms of creation and annihilation operators:
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.
Onsager's formula for spontaneous magnetization
obtained the following formula for the spontaneous magnetizationSpontaneous magnetizationSpontaneous magnetization is the term used to describe the appearance of an ordered spin state at zero applied magnetic field in a ferromagnetic or ferrimagnetic material below a critical point called the Curie temperature or .-Overview:...
M of a two-dimensional Ising ferromagnet
A complete derivation was later given by , using SzegőGábor SzegoGábor Szegő was a Hungarian mathematician. He was one of the foremost analysts of his generation and made fundamental contributions to the theory of Toeplitz matrices and orthogonal polynomials.-Life:...
's limit formulaSzegő limit theoremsIn mathematical analysis, the Szegő limit theorems describe the asymptotic behaviour of the determinants of large Toeplitz matrices. They were first proved by Gábor Szegő.-Notation:...
for Toeplitz determinants, proved in 1951 in response to Onsager's work. In this formula the total energy of Onsager's lattice model is given by
and β−1 =kT where k is Boltzmann's constant and T is the absolute temperature.
Three and four dimensions
In three dimensions, the Ising model was shown to have a representation in terms of non-interacting Fermionic lattice strings by Alexander PolyakovAlexander PolyakovAlexander Markovich Polyakov is a theoretical physicist, formerly at the Landau Institute in Moscow, at Princeton University.-Important discoveries:...
. In dimensions near four, the critical behavior of the model is understood to correspond to the renormalizationRenormalization groupIn theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...
behavior of the scalar phi-4 theory (see Kenneth Wilson).
More than four dimensions
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 depends on the values of H within one averaging volume from that region, the free energy contribution from each region only depends on the value of H there and in the neighboring regions. 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:
On a square lattice, symmetries guarantee that the coefficients
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
is a positive definite quadratic formQuadratic formIn mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
, 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
. Rotational symmetry emerges spontaneously at large distances just because there aren't very many
low order terms. At higher order multicritical points, this accidental symmetryAccidental symmetryIn physics, in renormalization theory, an accidental symmetry is a symmetry which is present in a renormalizable theory only because the terms which break it have too high a dimension to appear in the Lagrangian....
is lost.
Since
is a function of a slowly spatially varying field. The probability of any field configuration is:
The statistical average of any product of H's is equal to:
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
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 theoryQuantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
of a scalar field is that all the derivative terms enter with a positive sign, and there is no overall factor of i.
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,
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:
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
term or the 
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
is dimensionless, and Z has dimension 2 − d/2. In this scaling, the gradient term is only important at long distances for 
. 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
split in the following way:
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,
For t < 0, the minima are at H proportional to the square root of t. So Landau's catastropheCatastrophe theoryIn mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry....
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
term is 2, while the scale dimension of the 
term is 4 − d. For d < 4, the 
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
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:
valid when x − y is large. The function G(x − y) is the analytic continuation to imaginary time of the Feynman propagatorPropagatorIn quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. Propagators are used to represent the contribution of virtual particles on the internal...
, 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.
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:
This is valid at noncoincident 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 methodGaussian surfaceA Gaussian surface is a closed surface in three dimensional space through which the flux of an electromagnetic field is calculated. It is an arbitrary closed surface S=\partial V used in conjunction with Gauss's law in order to calculate the total enclosed electric charge by performing a surface...
from electrostatics. Define an electric field analog by
away from the origin:
since G is spherically symmetric in d dimensions, E is the radial gradient of G. Integrating over a large d − 1 dimensional sphere,
This gives:
and G can be found by integrating with respect to r.
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:
For r small compared with
, 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:
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.
This is the inverse of the operator
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
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
. The decay factor appropriate for position r is therefore 
.
A heuristic approximation for G(r) is:
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
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 dimensionFractal dimensionIn fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal...
of an ordinary random walk is 2. The number of balls of size ε required to cover the path increase as
. 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 SymanzikKurt SymanzikKurt Symanzik was a German physicist working in quantum field theory.- Life :Symanzik was born in Lyck , East Prussia, and spent his childhood in Königsberg. He started studying physics in 1946 at Universität München but after a short time moved to Werner Heisenberg at Göttingen...
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
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 expansionFeynman diagramFeynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
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:
This equation is an identity inside any correlation function away from other insertions. After integrating out the
modes with
, 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
term. In a Feynman diagram expansion, the 
term in
a correlation function inside a correlation has three dangling lines. Joining two of them at large wavenumber k
gives a change
with one dangling line, so proportional to H:
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:
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
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 productsOperator product expansion- 2D Euclidean quantum field theory :In quantum field theory, the operator product expansion is a Laurent series expansion of two operators...
:
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
splits into
three, which quickly joins with one of the other lines from the same vertex. The correction to the vertex is
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
These two equations together define the renormalization group equations in four dimensions:
The coefficient B is determined by the formula
And is proportional to the area of a three dimensional sphere of radius λ, times the width of the
integration region
divided by 
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
defined by:
In dimension
, the coupling λ has positive scale dimension ε, and this must be added to the flow.
The coefficient B is dimension dependent, but it will cancel. The fixed point for λ is no longer zero,
but at:
where the scale dimensions of t is altered by an amount
.
The magnetization exponent is altered proportionately to:
which is .333 in 3 dimensions (
) and .166 in 2 dimensions (
). This is not so far off from the measured exponent .308 and the Onsager two dimensional exponent .125.
