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{{short description|Mathematical model of ferromagnetism in statistical mechanics}}

{{Statistical mechanics|cTopic=Models}}

The '''Ising model''' ({{IPAc-en|ˈ|aɪ|s|ɪ|ŋ}}; {{IPA-de|ˈiːzɪŋ|lang}}), named after the physicist [[Ernst Ising]], is a [[mathematical models in physics|mathematical model]] of [[ferromagnetism]] in [[statistical mechanics]]. The model consists of [[discrete variables]] that represent [[Nuclear magnetic moment|magnetic dipole moments of atomic "spins"]] that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a [[lattice (group)|lattice]] (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. Neighboring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases. The model allows the identification of [[phase transition]]s, as a simplified model of reality. The two-dimensional [[square-lattice Ising model]] is one of the simplest statistical models to show a [[phase transition]].<ref>See {{harvtxt|Gallavotti|1999}}, Chapters VI-VII.</ref>

The Ising model (; ), named after the physicist Ernst Ising, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a lattice (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. Neighboring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases. The model allows the identification of phase transitions, as a simplified model of reality. The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition.

伊辛模型(; ;) ，以物理学家恩斯特 · 伊辛的名字命名，是统计力学时代铁磁性的数学模型。该模型由离散变量组成，它们表示处于两种状态之一(+ 1或-1)的原子“自旋”的磁偶极矩。自旋排列在一个图形中，通常是一个格子(其中局部结构在各个方向周期性地重复) ，允许每个自旋与它的邻居相互作用。相邻的同意的自旋的能量比不同意的自旋的能量要低; 这个系统倾向于最低的能量，但是热量扰乱了这种趋势，因此产生了不同结构相的可能性。该模型允许识别相变，作为一个简化的现实模型。二维方晶格易辛模型是显示相变的最简单的统计模型之一。

The Ising model was invented by the physicist {{harvs|txt|authorlink=Wilhelm Lenz|first=Wilhelm|last=Lenz|year=1920}}, who gave it as a problem to his student Ernst Ising. The one-dimensional Ising model was solved by {{harvtxt|Ising|1925}} himself in his 1924 thesis;<ref>[http://www.hs-augsburg.de/~harsch/anglica/Chronology/20thC/Ising/isi_fm00.html Ernst Ising, ''Contribution to the Theory of Ferromagnetism'']</ref> it has no phase transition. The two-dimensional square-lattice Ising model is much harder and was only given an analytic description much later, by {{harvs|txt|authorlink=Lars Onsager|first=Lars |last=Onsager|year=1944}}. It is usually solved by a [[transfer-matrix method]], although there exist different approaches, more related to [[quantum field theory]].

The Ising model was invented by the physicist , who gave it as a problem to his student Ernst Ising. The one-dimensional Ising model was solved by himself in his 1924 thesis; it has no phase transition. The two-dimensional square-lattice Ising model is much harder and was only 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.

伊辛模型是由物理学家发明的，他把这个问题交给了他的学生欧内斯特 · 伊辛。一维伊辛模型是他在1924年的论文中自己求解的，它没有相变。二维方晶格易辛模型要难得多，很久以后才得到分析性的描述。这个问题通常由传递矩阵法解决，尽管有不同的方法，更多的是与量子场论有关。

In dimensions greater than four, the phase transition of the Ising model is described by [[mean field theory]].

In dimensions greater than four, the phase transition of the Ising model is described by mean field theory.

在维数大于四的情况下，用平均场理论描述了伊辛模型的相变。

==Definition==

Consider a set Λ of lattice sites, each with a set of adjacent sites (e.g. a [[Graph (discrete mathematics)|graph]]) forming a ''d''-dimensional lattice. For each lattice site ''k'' ∈ Λ there is a discrete variable σ''k'' such that σ''k''  ∈ {+1, −1}, representing the site's spin. A ''spin configuration'', σ = (σ''k'')''k'' ∈ Λ is an assignment of spin value to each lattice site.

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

考虑一组格点，每个格点有一组相邻的格点(例如:。一个图形)形成 d 维格子。对于每个格点，k ∈存在一个离散变量子 k / sub，子 k / sub ∈{ + 1,-1} ，表示格点的自旋。自旋构型(sub k / sub)子 k ∈ / sub 是对每个格点的自旋值赋值。

For any two adjacent sites ''i'', ''j'' ∈ Λ there is an ''interaction'' ''J''''ij''. Also a site ''j'' ∈ Λ has an ''external magnetic field'' ''h''''j'' interacting with it. The ''energy'' of a configuration σ is given by the [[Hamiltonian function]]

For any two adjacent sites i, j ∈ Λ there is an interaction Jij. Also a site j ∈ Λ has an external magnetic field hj interacting with it. The energy of a configuration σ is given by the Hamiltonian function

对于任意两个相邻点 i，j ∈存在一个相互作用的 j 子 ij / 子。还有一个位置 j ∈有一个外磁场 h 子 j / 子与之相互作用。组态的能量由哈密顿函数给出

: <math>H(\sigma) = -\sum_{\langle i~j\rangle} J_{ij} \sigma_i \sigma_j - \mu \sum_j h_j \sigma_j,</math>

<math>H(\sigma) = -\sum_{\langle i~j\rangle} J_{ij} \sigma_i \sigma_j - \mu \sum_j h_j \sigma_j,</math>

数学 h (sigma)-sum (langle) i ~ j (j)-sigma (j)-mu-sum (j)-sigma (j) ，/ math

where the first sum is over pairs of adjacent spins (every pair is counted once). The notation ⟨''ij''⟩ indicates that sites ''i'' and ''j'' are nearest neighbors. The [[magnetic moment]] is given by µ. Note that the sign in the second term of the Hamiltonian above should actually be positive because the electron's magnetic moment is antiparallel to its spin, but the negative term is used conventionally.<ref>See {{harvtxt|Baierlein|1999}}, Chapter 16.</ref> The ''configuration probability'' is given by the [[Boltzmann distribution]] with [[inverse temperature]] β ≥ 0:

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

其中第一和超过对相邻的自旋(每对都计算一次)。符号 ij something 表示位置 i 和 j 是最近邻。磁矩是由。注意，上面哈密顿量第二项的符号实际上应该是正的，因为电子的磁矩与它的自旋是反平行的，但是负项是惯用的。在逆温度≥0时，由波兹曼分布给出其构型概率:

: <math>P_\beta(\sigma) = \frac{e^{-\beta H(\sigma)}}{Z_\beta},</math>

<math>P_\beta(\sigma) = \frac{e^{-\beta H(\sigma)}}{Z_\beta},</math>

数学 p beta ( sigma) frac { e ^ {- beta h ( sigma)}{ z beta } ，/ math

where β = (''k''B''T'')−1, and the normalization constant

where β = (kBT)−1, and the normalization constant

其中(k 子 b / 子 t) sup-1 / sup，和归一化常数

: <math>Z_\beta = \sum_\sigma e^{-\beta H(\sigma)}</math>

<math>Z_\beta = \sum_\sigma e^{-\beta H(\sigma)}</math>

数学 z beta sum sigma e ^ {- beta h ( sigma)} / math

is the [[partition function (statistical mechanics)|partition function]]. For a function ''f'' of the spins ("observable"), one denotes by

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

就是配分函数。对于自旋函数 f (“可观察”) ，一个表示为

: <math>\langle f \rangle_\beta = \sum_\sigma f(\sigma) P_\beta(\sigma)</math>

<math>\langle f \rangle_\beta = \sum_\sigma f(\sigma) P_\beta(\sigma)</math>

数学 langle f rangle beta sum sigma f ( sigma) p beta ( sigma) / math

the expectation (mean) value of ''f''.

the expectation (mean) value of f.

F 的期望(均值)。

The configuration probabilities ''P''β(σ) represent the probability that (in equilibrium) the system is in a state with configuration σ.

The configuration probabilities Pβ(σ) represent the probability that (in equilibrium) the system is in a state with configuration σ.

配置概率 p sub / sub ()表示系统(在平衡状态)处于配置状态的概率。

===Discussion===

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

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

哈密顿函数 h ()的每一项上的负号是常规的。使用这个符号约定，Ising 模型可以根据相互作用的符号进行分类: if，for a pair i，j

: <math>J_{ij} > 0</math>, the interaction is called [[ferromagnetic]],

<math>J_{ij} > 0</math>, the interaction is called ferromagnetic,

数学 j { ij }0 / math，这种相互作用称为铁磁,

: <math>J_{ij} < 0</math>, the interaction is called [[antiferromagnetic]],

<math>J_{ij} < 0</math>, the interaction is called antiferromagnetic,

0 / math，这种相互作用被称为反铁磁,

: <math>J_{ij} = 0</math>, the spins are ''noninteracting''.

<math>J_{ij} = 0</math>, the spins are noninteracting.

数学 j { ij }0 / 数学，自旋是不相互作用的。

The system is called ferromagnetic or antiferromagnetic if all interactions are ferromagnetic or all are antiferromagnetic. The original Ising models were ferromagnetic, and it is still often assumed that "Ising model" means a ferromagnetic Ising model.

