Ising model: Difference between revisions

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{{main|Lee–Yang theorem}}

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

==Applications==

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It is possible to view the Ising model as a [[Markov chain]], as the immediate probability ''P''_β(ν) of transitioning to a future state ν only depends on the present state μ. The Metropolis algorithm is actually a version of a [[Markov chain Monte Carlo]] simulation, and since we use single-spin-flip dynamics in the Metropolis algorithm, every state can be viewed as having links to exactly ''L'' other states, where each transition corresponds to flipping a single spin site to the opposite value.<ref>{{cite journal |last=Teif |first=Vladimir B.|title=General transfer matrix formalism to calculate DNA-protein-drug binding in gene regulation |journal=Nucleic Acids Res. |year=2007 |volume=35 |issue=11 |pages=e80 |doi=10.1093/nar/gkm268 |pmid=17526526 |pmc=1920246}}</ref> Furthermore, since the energy equation ''H''_σ change only depends on the nearest-neighbor interaction strength ''J'', the Ising model and its variants such the [[Sznajd model]] can be seen as a form of a [[Contact process (mathematics)#Voter model|voter model]] for opinion dynamics.

==One dimensionSolutions ==

{{splitto|Solutions to the Ising model}}

=== One dimension ===

The thermodynamic limit exists as long as the interaction decay is <math>J_{ij} \sim |i - j|^{-\alpha}</math> with α > 1.<ref name="Ruelle">{{cite book |first=David |last=Ruelle |title=Statistical Mechanics: Rigorous Results |url=https://books.google.com/books?id=2HPVCgAAQBAJ&pg=PR4 |date=1999 |publisher=World Scientific |isbn=978-981-4495-00-4 |orig-year=1969}}</ref>

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* 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.<ref>{{citation | last1=Baxter | first1=Rodney J. | title=Exactly solved models in statistical mechanics | url=http://tpsrv.anu.edu.au/Members/baxter/book | url-status=dead | publisher=Academic Press Inc. [Harcourt Brace Jovanovich Publishers] | ___location=London | isbn=978-0-12-083180-7 | mr=690578 | year=1982 | access-date=2009-10-25 | archive-date=2012-03-20 | archive-url=https://web.archive.org/web/20120320064257/http://tpsrv.anu.edu.au/Members/baxter/book }}</ref>

==== Ising's exact solution ====

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

<math display="block">H(\sigma) = -J \sum_{i=1,\ldots,L-1} \sigma_i \sigma_{i+1} - h \sum_i \sigma_i,</math>

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where ''C''(β) and ''c''(β) are positive functions for ''T'' > 0. For ''T'' → 0, though, the inverse correlation length ''c''(β) vanishes.

===== Proof =====

The proof of this result is a simple computation.

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

===== Comments =====

The energy of the lowest state is −''JL'', when all the spins are the same. For any other configuration, the extra energy is equal to 2''J'' times the number of sign changes that are encountered when scanning the configuration from left to right.

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which is [[Analytic function|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.

==== One-dimensional solution with transverse field ====

To express the Ising Hamiltonian using a quantum mechanical description of spins, we replace the spin variables with their respective [[Pauli matrices]]. However, depending on the direction of the magnetic field, we can create a transverse-field or longitudinal-field Hamiltonian. The [[transverse-field Ising model|transverse-field]] Hamiltonian is given by

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Since the roles of ''h'' and ''J'' are switched, the Hamiltonian undergoes a transition at ''J'' = ''h''.<ref name="Chakra">{{cite book |last1=Suzuki |first1= Sei |last2= Inoue |first2= Jun-ichi |last3= Chakrabarti |first3= Bikas K. |title=Quantum Ising Phases and Transitions in Transverse Ising Models |publisher=Springer |year=2012 |doi=10.1007/978-3-642-33039-1 |isbn=978-3-642-33038-4 |url= https://cds.cern.ch/record/1513030}}</ref>