Infinite dimensions – mean field
The behavior of an Ising model on a fully connected graph may be completely understood by mean field theoryMean field theoryMean field theory is a method to analyse physical systems with multiple bodies. A many-body system with interactions is generally very difficult to solve exactly, except for extremely simple cases . The n-body system is replaced by a 1-body problem with a chosen good external field...
. 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 + minus the average fraction of spins which are -. The energy cost of flipping a single spin in the mean field H is
2 JNH. It is convenient to redefine J to absorb the factor N, so that the limit 
is smooth. In terms of the new J, the energy cost for flipping a spin is 
2 JH.
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.
so that
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.
The solutions to this equation are the possible consistent mean fields. For
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
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
, just below the critical temperature, the value of H can be calculated from the Taylor expansion of the Hyperbolic tangent:
dividing by H to discard the unstable solution at H = 0, the stable solutions are:
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 LandauLev LandauLev Davidovich Landau was a prominent Soviet physicist who made fundamental contributions to many areas of theoretical physics...
to suspect that all Ising type phase transitions in all dimensions should follow this law.
The mean field exponent is universalUniversality (dynamical systems)In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system. Systems display universality in a scaling limit, when a large number of interacting parts come together...
because changes in the character of solutions of analytic equations are always described by catastrophesCatastrophe theoryIn mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry....
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:
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
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:
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
But there it was not a surprise, because it was predicted by OnsagerLars OnsagerLars Onsager was a Norwegian-born American physical chemist and theoretical physicist, winner of the 1968 Nobel Prize in Chemistry.He held the Gibbs Professorship of Theoretical Chemistry at Yale University....
.
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 KadanoffLeo KadanoffLeo Philip Kadanoff is an American physicist. He is a professor of physics at the University of Chicago and a former President of the American Physical Society . He has contributed to the fields of statistical physics, chaos theory, and theoretical condensed matter physics.-Biography: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.
Migdal-Kadanoff renormalization
Write the two dimensional Ising model with an infinite number of possible higher order interactions. To keep spin reflection symmetry, only even powers contribute:
By translation invariance,
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
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:
where
are the number of neighbors which are + and −. Ignoring the factor of 2, the free energy contribution from this odd site is:
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:
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:
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.
Magnetism
The original motivation for the model was the phenomenon of ferromagnetismFerromagnetismFerromagnetism is the basic mechanism by which certain materials form permanent magnets, or are attracted to magnets. In physics, several different types of magnetism are distinguished...
. Iron is magnetic; once it is magnetized it stays magnetized for a long time compared to any atomic
time.
In the 19th century, it was thought that magnetic fields are due to currents in matter, and AmpèreAndré-Marie AmpèreAndré-Marie Ampère was a French physicist and mathematician who is generally regarded as one of the main discoverers of electromagnetism. The SI unit of measurement of electric current, the ampere, is named after him....
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 LarmorJoseph LarmorSir Joseph Larmor , a physicist and mathematician who made innovations in the understanding of electricity, dynamics, thermodynamics, and the electron theory of matter...
. In order to have ferromagnetism, the atoms must have permanent magnetic momentMagnetic momentThe magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will exert on it...
s 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
, 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 
for each occupied neighboring pair.
The density of the atoms can be controlled by adding a chemical potentialChemical potentialChemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system...
, 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
. The probability cost of one more atom is a factor of 
.
So the energy of the lattice gas is:
Rewriting the bits in terms of spins,
.
For lattices where every site has an equal number of neighbors, this is the Ising model with a magnetic field
, where 
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 approximationHopfield netA Hopfield network is a form of recurrent artificial neural network invented by John Hopfield. Hopfield nets serve as content-addressable memory systems with binary threshold units. They are guaranteed to converge to a local minimum, but convergence to one of the stored patterns is not guaranteed...
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 entropyPrinciple of maximum entropyIn Bayesian probability, the principle of maximum entropy is a postulate which states that, subject to known constraints , the probability distribution which best represents the current state of knowledge is the one with largest entropy.Let some testable information about a probability distribution...
. 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:
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:
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
With the Ising model the so-called spin glasses can also be described, by the usual Hamiltonian
where the S-variables describe the Ising spins, while the Ji,k are taken from a random distribution. 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.
External links
- Barry A. Cipra, "The Ising model is NP-complete", SIAM News, Vol. 33, No. 6; online edition (.pdf)
- Reports why the Ising model can't be solved exactly in general, since non-planar Ising models are NP-completeNP-completeIn computational complexity theory, the complexity class NP-complete is a class of decision problems. A decision problem L is NP-complete if it is in the set of NP problems so that any given solution to the decision problem can be verified in polynomial time, and also in the set of NP-hard...
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- Reports why the Ising model can't be solved exactly in general, since non-planar Ising models are NP-complete
- Science World article on the Ising Model
- An Ising Applet by Syracuse University
- A nice dynamical 2D Ising Applet
- A larger/more complicated 2D Ising Applet
- A nice HTML5 Ising Model simulation
- Ising Model simulation by Enrique Zeleny, the Wolfram Demonstrations ProjectWolfram Demonstrations ProjectThe Wolfram Demonstrations Project is hosted by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. It consists of an organized, open-source collection of small interactive programs called Demonstrations, which are meant to visually and...
- Phase transitions on lattices
- Three-dimensional proof for Ising Model impossible, Sandia researcher claims
- Multi-GPU accelerated multi-spin Monte Carlo simulations of the 2D Ising model
- Interactive dynamical simulation for MacOs of the 2D ising model on a square lattice
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