The system is called ferromagnetic or antiferromagnetic if all interactions are ferromagnetic or all are antiferromagnetic. The original Ising models were ferromagnetic, and it is still often assumed that "Ising model" means a ferromagnetic Ising model.

如果所有的相互作用都是铁磁性的或者都是反铁磁性的，那么这个系统就叫做铁磁或反铁磁性系统。最初的伊辛模型是铁磁性的，人们仍然经常假设“伊辛模型”是指铁磁性的伊辛模型。

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

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

在铁磁性伊辛模型中，自旋希望被对齐: 相邻自旋具有相同符号的构型有更高的可能性。在反铁磁模型中，相邻的自旋往往有相反的符号。

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

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

H ()的符号约定也解释了自旋位置 j 如何与外场相互作用。也就是说，自旋站点想要与外部场保持一致。如果:

: <math>h_j > 0</math>, the spin site ''j'' desires to line up in the positive direction,

<math>h_j > 0</math>, the spin site j desires to line up in the positive direction,

数学公式，自旋网站 j 希望排成正方向,

: <math>h_j < 0</math>, the spin site ''j'' desires to line up in the negative direction,

<math>h_j < 0</math>, the spin site j desires to line up in the negative direction,

数学公式，自旋站点 j 希望排成负方向,

: <math>h_j = 0</math>, there is no external influence on the spin site.

<math>h_j = 0</math>, there is no external influence on the spin site.

数学公式，旋转网站上没有外部影响。

===Simplifications===

Ising models are often examined without an external field interacting with the lattice, that is, ''h'' = 0 for all ''j'' in the lattice Λ. Using this simplification, the Hamiltonian becomes

Ising models are often examined without an external field interacting with the lattice, that is, h = 0 for all j in the lattice Λ. Using this simplification, the Hamiltonian becomes

伊辛模型通常在没有外场与晶格相互作用的情况下进行研究，即对晶格中的所有 j 都采用 h0。利用这种简化，哈密顿量变成了

: <math>H(\sigma) = -\sum_{\langle i~j\rangle} J_{ij} \sigma_i \sigma_j.</math>

<math>H(\sigma) = -\sum_{\langle i~j\rangle} J_{ij} \sigma_i \sigma_j.</math>

数学 h (sigma)-总和 i ~ j (ij)-sigma i-sigma j. / math

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 nonzero field breaks this symmetry.

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 nonzero field breaks this symmetry.

当外部电场处处为零，h0时，伊辛模型在切换所有晶格点的自旋值时是对称的; 一个非零电场打破了这种对称性。

Another common simplification is to assume that all of the nearest neighbors ⟨''ij''⟩ have the same interaction strength. Then we can set ''Jij'' = ''J'' for all pairs ''i'', ''j'' in Λ. In this case the Hamiltonian is further simplified to

Another common simplification is to assume that all of the nearest neighbors ⟨ij⟩ have the same interaction strength. Then we can set Jij = J for all pairs i, j in Λ. In this case the Hamiltonian is further simplified to

另一种常见的简化方法是假定所有的最近邻居 something ij something 具有相同的相互作用强度。然后我们可以设置所有对 i，j in 的 j 次方 ij / 次方 j。在这种情况下，哈密顿量进一步简化为

: <math>H(\sigma) = -J \sum_{\langle i~j\rangle} \sigma_i \sigma_j.</math>

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

数学 h (sigma)-j (langle i)-j (rangle)-sigma i-sigma j. / math

===Questions===

A significant number of statistical questions to ask about this model are in the limit of large numbers of spins:

A significant number of statistical questions to ask about this model are in the limit of large numbers of spins:

关于这个模型，有许多统计学上的问题需要提出，这些问题都是在大量自旋的极限下:

* 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 Λ, what is the fractal dimension of the shape of a large cluster of +1 spins?

==Basic properties and history==

[[File:Ising-tartan.png|thumb|right|Visualization of the translation-invariant probability measure of the one-dimensional Ising model]]

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

一维 Ising 模型平移不变机率量测的可视化

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

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.

伊辛模型的研究最多的是 d 维晶格上的平移不变铁磁零场模型，即 z sup d / sup，j sub ij / sub 1，h 0。

In his 1924 PhD thesis, Ising solved the model for the ''d'' = 1 case, which can be thought of as a linear horizontal lattice where each site only interacts with its left and right neighbor. In one dimension, the solution admits no [[phase transition]].<ref>{{Cite journal |url=http://users-phys.au.dk/fogedby/statphysII/no-PT-in-1D.pdf |title=Solving the 3d Ising Model with the Conformal Bootstrap II. C -Minimization and Precise Critical Exponents |journal=Journal of Statistical Physics |volume=157 |issue=4–5 |pages=869–914 |last1=El-Showk |first1=Sheer |last2=Paulos |first2=Miguel F. |last3=Poland |first3=David |last4=Rychkov |first4=Slava |last5=Simmons-Duffin |first5=David |last6=Vichi |first6=Alessandro |year=2014 |doi=10.1007/s10955-014-1042-7 |arxiv=1403.4545 |access-date=2013-04-21 |archive-url=https://web.archive.org/web/20140407154639/http://users-phys.au.dk/fogedby/statphysII/no-PT-in-1D.pdf |archive-date=2014-04-07 |url-status=dead}}</ref> Namely, for any positive β, the correlations ⟨σ''i''σ''j''⟩ decay exponentially in |''i'' − ''j''|:

In his 1924 PhD thesis, Ising solved the model for the d = 1 case, which can be thought of as a linear horizontal lattice where each site only interacts with its left and right neighbor. In one dimension, the solution admits no phase transition. Namely, for any positive β, the correlations ⟨σiσj⟩ decay exponentially in |i − j|:

在他1924年的博士论文中，伊辛解决了 d 1情况的模型，这个模型可以被认为是一个线性的水平晶格，每个位置只与其左右相邻的位置相互作用。在一维情况下，解不存在相变。也就是说，对于任何正数，关联项 i / sub j / sub something 在 | i-j | 中呈指数衰减:

: <math>\langle \sigma_i \sigma_j \rangle_\beta \leq C \exp\big(-c(\beta) |i - j|\big),</math>

<math>\langle \sigma_i \sigma_j \rangle_\beta \leq C \exp\big(-c(\beta) |i - j|\big),</math>

数学 langle sigma i sigma j rangle beta leq c exp big (- c ( beta) | i-j | big) / math

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.

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.

系统是无序的。在这个结果的基础上，他错误地得出结论，认为这个模型在任何维度上都没有表现出阶段行为。

The Ising model undergoes a [[phase transition]] between an [[ordered phase|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:

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:

伊辛模型在二维或二维以上经历了有序相和无序相之间的相变。也就是说，系统在小的情况下是无序的，而在大的情况下，系统呈现出铁磁有序:

: <math>\langle \sigma_i \sigma_j \rangle_\beta \geq c(\beta) > 0.</math>

<math>\langle \sigma_i \sigma_j \rangle_\beta \geq c(\beta) > 0.</math>

数学 langle sigma i sigma j rangle beta geq c ( beta)0. / math

This was first proven by [[Rudolf Peierls]] in 1936,<ref>{{Cite journal |doi=10.1017/S0305004100019174 |title=On Ising's model of ferromagnetism |journal=Mathematical Proceedings of the Cambridge Philosophical Society |volume=32 |issue=3 |pages=477 |year=1936 |last1=Peierls |first1=R. |last2=Born |first2=M.}}</ref> using what is now called a '''Peierls argument'''.

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

这是鲁道夫 · 佩尔斯在1936年首次证明的，用的是现在所谓的佩尔斯论证。

The Ising model on a two-dimensional square lattice with no magnetic field was analytically solved by {{harvs|txt|authorlink=Lars Onsager|first=Lars |last=Onsager|year=1944}}. Onsager showed that the [[correlation function]]s and [[Thermodynamic free energy|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. {{harvtxt|Yang|1952}} gave the first published proof of this formula, using a [[Szegő limit theorems|limit formula]] for [[Fredholm determinant]]s, proved in 1951 by [[Gábor Szegő|Szegő]] in direct response to Onsager's work.<ref name="Montroll 1963 pages=308-309">{{harvnb|Montroll|Potts|Ward|1963|pages=308–309}}</ref>

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

本文用解析方法求解了无磁场二维正方晶格上的 Ising 模型。昂萨格表明，伊辛模型的关联函数和自由能是由非相互作用的格子费米子决定的。昂萨格在1949年宣布了二维模型的自发磁化公式，但没有给出推导。利用弗雷德霍尔姆行列式的一个极限公式，给出了这个公式的第一个公开证明，szeg 在1951年直接响应昂萨格的工作证明了这个公式。

==Historical significance==

One of [[Democritus]]' arguments in support of [[atomism]] was that atoms naturally explain the sharp phase boundaries observed in materials{{citation needed|date=July 2014}}, 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.