==== Renormalization ====

When there is no external field, we can derive a functional equation that <math>f(\beta, 0) = f(\beta)</math> satisfies using renormalization.<ref>{{Cite journal |last1=Maris |first1=Humphrey J. |last2=Kadanoff |first2=Leo P. |date=June 1978 |title=Teaching the renormalization group |url=https://pubs.aip.org/aapt/ajp/article/46/6/652-657/1045608 |journal=American Journal of Physics |language=en |volume=46 |issue=6 |pages=652–657 |doi=10.1119/1.11224 |issn=0002-9505}}</ref> Specifically, let <math>Z_N(\beta, J)</math> be the partition function with <math>N</math> sites. Now we have:<math display="block">Z_N(\beta, J) = \sum_{\sigma} e^{K \sigma_2(\sigma_1 + \sigma_3)}e^{K \sigma_4(\sigma_3 + \sigma_5)}\cdots</math>where <math>K := \beta J</math>. We sum over each of <math>\sigma_2, \sigma_4, \cdots</math>, to obtain<math display="block">Z_N(\beta, J) = \sum_{\sigma} (2\cosh(K(\sigma_1 + \sigma_3))) \cdot (2\cosh(K(\sigma_3 + \sigma_5))) \cdots</math>Now, since the cosh function is even, we can solve <math>Ae^{K'\sigma_1\sigma_3} = 2\cosh(K(\sigma_1+\sigma_3))</math> as <math display="inline">A = 2\sqrt{\cosh(2K)}, K' = \frac 12 \ln\cosh(2K)</math>. Now we have a self-similarity relation:<math display="block">\frac 1N \ln Z_N(K) = \frac 12 \ln\left(2\sqrt{\cosh(2K)}\right) + \frac 12 \frac{1}{N/2} \ln Z_{N/2}(K')</math>Taking the limit, we obtain<math display="block">f(\beta) = \frac 12 \ln\left(2\sqrt{\cosh(2K)}\right) + \frac 12 f(\beta')</math>where <math>\beta' J = \frac 12 \ln\cosh(2\beta J)</math>.

When <math>\beta</math> is small, we have <math>f(\beta)\approx \ln 2</math>, so we can numerically evaluate <math>f(\beta)</math> by iterating the functional equation until <math>K</math> is small.

=== Two dimensions ===

* In the ferromagnetic case there is a phase transition. At low temperature, the [[Peierls argument]] proves positive magnetization for the nearest neighbor case and then, by the [[Griffiths inequality]], also when longer range interactions are added. Meanwhile, at high temperature, the [[cluster expansion]] gives analyticity of the thermodynamic functions.

* In the nearest-neighbor case, the free energy was exactly computed by Onsager, through the equivalence of the model with free fermions on lattice. The spin-spin correlation functions were computed by McCoy and Wu.

==== Onsager's exact solution ====

{{main|Square lattice Ising model}}

{{harvtxt|Onsager|1944}} obtained the following analytical expression for the free energy of the Ising model on the anisotropic square lattice when the magnetic field <math>h=0</math> in the thermodynamic limit as a function of temperature and the horizontal and vertical interaction energies <math>J_1</math> and <math>J_2</math>, respectively

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When the interaction energies <math>J_1</math>, <math>J_2</math> are both negative, the Ising model becomes an antiferromagnet. Since the square lattice is bi-partite, it is invariant under this change when the magnetic field <math>h=0</math>, so the free energy and critical temperature are the same for the antiferromagnetic case. For the triangular lattice, which is not bi-partite, the ferromagnetic and antiferromagnetic Ising model behave notably differently. Specifically, around a triangle, it is impossible to make all 3 spin-pairs antiparallel, so the antiferromagnetic Ising model cannot reach the minimal energy state. This is an example of [[geometric frustration]].

===== Transfer matrix =====

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

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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:

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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 σ^''z'' term in ''T'' counts the number of spin flips, which we can write in terms of spin-flip creation and annihilation operators:

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Carrying out the diagonalization produces the Onsager free energy.

===== Onsager's formula for spontaneous magnetization =====

Onsager famously announced the following expression for the [[spontaneous magnetization]] ''M'' of a two-dimensional Ising ferromagnet on the square lattice at two different conferences in 1948, though without proof<ref name="Montroll 1963 pages=308-309"/>

<math display="block">M = \left(1 - \left[\sinh 2\beta J_1 \sinh 2\beta J_2\right]^{-2}\right)^{\frac{1}{8}}</math>

where <math>J_1</math> and <math>J_2</math> are horizontal and vertical interaction energies.

A complete derivation was only given in 1951 by {{harvtxt|Yang|1952}} using a limiting process of transfer matrix eigenvalues. The proof was subsequently greatly simplified in 1963 by Montroll, Potts, and Ward<ref name="Montroll 1963 pages=308-309"/> using [[Gábor Szegő|Szegő]]'s [[Szegő limit theorems|limit formula]] for [[Toeplitz determinant]]s by treating the magnetization as the limit of correlation functions.

==== Minimal model ====

{{main|Two-dimensional critical Ising model}}

At the critical point, the two-dimensional Ising model is a [[two-dimensional conformal field theory]]. The spin and energy correlation functions are described by a [[Minimal model (physics)|minimal model]], which has been exactly solved.