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

德谟克利特支持原子论的论据之一是，原子自然地解释了在材料中观察到的尖锐的相界，就像冰融化成水或水变成蒸汽一样。他的想法是，原子尺度特性的微小变化将导致聚合行为的巨大变化。其他人认为物质本质上是连续的，而不是原子的，物质的大尺度属性不能还原为基本的原子属性。

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 mechanics|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.

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.

虽然化学结合定律使十九世纪的化学家清楚地认识到原子是真实存在的，但物理学家之间的争论一直持续到二十世纪早期。原子论者，尤其是詹姆斯·克拉克·麦克斯韦和路德维希·玻尔兹曼，将汉密尔顿的牛顿定律公式应用于大系统，发现原子的统计行为正确地描述了室温气体。但是经典的统计力学不能解释液体和固体的所有性质，也不能解释低温下气体的所有性质。

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 (disambiguation)|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.

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 laws 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.

一旦现代量子力学被制定出来，原子主义就不再与实验相冲突，但这并没有导致普遍接受超越原子主义的统计力学。约西亚·威拉德·吉布斯给出了一个完整的形式主义来再现力学定律中的热力学定律。但是许多错误的观点从19世纪流传下来，当时统计力学被认为是可疑的。直觉上的失误主要源于这样一个事实，即无限统计系统的极限有许多在有限系统中不存在的0-1定律: 正如德谟克利特所预料的那样，一个参数的微小变化可能导致总体行为的巨大差异。

===No phase transitions in finite volume===

In the early part of the twentieth century, some believed that the [[partition function (statistical mechanics)|partition function]] could never describe a phase transition, based on the following argument:

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

在20世纪早期，基于以下论点，一些人认为配分函数不可能描述相变:

# The partition function is a sum of ''e''−β''E'' over all configurations.

The partition function is a sum of e−βE over all configurations.

配分函数是 e sup-e / sup 超过所有配置的总和。

# The exponential function is everywhere [[analytic function|analytic]] as a function of β.

The exponential function is everywhere analytic as a function of β.

指数函数是随处可见的分析函数。

# The sum of analytic functions is an analytic function.

The sum of analytic functions is an analytic function.

分析函数之和是解析函数。

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 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.

这个论证适用于有限的指数和，并且正确地证明了一个有限大小的系统的自由能中没有奇点。对于热力学极限系统(即无穷系统) ，无穷和可以导致奇点。收敛到热力学极限的速度很快，因此相位行为在一个相对较小的点阵上已经很明显了，即使奇点被系统的有限尺寸平滑了。

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

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

这是鲁道夫 · 佩尔斯在伊辛模型中首次确立的。

===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.

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

在 Lenz 和 Ising 构建 Ising 模型后不久，Peierls 就能够明确地证明在二维空间中发生了相变。

To do this, he compared the high-temperature and low-temperature limits. At infinite temperature (β = 0) 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 [[noise (video)|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 randomly looking, and there is no net excess of black or white.

To do this, he compared the high-temperature and low-temperature limits. At infinite temperature (β = 0) 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 randomly looking, and there is no net excess of black or white.

为了做到这一点，他比较了高温和低温极限。在无限温度下(0)所有构型的概率相等。每个自旋都是完全独立于其他任何自旋的，如果在无限温度下绘制出典型的构型，使得正负号用黑色和白色表示，它们看起来就像电视上的雪。对于高温，但不是无限的温度，相邻位置之间有小的相关性，雪倾向于聚集一点，但屏幕看起来是随机的，没有过多的黑色或白色。

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

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

超量的一个定量度量是磁化强度，也就是自旋的平均值:

: <math>M = \frac{1}{N} \sum_{i=1}^N \sigma_i.</math>

<math>M = \frac{1}{N} \sum_{i=1}^N \sigma_i.</math>

数学模型{1}{ n }{1} ^ n sigma i. /

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

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

一个类似于最后一节的论证的伪论证，证明了伊辛模型中的磁化强度总是为零。

# Every configuration of spins has equal energy to the configuration with all spins flipped.

Every configuration of spins has equal energy to the configuration with all spins flipped.

每个自旋构型的能量与所有自旋构型的能量相等。

# So for every configuration with magnetization ''M'' there is a configuration with magnetization −''M'' with equal probability.

So for every configuration with magnetization M there is a configuration with magnetization −M with equal probability.

所以对于每一个磁化强度为 m 的组态，都有一个磁化强度为-m 的组态，其概率是相等的。

# The system should therefore spend equal amounts of time in the configuration with magnetization ''M'' as with magnetization −''M''.

The system should therefore spend equal amounts of time in the configuration with magnetization M as with magnetization −M.

因此，系统在磁化强度 m 和磁化强度 m 的配置中花费的时间应该是相等的。

# So the average magnetization (over all time) is zero.

So the average magnetization (over all time) is zero.

所以平均磁化强度(总体上)是零。

As before, this only proves that the average 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 a nonzero probability.

As before, this only proves that the average 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 a nonzero probability.

和以前一样，这只能证明在任何有限体积下的平均磁化强度为零。对于一个无限系统来说，涨落可能不能以非零概率将系统从大部分正态推到大部分负态。

For very high temperatures, the magnetization is zero, as it is at infinite temperature. To see this, note that if spin A has only a small correlation ε with spin B, and B is only weakly correlated with C, but C is otherwise independent of A, the amount of correlation of A and C goes like ε2. For two spins separated by distance ''L'', the amount of correlation goes as ε''L'', but if there is more than one path by which the correlations can travel, this amount is enhanced by the number of paths.

For very high temperatures, the magnetization is zero, as it is at infinite temperature. To see this, note that if spin A has only a small correlation ε with spin B, and B is only weakly correlated with C, but C is otherwise independent of A, the amount of correlation of A and C goes like ε2. For two spins separated by distance L, the amount of correlation goes as εL, but if there is more than one path by which the correlations can travel, this amount is enhanced by the number of paths.

对于非常高的温度，磁化强度为零，因为它处于无限的温度。为了证明这一点，请注意，如果自旋 a 与自旋 b 只有很小的相关性，而 b 与 c 只有很弱的相关性，而 c 与 a 无关，那么 a 与 c 的相关性就像 sup 2 / sup。对于由距离 l 分开的两个自旋，相关系数的大小为 sup l / sup，但是如果相关系数可以通过多个路径传递，则相关系数随路径数的增加而增加。

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

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

D 维正方格上长度 l 的路数为

: <math>N(L) = (2d)^L,</math>

<math>N(L) = (2d)^L,</math>

数学 n (l)(2d) ^ l，/ 数学

since there are 2''d'' choices for where to go at each step.

since there are 2d choices for where to go at each step.

因为每个步骤都有2d 选择。

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 paths of length ''L'' divided by

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 paths of length L divided by

通过对连接两点的所有路径进行求和，给出了总相关性的一个界，这个界由长度 l 除以的所有路径的和来界定

: <math>\sum_L (2d)^L \varepsilon^L,</math>

<math>\sum_L (2d)^L \varepsilon^L,</math>

<math>\sum_L (2d)^L \varepsilon^L,</math>

which goes to zero when ε is small.

which goes to zero when ε is small.

当它小的时候，它就会变成零。

At low temperatures (β ≫ 1) the configurations are near the lowest-energy configuration, the one where all the spins are plus or all the spins are minus. 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.

At low temperatures (β ≫ 1) the configurations are near the lowest-energy configuration, the one where all the spins are plus or all the spins are minus. 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.

在低温情况下，这些构型接近最低能量构型，即所有的自旋都是正的或者所有的自旋都是负的。佩尔斯提出的问题是，从统计学的角度来看，是否有可能在低温下，从所有的自旋负开始，到大多数自旋正负起伏的状态。要做到这一点，加旋液滴必须能够凝结成加旋状态。

The energy of a droplet of plus spins in a minus background is proportional to the perimeter of the droplet L, where plus spins and minus spins neighbor each other. For a droplet with perimeter ''L'', the area is somewhere between (''L'' − 2)/2 (the straight line) and (''L''/4)2 (the square box). The probability cost for introducing a droplet has the factor ''e''−β''L'', but this contributes to the partition function multiplied by the total number of droplets with perimeter ''L'', which is less than the total number of paths of length ''L'':

The energy of a droplet of plus spins in a minus background is proportional to the perimeter of the droplet L, where plus spins and minus spins neighbor each other. For a droplet with perimeter L, the area is somewhere between (L − 2)/2 (the straight line) and (L/4)2 (the square box). The probability cost for introducing a droplet has the factor e−βL, but this contributes to the partition function multiplied by the total number of droplets with perimeter L, which is less than the total number of paths of length L:

在负背景下，正自旋液滴的能量与液滴 l 的周长成正比，l 中的正自旋和负自旋彼此相邻。对于周长为 l 的液滴，其面积介于(l-2) / 2(直线)和(l / 4) sup 2 / sup (方盒)之间。引入液滴的概率成本具有因子 e sup-l / sup，但是这个因子贡献了配分函数乘以周长为 l 的液滴总数，小于长度为 l 的路径总数:

: <math>N(L) < 4^{2L}.</math>

<math>N(L) < 4^{2L}.</math>

4 ^ {2L } . / math

So that the total spin contribution from droplets, even overcounting by allowing each site to have a separate droplet, is bounded above by

So that the total spin contribution from droplets, even overcounting by allowing each site to have a separate droplet, is bounded above by

因此，来自液滴的总自旋贡献，即使通过允许每个位置有一个单独的液滴来过度计算，也是在

: <math>\sum_L L^2 4^{2L} e^{-4\beta L},</math>

<math>\sum_L L^2 4^{2L} e^{-4\beta L},</math>

数学 l ^ 24 ^ {2L } e ^ {-4 beta l } ，/ math

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

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

最终归于零。对于足够大来说，这种指数形式抑制了长的循环，所以它们不会发生，磁化强度也不会在 -1附近波动太大。

So Peierls established that the magnetization in the Ising model eventually defines [[superselection sector]]s, separated domains not linked by finite fluctuations.