=== Three dimensions ===

In three as in two dimensions, the most studied case of the Ising model is the translation-invariant model on a cubic lattice with nearest-neighbor coupling in the zero magnetic field. Many theoreticians searched for an analytical three-dimensional solution for many decades, which would be analogous to Onsager's solution in the two-dimensional case.'''<ref>{{Cite web|last=Wood|first=Charlie|title=The Cartoon Picture of Magnets That Has Transformed Science|url=https://www.quantamagazine.org/the-cartoon-picture-of-magnets-that-has-transformed-science-20200624/|access-date=2020-06-26|website=Quanta Magazine|date=24 June 2020|language=en}}</ref>''' <ref>{{Cite web |title=Ken Wilson recalls how Murray Gell-Mann suggested that he solve the three-dimensional Ising model |url=https://authors.library.caltech.edu/5456/1/hrst.mit.edu/hrs/renormalization/Wilson/index.htm}}</ref> Such a solution has not been found until now, although there is no proof that it may not exist. In three dimensions, the Ising model was shown to have a representation in terms of non-interacting fermionic strings by [[Alexander Markovich Polyakov|Alexander Polyakov]] and [[Vladimir Dotsenko]]. This construction has been carried on the lattice, and the [[continuum limit]], conjecturally describing the critical point, is unknown.

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In 2000, [[Sorin Istrail]] of [[Sandia National Laboratories]] proved that the spin glass Ising model on a [[nonplanar]] lattice is [[NP-completeness|NP-complete]]. That is, assuming '''P''' ≠ '''NP,''' the general spin glass Ising model is exactly solvable only in [[Planar graph|planar]] cases, so solutions for dimensions higher than two are also intractable.<ref>{{cite journal |last=Cipra |first=Barry A. |year=2000 |title=The Ising Model Is NP-Complete |url=https://archive.siam.org/pdf/news/654.pdf |journal=SIAM News |volume=33 |issue=6}}</ref> Istrail's result only concerns the spin glass model with spatially varying couplings, and tells nothing about Ising's original ferromagnetic model with equal couplings.

=== Four dimensions and above ===

{{unreferenced|section}}

{{overly detailed|section}}

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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''.

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The denominator in this expression is called the ''partition function'':<math display="block">Z = \int DH \, e^{ - \int d^dx \left[ A H^2 + Z |\nabla H|^2 + \lambda H^4 \right]}</math>and the integral over all possible values of ''H'' is a statistical path integral. It integrates exp(β''F'') over all values of ''H'', over all the long wavelength fourier components of the spins. ''F'' is a "Euclidean" Lagrangian for the field ''H''. It is similar to the Lagrangian in of a scalar field in [[quantum field theory]], the difference being that all the derivative terms enter with a positive sign, and there is no overall factor of ''i'' (thus "Euclidean").

==== 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.

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Since ''t'' is vanishing, fixing the scale of the field using this term makes the other terms blow up. Once ''t'' is small, the scale of the field can either be set to fix the coefficient of the ''H''⁴ term or the (∇''H'')² term to 1.

==== Magnetization ====

To find the magnetization, fix the scaling of ''H'' so that λ is one. Now the field ''H'' has dimension −''d''/4, so that ''H''⁴''d^dx'' is dimensionless, and ''Z'' has dimension 2&nbsp;−&nbsp;''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.

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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 fluctuations 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&nbsp;−&nbsp;''d''/2. Now the field has constant quadratic spatial fluctuations at all temperatures. The scale dimension of the ''H''² term is 2, while the scale dimension of the ''H''⁴ term is 4&nbsp;−&nbsp;''d''. For ''d'' < 4, the ''H''⁴ term has positive scale dimension. In dimensions higher than 4 it has negative scale dimensions.

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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:

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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:

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

==== Symanzik polymer interpretation ====

The interpretation of the correlations as fixed size quanta travelling along random walks gives a way of understanding why the critical dimension of the ''H''⁴ interaction is 4. The term ''H''⁴ 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 ''H''² coefficient by a constant. But the intersection probability of random walks depends on the dimension, and random walks in dimension higher than 4 do not intersect.

The [[fractal dimension]] of an ordinary random walk is 2. The number of balls of size ε required to cover the path increase as ε⁻². Two objects of fractal dimension 2 will intersect with reasonable probability only in a space of dimension 4 or less, the same condition as for a generic pair of planes. [[Kurt Symanzik]] argued that this implies that the critical Ising fluctuations in dimensions higher than 4 should be described by a free field. This argument eventually became a mathematical proof.

==== 4&nbsp;−&nbsp;''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.

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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.

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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 fixed 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 does not change very much when the dimension is changed.

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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 ====

{{Main|Mean-field theory}}

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But there it was not a surprise, because it was predicted by [[Lars Onsager|Onsager]].

==== Low dimensions&nbsp;– 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.

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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:

<math display="block">E = \sum_{ij} J_{ij} S_i S_j + \sum J_{ijkl} S_i S_j S_k S_l \ldots.</math>