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

因此 Peierls 确定了伊辛模型中的磁化最终定义了超选区，即没有有限涨落联系的分离区域。

===Kramers–Wannier duality===

{{main|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).

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).

克拉默斯和万尼尔能够证明，模型的高温膨胀和低温膨胀相当于自由能的总体重新标度。这使得二维模型中的相变点可以被精确地确定(假设存在唯一的临界点)。

===Yang–Lee zeros===

{{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.

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

在昂萨格的解决方案之后，Yang 和 Lee 研究了当温度接近临界温度时配分函数变得奇异的方式。

==Monte Carlo methods for numerical simulation==

[[File:Ising quench b10.gif|framed|right|Quench of an Ising system on a two-dimensional square lattice (500 × 500) with inverse temperature ''β'' = 10, starting from a random configuration]]

Quench of an Ising system on a two-dimensional square lattice (500 × 500) with inverse temperature β = 10, starting from a random configuration

二维正方晶格(500500)上伊辛系统的淬火，从随机构型出发，逆温度10

===Definitions===

The Ising model can often be difficult to evaluate numerically if there are many states in the system. Consider an Ising model with

The Ising model can often be difficult to evaluate numerically if there are many states in the system. Consider an Ising model with

如果系统中有许多状态，伊辛模型通常难以用数值方法进行计算。考虑一个伊辛模型

: ''L'' = |Λ|: the total number of sites on the lattice,

L = |Λ|: the total number of sites on the lattice,

L | | : 点阵上的总数,

: σ''j'' ∈ {−1, +1}: an individual spin site on the lattice, ''j'' = 1, ..., ''L'',

σj ∈ {−1, +1}: an individual spin site on the lattice, j = 1, ..., L,

子 j / 子∈{-1，+ 1} : 格子上的单个自旋位置，j1，... ，l,

: ''S'' ∈ {−1, +1}''L'': state of the system.

S ∈ {−1, +1}L: state of the system.

S ∈ {−1, +1}L: state of the system.

Since every spin site has ±1 spin, there are ''2''''L'' different states that are possible.<ref name = "Newman">Newman M. E. J., Barkema G. T., "Monte Carlo Methods in Statistical Physics", Clarendon Press, 1999.</ref> This motivates the reason for the Ising model to be simulated using [[Monte Carlo methods]].<ref name="Newman" />

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

由于每个旋转站有1个旋转，有2个支撑 / 支撑不同的状态是可能的。这就促使伊辛模型使用蒙特卡罗方法进行模拟。

The [[Hamiltonian mechanics|Hamiltonian]] that is commonly used to represent the energy of the model when using Monte Carlo methods is

The Hamiltonian that is commonly used to represent the energy of the model when using Monte Carlo methods is

蒙特卡罗方法中常用来表示模型能量的哈密顿量是

: <math>H(\sigma) = -J \sum_{\langle i~j\rangle} \sigma_i \sigma_j - h \sum_j \sigma_j.</math>

<math>H(\sigma) = -J \sum_{\langle i~j\rangle} \sigma_i \sigma_j - h \sum_j \sigma_j.</math>

数学 h (sigma)-jsum { langle i ~ j rangle } sigma j-h sum j sigma j. / math

Furthermore, the Hamiltonian is further simplified by assuming zero external field ''h'', since many questions that are posed to be solved using the model can be answered in absence of an external field. This leads us to the following energy equation for state σ:

Furthermore, the Hamiltonian is further simplified by assuming zero external field h, since many questions that are posed to be solved using the model can be answered in absence of an external field. This leads us to the following energy equation for state σ:

此外，假设外场为零，可以进一步简化哈密顿量，因为许多用该模型求解的问题可以在没有外场的情况下得到解答。这使我们得到以下状态能量方程:

: <math>H(\sigma) = -J \sum_{\langle i~j\rangle} \sigma_i \sigma_j.</math>

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

数学 h (sigma)-jsum { langle i ~ j rangle } sigma i sigma j. / math

Given this Hamiltonian, quantities of interest such as the specific heat or the magnetization of the magnet at a given temperature can be calculated.<ref name="Newman" />

Given this Hamiltonian, quantities of interest such as the specific heat or the magnetization of the magnet at a given temperature can be calculated.

根据这个哈密顿量，可以计算出一些有意义的量，例如磁体在给定温度下的比热或磁化强度。

===Metropolis algorithm===

====Overview====

The [[Metropolis–Hastings algorithm]] is the most commonly used Monte Carlo algorithm to calculate Ising model estimations.<ref name="Newman" /> The algorithm first chooses ''selection probabilities'' ''g''(μ, ν), which represent the probability that state ν is selected by the algorithm out of all states, given that one is in state μ. It then uses acceptance probabilities ''A''(μ, ν) so that [[detailed balance]] is satisfied. If the new state ν is accepted, then we move to that state and repeat with selecting a new state and deciding to accept it. If ν is not accepted then we stay in μ. This process is repeated until some stopping criterion is met, which for the Ising model is often when the lattice becomes [[ferromagnetic]], meaning all of the sites point in the same direction.<ref name="Newman" />

The Metropolis–Hastings algorithm is the most commonly used Monte Carlo algorithm to calculate Ising model estimations. The algorithm first chooses selection probabilities g(μ, ν), which represent the probability that state ν is selected by the algorithm out of all states, given that one is in state μ. It then uses acceptance probabilities A(μ, ν) so that detailed balance is satisfied. If the new state ν is accepted, then we move to that state and repeat with selecting a new state and deciding to accept it. If ν is not accepted then we stay in μ. This process is repeated until some stopping criterion is met, which for the Ising model is often when the lattice becomes ferromagnetic, meaning all of the sites point in the same direction.

Metropolis-Hastings 算法是最常用的 Monte Carlo 算法来计算 Ising 模型估计。算法首先选择选择概率 g (,) ，它表示算法从所有状态中选择状态的概率。然后它使用接受概率 a (,) ，以满足详细的平衡。如果新的状态被接受，那么我们移动到该状态，并通过选择一个新的状态并决定接受它来重复这个过程。如果不被接受，那么我们就呆在家里。这个过程不断重复，直到达到某种停止标准，对于伊辛模型来说，通常是当晶格变成铁磁性时，这意味着所有的位点都指向同一个方向。

When implementing the algorithm, one must ensure that ''g''(μ, ν) is selected such that [[ergodicity]] is met. In [[thermal equilibrium]] a system's energy only fluctuates within a small range.<ref name="Newman" /> This is the motivation behind the concept of '''single-spin-flip dynamics''', which states that in each transition, we will only change one of the spin sites on the lattice.<ref name="Newman" /> Furthermore, by using single- spin-flip dynamics, one can get from any state to any other state by flipping each site that differs between the two states one at a time.

When implementing the algorithm, one must ensure that g(μ, ν) is selected such that ergodicity is met. In thermal equilibrium a system's energy only fluctuates within a small range. This is the motivation behind the concept of single-spin-flip dynamics, which states that in each transition, we will only change one of the spin sites on the lattice. Furthermore, by using single- spin-flip dynamics, one can get from any state to any other state by flipping each site that differs between the two states one at a time.

在实现算法时，必须确保选择 g (,)以满足遍历性。在21世纪热平衡，一个系统的能量只在很小的范围内波动。这就是单自旋翻转动力学概念背后的动机，它指出在每个跃迁中，我们只改变晶格上的一个自旋位置。此外，通过使用单自旋翻转动力学，我们可以从任何状态翻转到任何其他状态，只要翻转两个状态之间每次一个不同的位置。

The maximum amount of change between the energy of the present state, ''H''μ and any possible new state's energy ''H''ν (using single-spin-flip dynamics) is 2''J'' between the spin we choose to "flip" to move to the new state and that spin's neighbor.<ref name="Newman" /> Thus, in a 1D Ising model, where each site has two neighbors (left and right), the maximum difference in energy would be 4''J''.

The maximum amount of change between the energy of the present state, Hμ and any possible new state's energy Hν (using single-spin-flip dynamics) is 2J between the spin we choose to "flip" to move to the new state and that spin's neighbor. Thus, in a 1D Ising model, where each site has two neighbors (left and right), the maximum difference in energy would be 4J.

当前态、 h 亚 / 亚态和任何可能的新态的能量 h 亚 / 亚态(使用单自旋翻转动力学)之间的最大变化量为2J，我们选择翻转到新态和该自旋的邻居之间的自旋。因此，在一维伊辛模型中，每个站点有两个邻居(左边和右边) ，能量的最大差异是4J。

Let ''c'' represent the '''lattice coordination number'''; the number of nearest neighbors that any lattice site has. We assume that all sites have the same number of neighbors due to [[periodic boundary conditions]].<ref name="Newman" /> It is important to note that the Metropolis–Hastings algorithm does not perform well around the critical point due to critical slowing down. Other techniques such as multigrid methods, Niedermayer's algorithm, Swendsen–Wang algorithm, or the Wolff algorithm are required in order to resolve the model near the critical point; a requirement for determining the critical exponents of the system.

Let c represent the lattice coordination number; the number of nearest neighbors that any lattice site has. We assume that all sites have the same number of neighbors due to periodic boundary conditions. It is important to note that the Metropolis–Hastings algorithm does not perform well around the critical point due to critical slowing down. Other techniques such as multigrid methods, Niedermayer's algorithm, Swendsen–Wang algorithm, or the Wolff algorithm are required in order to resolve the model near the critical point; a requirement for determining the critical exponents of the system.

设 c 表示格协调数，即任何格点所具有的最近邻居的数目。由于周期性的边界条件，我们假设所有的网站有相同数量的邻居。值得注意的是，由于临界减速，Metropolis-Hastings 算法在临界点附近并不能很好地运行。为了求解临界点附近的模型，需要使用其他技术，如多重网格法、尼德迈尔算法、 Swendsen-Wang 算法或沃尔夫算法，这是确定系统临界指数的一个要求。

====Specification====

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

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

具体针对伊辛模型和使用单自旋翻转动力学，可以建立如下。

Since there are ''L'' total sites on the lattice, using single-spin-flip as the only way we transition to another state, we can see that there are a total of ''L'' new states ν from our present state μ. The algorithm assumes that the selection probabilities are equal to the ''L'' states: ''g''(μ, ν) = 1/''L''. [[Detailed balance]] tells us that the following equation must hold:

Since there are L total sites on the lattice, using single-spin-flip as the only way we transition to another state, we can see that there are a total of L new states ν from our present state μ. The algorithm assumes that the selection probabilities are equal to the L states: g(μ, ν) = 1/L. Detailed balance tells us that the following equation must hold:

由于在晶格上有 l 个总位点，使用单自旋翻转作为我们过渡到另一个状态的唯一途径，我们可以看到从我们目前的状态总共有 l 个新状态。该算法假定选择概率等于 l 状态: g (,)1 / l。详细的平衡告诉我们，下面的等式必须成立:

: <math>\frac{P(\mu, \nu)}{P(\nu, \mu)} =

<math>\frac{P(\mu, \nu)}{P(\nu, \mu)} =

Math frac { p (mu，nu)}{ p (nu，mu)}

\frac{g(\mu, \nu) A(\mu, \nu)}{g(\nu, \mu) A(\nu, \mu)} =

\frac{g(\mu, \nu) A(\mu, \nu)}{g(\nu, \mu) A(\nu, \mu)} =

{ mu， nu) a (mu， nu)}{ g ( nu， mu) a ( nu，mu)}

\frac{A(\mu, \nu)}{A(\nu, \mu)} =

\frac{A(\mu, \nu)}{A(\nu, \mu)} =

{ mu， nu， mu }{ a ( nu， mu)}

\frac{P_\beta(\nu)}{P_\beta(\mu)} =

\frac{P_\beta(\nu)}{P_\beta(\mu)} =

Frac { p beta ( nu)}{ p beta ( mu)}

\frac{\frac{1}{Z} e^{-\beta(H_\nu)}}{\frac{1}{Z} e^{-\beta(H_\mu)}} =

\frac{\frac{1}{Z} e^{-\beta(H_\nu)}}{\frac{1}{Z} e^{-\beta(H_\mu)}} =

Frac {1}{ z } e ^ {- beta (h nu)}{ frac {1}{ z } e ^ {- beta (h mu)}}

e^{-\beta(H_\nu - H_\mu)}.</math>

e^{-\beta(H_\nu - H_\mu)}.</math>

E ^ {- beta (h nu-h mu)} . / math

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

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

因此，我们要选择我们的算法满足的接受概率

: <math>\frac{A(\mu, \nu)}{A(\nu, \mu)} = e^{-\beta(H_\nu - H_\mu)}.</math>

<math>\frac{A(\mu, \nu)}{A(\nu, \mu)} = e^{-\beta(H_\nu - H_\mu)}.</math>

Math frac { a ( mu， nu)} e ^ {- beta (h nu-h mu)} . / math

If ''H''ν > ''H''μ, then ''A''(ν, μ) > ''A''(μ, ν). Metropolis sets the larger of ''A''(μ, ν) or ''A''(ν, μ) to be 1. By this reasoning the acceptance algorithm is:<ref name="Newman" />

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

If Hν > Hμ, then A(ν, μ) > A(μ, ν).Metropolis 将 a (,)或 a (,)中较大的一个设置为1。根据这种推理，验收算法是:

: <math>A(\mu, \nu) = \begin{cases}

<math>A(\mu, \nu) = \begin{cases}

Math a ( mu， nu) begin { cases }

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

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

E ^ {- beta (h nu-h mu)} ，& text { if } h nu-h mu 0，

1 & \text{otherwise}.

1 & \text{otherwise}.

1 & text { otherwise }.

\end{cases}</math>

\end{cases}</math>

End { cases } / math

The basic form of the algorithm is as follows:

The basic form of the algorithm is as follows:

该算法的基本形式如下:

# Pick a spin site using selection probability ''g''(μ, ν) and calculate the contribution to the energy involving this spin.

Pick a spin site using selection probability g(μ, ν) and calculate the contribution to the energy involving this spin.

使用选择概率 g (,)选择一个自旋位置，并计算该自旋对能量的贡献。

# Flip the value of the spin and calculate the new contribution.

Flip the value of the spin and calculate the new contribution.

翻转旋转的值并计算新的贡献。

# If the new energy is less, keep the flipped value.

If the new energy is less, keep the flipped value.

如果新能量较少，则保持反转值。

# If the new energy is more, only keep with probability <math>e^{-\beta(H_\nu - H_\mu)}.</math>

If the new energy is more, only keep with probability <math>e^{-\beta(H_\nu - H_\mu)}.</math>

如果新能量更多，只保留 e ^ {- beta (h nu-h mu)} . / math

# Repeat.

Repeat.

重复。

The change in energy ''H''ν − ''H''μ 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.

The change in energy Hν − Hμ 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.

能量 h sub / sub-h sub / sub 的变化仅取决于自旋及其最近邻图的值。因此，如果图不是太连通，算法是快速的。这个过程最终将产生一个从分配选择。

===Viewing the Ising model as a Markov chain===

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.

It is possible to view the Ising model as a Markov chain, as the immediate probability Pβ(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. Furthermore, since the energy equation Hσ change only depends on the nearest-neighbor interaction strength J, the Ising model and its variants such the Sznajd model can be seen as a form of a voter model for opinion dynamics.

可以把 Ising 模型看作是一个马尔可夫链，因为过渡到未来状态的即时概率 p 子 / 子()只取决于当前状态。Metropolis–Hastings 演算法实际上是马尔科夫蒙特卡洛模拟的一个版本，由于我们在 Metropolis–Hastings 演算法中使用了单自旋翻转动力学，每个状态都可以被看作是与 l 其他状态的链接，其中每个跃迁都对应于翻转一个自旋位置与相反的值。此外，由于能量方程 h 子 / 子变化仅依赖于最近邻相互作用强度 j，伊辛模型及其变体如 Sznajd 模型可以被看作是观点动力学选民模型的一种形式。

==One dimension==

The thermodynamic limit exists as soon as the interaction decay is <math>J_{ij} \sim |i - j|^{-\alpha}</math> with α > 1.<ref name="Ruelle">{{cite book |last1=Ruelle |title=Statistical Mechanics:Rigorous Results |publisher=W. A. Benjamin Inc. |year=1969 |location=New York}}</ref>

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

一旦相互作用衰变为 math j { ij } sim | i-j | ^ {- alpha } / math 1，热力学极限就会存在。

* In the case of ''ferromagnetic'' interaction <math>J_{ij} \sim |i - j|^{-\alpha} </math> with 1 < α < 2, Dyson proved, by comparison with the hierarchical case, that there is phase transition at small enough temperature.<ref>{{cite journal |last=Dyson |first=F. J. |title=Existence of a phase-transition in a one-dimensional Ising ferromagnet |journal=Comm. Math. Phys. |year=1969 |volume=12 |issue=2 |pages=91–107 |doi=10.1007/BF01645907 |bibcode = 1969CMaPh..12...91D }}</ref>

* In the case of ''ferromagnetic'' interaction <math>J_{ij} \sim |i - j|^{-2}</math>, Fröhlich and Spencer proved that there is phase transition at small enough temperature (in contrast with the hierarchical case).<ref>{{cite journal |last1=Fröhlich |first1=J. |last2=Spencer |first2=T. |title=The phase transition in the one-dimensional Ising model with 1/''r''2 interaction energy |journal=Comm. Math. Phys. |year=1982 |volume=84 |issue=1 |doi=10.1007/BF01208373 |pages=87–101 |bibcode = 1982CMaPh..84...87F }}</ref>

* In the case of interaction <math>J_{ij} \sim |i - j|^{-\alpha}</math> with α > 2 (which includes the case of finite-range interactions), there is no phase transition at any positive temperature (i.e. finite β), since the [[Thermodynamic free energy|free energy]] is analytic in the thermodynamic parameters.<ref name="Ruelle"/>

* 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}}</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 periodic boundary conditions is

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 periodic boundary conditions is

在最近邻情况下(有周期或自由边界条件) ，可得到精确解。本文研究了具有周期边界条件的 l 网格上的一维伊辛模型的哈密顿量

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

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

数学 h (sigma)-jsum { i 1，ldots，l-1} sigma i sigma { i + 1}-h sigma i，/ math

where ''J'' and ''h'' can be any number, since in this simplified case ''J'' is a constant representing the interaction strength between the nearest neighbors and ''h'' is the constant external magnetic field applied to lattice sites. Then the

where J and h can be any number, since in this simplified case J is a constant representing the interaction strength between the nearest neighbors and h is the constant external magnetic field applied to lattice sites. Then the

其中 j 和 h 可以是任意数，因为在这种简化的情况下，j 是表示最近邻之间相互作用强度的常数，而 h 是施加在晶格点上的恒定外磁场。然后

[[Thermodynamic free energy|free energy]] is

free energy is

自由能是

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

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

数学 f ( beta，h)- lim { l to infty } frac {1}{ beta l } ln z ( beta)- frac {1}{ beta } ln 左(e ^ { beta j } cosh beta h + sqrt { e ^ beta }( sinh h) ^ 2 + e {-2 beta j }右) ,

</math>

</math>

数学

and the spin-spin correlation (i.e. the covariance) is

and the spin-spin correlation (i.e. the covariance) is

和自旋-自旋相关性(即自旋-自旋相关性。协方差)是

: <math>\langle\sigma_i \sigma_j\rangle - \langle\sigma_i\rangle \langle\sigma_j\rangle = C(\beta) e^{-c(\beta)|i - j|},</math>

<math>\langle\sigma_i \sigma_j\rangle - \langle\sigma_i\rangle \langle\sigma_j\rangle = C(\beta) e^{-c(\beta)|i - j|},</math>

数学 langle sigma i sigma j rangle langle sigma i rangle langle sigma j rangle c ( beta) e ^ {-c ( beta) | i-j | } ，/ math

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

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

其中 c ()和 c ()是 t0的正函数。然而对于 t →0，逆相关长度 c ()消失了。

====Proof====

The proof of this result is a simple computation.

The proof of this result is a simple computation.

这个结果的证明是一个简单的计算。

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

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

如果 h 为0，在自由边界条件下很容易得到自由能。当

: <math>H(\sigma) = -J(\sigma_1 \sigma_2 + \cdots + \sigma_{L-1} \sigma_L).</math>

<math>H(\sigma) = -J(\sigma_1 \sigma_2 + \cdots + \sigma_{L-1} \sigma_L).</math>

数学 h (sigma)-j (sigma 1-sigma 2 + cdots + sigma { L-1}-sigma l) . /

Then the model factorizes under the change of variables

Then the model factorizes under the change of variables

然后模型在变量变化的情况下进行因式分解

: <math>\sigma'_j = \sigma_j \sigma_{j-1}, \quad j \ge 2.</math>

<math>\sigma'_j = \sigma_j \sigma_{j-1}, \quad j \ge 2.</math>

数学，数学，数学

This gives

This gives

这给了

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

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

数学 z (beta) sum { sigma 1， ldots， sigma l } e ^ { beta j sigma 1 sigma 2} e ^ { beta j sigma 2 sigma 3} cdots e ^ { beta j sigma l }2 prod { j2∑ j’ j } e ^ { beta j j’ j } j ^ 2左[ beta j } + e ^-beta j }右]{ L-1}。数学

Therefore, the free energy is

Therefore, the free energy is

因此，自由能是

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

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

Math f ( beta，0)- frac {1}{ beta } ln left [ e ^ { beta j } + e ^ {- beta j } right ] . / math

With the same change of variables

With the same change of variables

变量的变化相同

: <math>\langle\sigma_j\sigma_{j+N}\rangle = \left[\frac{e^{\beta J} - e^{-\beta J}}{e^{\beta J} + e^{-\beta J}}\right]^N,</math>

<math>\langle\sigma_j\sigma_{j+N}\rangle = \left[\frac{e^{\beta J} - e^{-\beta J}}{e^{\beta J} + e^{-\beta J}}\right]^N,</math>

[数学] ，[数学] ，[数学] ，[数学]

hence it decays exponentially as soon as ''T'' ≠ 0; but for ''T'' = 0, i.e. in the limit β → ∞ there is no decay.

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

因此，它在 t ≠0时指数衰减; 但对于 t 0，即。在极限→∞中不存在衰变。

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

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

如果 h ≠0，我们需要传递矩阵法。对于周期边界条件，情况如下。配分函数

: <math>Z(\beta) = \sum_{\sigma_1,\ldots,\sigma_L} e^{\beta h \sigma_1} e^{\beta J\sigma_1\sigma_2} e^{\beta h \sigma_2} e^{\beta J\sigma_2\sigma_3} \cdots e^{\beta h \sigma_L} e^{\beta J\sigma_L\sigma_1} = \sum_{\sigma_1,\ldots,\sigma_L} V_{\sigma_1,\sigma_2} V_{\sigma_2,\sigma_3} \cdots V_{\sigma_L,\sigma_1}.</math>

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

数学 z (beta) sum { sigma 1，ldots， sigma l } e ^ { beta h sigma 1} e ^ { beta j1 sigma 2} e ^ { beta h sigma 2} e ^ { beta j2 sigma 3} cdots e ^ beta h sigma { j sigma l sigma 1}和 { sigma 1，ldots，sigma l } V { sigma 1，sigma 2} v { sigma 2，sigma 3} cdots v { sigma l，sigma 1}. 数学

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

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

系数数学 v { sigma，sigma’} / math 可以看作是矩阵的项。有不同的选择: 方便的选择(因为矩阵是对称的)是

: <math>V_{\sigma, \sigma'} = e^{\frac{\beta h}{2} \sigma} e^{\beta J\sigma\sigma'} e^{\frac{\beta h}{2} \sigma'}</math>

<math>V_{\sigma, \sigma'} = e^{\frac{\beta h}{2} \sigma} e^{\beta J\sigma\sigma'} e^{\frac{\beta h}{2} \sigma'}</math>

数学五西格玛，西格玛’} e ^ {贝塔 · 西格玛} e ^ {贝塔 · 西格玛’} e ^ {贝塔 · 西格玛’} / 数学

or

or

或

: <math>V = \begin{bmatrix}

<math>V = \begin{bmatrix}

数学 v 开始{ bmatrix }

e^{\beta(h+J)} & e^{-\beta J} \\

e^{\beta(h+J)} & e^{-\beta J} \\

E ^ { beta (h + j)} & e ^ {- beta j }

e^{-\beta J} & e^{-\beta(h-J)}

e^{-\beta J} & e^{-\beta(h-J)}

E ^ {- beta j } & e ^ {- beta (h-J)}

\end{bmatrix}.</math>

\end{bmatrix}.</math>

{ bmatrix } . / math

In matrix formalism

In matrix formalism

在矩阵形式中

: <math>Z(\beta) = \operatorname{Tr} \left(V^L\right) = \lambda_1^L + \lambda_2^L = \lambda_1^L \left[1 + \left(\frac{\lambda_2}{\lambda_1}\right)^L\right],</math>

<math>Z(\beta) = \operatorname{Tr} \left(V^L\right) = \lambda_1^L + \lambda_2^L = \lambda_1^L \left[1 + \left(\frac{\lambda_2}{\lambda_1}\right)^L\right],</math>

数学 z ( beta) operatorname { Tr } left (v ^ l right) lambda 1 ^ l + lambda 2 ^ l lambda 1 ^ l left [1 + 左(frac { lambda 2}{ lambda 1}右) ^ l right ] ，/ math

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

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

其中子1 / 子是 v 的最高特征值，而子2 / 子是另一个特征值:

: <math>\lambda_1 = e^{\beta J} \cosh \beta h + \sqrt{e^{2\beta J} (\sinh\beta h)^2 + e^{-2\beta J}},</math>

<math>\lambda_1 = e^{\beta J} \cosh \beta h + \sqrt{e^{2\beta J} (\sinh\beta h)^2 + e^{-2\beta J}},</math>

1 e ^ { beta j } cosh beta h + sqrt { e ^ {2 beta j }( sinh beta h) ^ 2 + e ^ {-2 beta j } ，/ math

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

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

and |λ2| < λ1.这就给出了自由能的公式。

====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.

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 2J times the number of sign changes that are encountered when scanning the configuration from left to right.

当所有的自旋都相同时，最低态的能量是 -JL。对于任何其他配置，额外能量等于2J 乘以从左向右扫描配置时遇到的符号更改次数。

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

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

如果我们把一个组态中符号变化的数目指定为 k，那么最低能态的能量差就是2 k。由于能量是翻转次数的附加，因此在每个位置发生自旋翻转的概率 p 是独立的。找到翻转的概率与找不到翻转的概率的比率是玻尔兹曼因子:

: <math>\frac{p}{1 - p} = e^{-2\beta J}.</math>

<math>\frac{p}{1 - p} = e^{-2\beta J}.</math>

1-p } e ^ {-2 beta j } . / math

The problem is reduced to independent biased [[coin toss]]es. This essentially completes the mathematical description.

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

这个问题归结为独立的有偏见的硬币投掷。这基本上完成了数学描述。

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

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

通过对模型的独立向量的描述，可以理解长线模型的统计特性。该行分成多个域。每个域的平均长度为 exp (2)。域的长度呈指数分布，因为碰到翻转的任何一步都有一个常数概率。磁畴永远不会变成无限的，所以长系统永远不会被磁化。每一步都将自旋与其邻居之间的相关性降低一个与 p 成正比的数量，因此相关性呈指数下降。

: <math>\langle S_i S_j \rangle \propto e^{-p|i-j|}.</math>

<math>\langle S_i S_j \rangle \propto e^{-p|i-j|}.</math>

数学，数学，数学，数学

The [[partition function (statistical mechanics)|partition function]] 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 partition function 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:

配分函数是配置的体积，每个配置以其 Boltzmann 重量加权。因为每个配置都是由符号变化来描述的，所以配分函数分解因子分解:

: <math>Z = \sum_{\text{configs}} e^{\sum_k S_k} = \prod_k (1 + p ) = (1 + p)^L.</math>

<math>Z = \sum_{\text{configs}} e^{\sum_k S_k} = \prod_k (1 + p ) = (1 + p)^L.</math>

<math>Z = \sum_{\text{configs}} e^{\sum_k S_k} = \prod_k (1 + p ) = (1 + p)^L.</math>

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

The logarithm divided by L is the free energy density:

对数除以 l 是自由能密度:

: <math>\beta f = \log(1 + p) = \log\left(1 + \frac{e^{-2\beta J}}{1 + e^{-2\beta J}}\right),</math>

<math>\beta f = \log(1 + p) = \log\left(1 + \frac{e^{-2\beta J}}{1 + e^{-2\beta J}}\right),</math>

Math beta f log (1 + p) log left (1 + frac { e ^ {-2 beta j }{1 + e ^ {-2 beta j } right) ，/ math

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.

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

这是一个偏离∞的解析过程。相变的标志是一个非解析的自由能，所以一维模型没有相变。

===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

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 Hamiltonian is given by

为了用自旋的量子力学描述来表示伊辛哈密顿量，我们将自旋变量替换为它们各自的泡利矩阵。然而，根据磁场的方向，我们可以建立横向磁场或纵向磁场的哈密顿量。给出了横向场的哈密顿量

: <math>H(\sigma) = -J \sum_{i=1,\ldots,L} \sigma_i^z \sigma_{i+1}^z - h \sum_i \sigma_i^x.</math>

<math>H(\sigma) = -J \sum_{i=1,\ldots,L} \sigma_i^z \sigma_{i+1}^z - h \sum_i \sigma_i^x.</math>

数学 h (sigma)-jsum { i 1，ldots，l } | sigma i ^ z { i + 1} ^ z-h sum i ^ sigma i ^ x / math

The transverse-field model experiences a phase transition between an ordered and disordered regime at ''J'' ~ ''h''. This can be shown by a mapping of Pauli matrices

The transverse-field model experiences a phase transition between an ordered and disordered regime at J ~ h. This can be shown by a mapping of Pauli matrices

横场模型在 j ~ h 处经历了有序区和无序区之间的相变过程。这可以用 Pauli 矩阵的映射来表示

: <math>\sigma_n^z = \prod_{i=1}^n T_i^x,</math>

<math>\sigma_n^z = \prod_{i=1}^n T_i^x,</math>

数学，数学，数学，数学

: <math>\sigma_n^x = T_n^z T_{n+1}^z.</math>

<math>\sigma_n^x = T_n^z T_{n+1}^z.</math>

数学 n ^ x n ^ z t { n + 1} ^ z / math

Upon rewriting the Hamiltonian in terms of this change-of-basis matrices, we obtain

Upon rewriting the Hamiltonian in terms of this change-of-basis matrices, we obtain

利用基变换矩阵重写哈密顿量，我们得到了哈密顿量

: <math>H(\sigma) = -h \sum_{i=1,\ldots,L} T_i^z T_{i+1}^z - J \sum_i T_i^x.</math>

<math>H(\sigma) = -h \sum_{i=1,\ldots,L} T_i^z T_{i+1}^z - J \sum_i T_i^x.</math>

数学 h (sigma)-h (sum { i 1，ldots，l } t ^ z ^ t (i + 1) ^ z-j (sum i) t i ^ x / math

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= http://cds.cern.ch/record/1513030}}</ref>

Since the roles of h and J are switched, the Hamiltonian undergoes a transition at J = h.

由于 h 和 j 的角色互换，哈密顿量在 j h 处发生了转变。

==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

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

得到了各向异性方晶格上 Ising 模型自由能的解析表达式，其中磁场的数学公式 h0 / math 分别是温度的函数和水平和垂直相互作用能的数学公式 j1 / math 和数学公式 j2 / math

:<math> -\beta f = \ln 2 + \frac{1}{8\pi^2}\int_0^{2\pi}d\theta_1\int_0^{2\pi}d\theta_2 \ln[\cosh(2\beta J_1)\cosh(2\beta J_2) -\sinh(2\beta J_1)\cos(\theta_1)-\sinh(2\beta J_2)\cos(\theta_2)]. </math>

<math> -\beta f = \ln 2 + \frac{1}{8\pi^2}\int_0^{2\pi}d\theta_1\int_0^{2\pi}d\theta_2 \ln[\cosh(2\beta J_1)\cosh(2\beta J_2) -\sinh(2\beta J_1)\cos(\theta_1)-\sinh(2\beta J_2)\cos(\theta_2)]. </math>

Math-beta f ln 2 + frac {1}8 pi ^ 2} int 0 ^ {2 pi } d theta 2 ln [ cosh (2 beta j1) cosh (2 beta j2)- sinh (2 beta j1) cos ( theta 1)-sinh (2 beta j2) cos ( theta 2)].数学

From this expression for the free energy, all thermodynamic functions of the model can be calculated by using an appropriate derivative. The 2D Ising model was the first model to exhibit a continuous phase transition at a positive temperature. It occurs at the temperature <math>T_c</math> which solves the equation

From this expression for the free energy, all thermodynamic functions of the model can be calculated by using an appropriate derivative. The 2D Ising model was the first model to exhibit a continuous phase transition at a positive temperature. It occurs at the temperature <math>T_c</math> which solves the equation

根据自由能的表达式，可以用适当的导数计算出模型的所有热力学函数。2D Ising 模型是第一个在正温度下表现出连续相变的模型。它发生在求解方程的温度数学中

:<math> \sinh\left(\frac{2J_1}{kT_c}\right)\sinh\left(\frac{2J_2}{kT_c}\right) = 1. </math>

<math> \sinh\left(\frac{2J_1}{kT_c}\right)\sinh\left(\frac{2J_2}{kT_c}\right) = 1. </math>

数学 sinh ( frac {2j1}{ kT c }右) sinh ( frac {2j2}{ kT c }右)1。数学

In the isotropic case when the horizontal and vertical interaction energies are equal <math>J_1=J_2=J</math>, the critical temperature <math>T_c</math> occurs at the following point

In the isotropic case when the horizontal and vertical interaction energies are equal <math>J_1=J_2=J</math>, the critical temperature <math>T_c</math> occurs at the following point

在各向同性情况下，当水平相互作用能和垂直相互作用能等于 j1j2j / math 时，临界温度数学 t / math 发生在下一点

:<math> T_c = \frac{2J}{k\ln(1+\sqrt{2})} </math>

<math> T_c = \frac{2J}{k\ln(1+\sqrt{2})} </math>

2J }{ k ln (1 + sqrt {2})} / math

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.

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.

当相互作用能数学 j1 / math，数学 j2 / math 都是负值时，伊辛模型就变成了反铁磁体。由于正方晶格是双体的，在磁场数为 h0 / math 的情况下它是不变的，所以反铁磁情况下的自由能和临界温度是相同的。对于不是双体的三角晶格，铁磁和反铁磁 Ising 模型表现出明显的不同。

====Transfer matrix====

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

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

我们可以以量子力学为例。长周期点阵上的伊辛模型有一个配分函数

:<math>\sum_S \exp\biggl(\sum_{ij} S_{i,j} S_{i,j+1} + S_{i,j} S_{i+1,j}\biggr).</math>

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

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

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

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

把 i 的方向想象成空间，把 j 的方向想象成时间。这是自旋在每个时刻所能取得的所有值的独立和。这是一种路径积分，它是所有自旋历史的和。

A path integral can be rewritten as a Hamiltonian evolution. The Hamiltonian steps through time by performing a unitary rotation between time ''t'' and time ''t'' + Δ''t'':

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

路径积分可以重写为哈密顿演化。哈密顿量通过在时间 t 和时间 t + t 之间执行幺正旋转来逐步通过时间:

:<math> U = e^{i H \Delta t}</math>

<math> U = e^{i H \Delta t}</math>

数学 u e ^ { i h Delta t } / 数学

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

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

U 矩阵的乘积，一个接一个，是总时间演化算符，也就是我们开始讨论的路径积分。

:<math> U^N = (e^{i H \Delta t})^N = \int DX e^{iL}</math>

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

数学 u ^ n (e ^ { i h Delta t }) ^ n int DX e ^ { iL } / math

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.

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.

其中 n 是时间片的数量。所有路径的和是由矩阵的乘积给出的，每个矩阵元素是从一个切片到下一个切片的转移概率。

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

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

类似地，我们可以将所有配分函数配置的总和分成片，其中每个片在时间1时是一维配置。这定义了转移矩阵:

:<math>T_{C_1 C_2}.</math>

<math>T_{C_1 C_2}.</math>

数学 t { c1c2} . / 数学

The configuration in each slice is a one-dimensional collection of spins. At each time slice, ''T'' has matrix elements between two configurations of spins, one in the immediate future and one in the immediate past. These two configurations are ''C''1 and ''C''2, and they are all one-dimensional spin configurations. We can think of the vector space that ''T'' acts on as all complex linear combinations of these. Using quantum mechanical notation:

The configuration in each slice is a one-dimensional collection of spins. At each time slice, T has matrix elements between two configurations of spins, one in the immediate future and one in the immediate past. These two configurations are C1 and C2, and they are all one-dimensional spin configurations. We can think of the vector space that T acts on as all complex linear combinations of these. Using quantum mechanical notation:

每个薄片中的构型都是一维自旋的集合。在每一个时间片上，t 有两种自旋构型之间的矩阵元素，一种在不久的将来，另一种在不久的过去。这两种构型是 c 子1 / 子和 c 子2 / 子，它们都是一维自旋构型。我们可以把 t 作用的向量空间，看作是所有这些的复杂线性组合。使用量子力学记数法:

:<math>|A\rangle = \sum_S A(S) |S\rangle</math>

<math>|A\rangle = \sum_S A(S) |S\rangle</math>

数学 | a rangle sum s a (s) | s rangle / math

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

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

其中每个基向量数学 | s rangle / math 是一维伊辛模型的自旋结构。

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:

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:

与哈密顿量一样，转移矩阵作用于所有状态的线性组合。配分函数是 t 的矩阵函数，定义为 n 步后所有历史回到原始构型的和:

:<math>Z= \mathrm{tr}(T^N).</math>

<math>Z= \mathrm{tr}(T^N).</math>

(t ^ n) . / math

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''.

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.

因为这是一个矩阵方程，它可以在任何基础上进行评估。所以如果我们能对角化矩阵 t，我们就能找到 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:

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:

切片上过去 / 将来每对配置对配分函数的贡献是两项之和。在过去的切片中有多少次旋转翻转，在过去和未来的切片之间有多少次旋转翻转。定义一个操作符的配置翻转在网站 i 的旋转:

:<math>\sigma^x_i.</math>

<math>\sigma^x_i.</math>

数学，数学

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.

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.

在通常的伊辛基础上，作用于任何线性组合的过去构型，它产生相同的线性组合，但在每个基矢的位置 i 自旋翻转。

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

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

定义第二个算符，根据位置 i 的自旋将基矢量乘以 + 1和-1:

:<math>\sigma^z_i.</math>

<math>\sigma^z_i.</math>

数学，数学

''T'' can be written in terms of these:

T can be written in terms of these:

可以这样写:

:<math>\sum_i A \sigma^x_i + B \sigma^z_i \sigma^z_{i+1}</math>

<math>\sum_i A \sigma^x_i + B \sigma^z_i \sigma^z_{i+1}</math>

数学总和 i a / sigma ^ si + b / sigma ^ z ^ i + 1} /

where ''A'' and ''B'' are constants which are to be determined so as to reproduce the partition function. The interpretation is that the statistical configuration at this slice contributes according to both the number of spin flips in the slice, and whether or not the spin at position ''i' has flipped.

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.

其中 a 和 b 是常数，这些常数需要确定以便重现配分函数。这种解释是，这个切片上的统计构型取决于切片上自旋翻转的次数，以及位置 i’处的自旋是否翻转。

====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:

Just as in the one-dimensional case, we will shift attention from the spins to the spin-flips. The σz term in T counts the number of spin flips, which we can write in terms of spin-flip creation and annihilation operators:

正如在一维情况下，我们将把注意力从旋转转移到旋转翻转。T 中的 sup z / sup 项计算自旋翻转的次数，我们可以用自旋翻转创生及消灭算符来表示:

:<math> \sum C \psi^\dagger_i \psi_i. \,</math>

<math> \sum C \psi^\dagger_i \psi_i. \,</math>

<math> \sum C \psi^\dagger_i \psi_i.数学

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

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

第一个术语翻转一个旋转，所以根据基本状态它可以是:

#moves a spin-flip one unit to the right

moves a spin-flip one unit to the right

向右移动一个单位

#moves a spin-flip one unit to the left

moves a spin-flip one unit to the left

向左移动一个单位

#produces two spin-flips on neighboring sites

produces two spin-flips on neighboring sites

在邻近的网站上产生两个自旋翻转

#destroys two spin-flips on neighboring sites.

destroys two spin-flips on neighboring sites.

摧毁了邻近网站的两个自旋翻转。

Writing this out in terms of creation and annihilation operators:

Writing this out in terms of creation and annihilation operators:

以创生及消灭算符的方式写下这些:

:<math> \sigma^x_i = D {\psi^\dagger}_i \psi_{i+1} + D^* {\psi^\dagger}_i \psi_{i-1} + C\psi_i \psi_{i+1} + C^* {\psi^\dagger}_i {\psi^\dagger}_{i+1}.</math>

<math> \sigma^x_i = D {\psi^\dagger}_i \psi_{i+1} + D^* {\psi^\dagger}_i \psi_{i-1} + C\psi_i \psi_{i+1} + C^* {\psi^\dagger}_i {\psi^\dagger}_{i+1}.</math>

Math sigma ^ x i d { psi ^ i + 1} + d ^ * psi ^ dagger { i-1} + cpsi i + 1} + c ^ psi ^ dagger { i + 1} . / math

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.

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.

忽略常数系数，把注意力集中在形式上。它们都是二次方程式。由于系数是常数，这意味着 t 矩阵可以通过傅里叶变换对角化。

Carrying out the diagonalization produces the Onsager free energy.

Carrying out the diagonalization produces the Onsager free energy.

实现对角化生成昂萨格自由能。

====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"/>

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

昂萨格在1948年的两次不同的会议上，对方形晶格上的二维伊辛铁磁体的自发磁化 m 做了一个著名的表述，尽管没有证据

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

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

数学 m 左(1- 左[1-sinh 2 beta j1 sinh 2 beta j2右] ^ {-2}右) ^ { frac {1} / math

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

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

其中数学 j1 / math 和数学 j2 / math 是水平和垂直相互作用的能量。

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.

A complete derivation was only given in 1951 by using a limiting process of transfer matrix eigenvalues. The proof was subsequently greatly simplified in 1963 by Montroll, Potts, and Ward using Szegő's limit formula for Toeplitz determinants by treating the magnetization as the limit of correlation functions.

直到1951年才利用转移矩阵特征值的极限过程给出了完整的推导。1963年，Montroll、 Potts 和 Ward 用 szeg 的 Toeplitz 行列式极限公式，将磁化强度作为相关函数的极限，大大简化了这一证明。

===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 models|minimal model]], which has been exactly solved.

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, which has been exactly solved.

在临界点，二维 Ising 模型是一个二维的共形场论。用极小模型描述了自旋和能量关联函数，并得到了精确解。

==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 Markovich Polyakov|Alexander Polyakov]]. The [[Critical point (thermodynamics)|critical point]] of the three-dimensional Ising model is described by a [[conformal field theory]], as evidenced by [[Metropolis–Hastings algorithm|Monte Carlo]] simulations<ref>{{Cite journal| doi = 10.1007/JHEP07(2013)055| volume = 1307| issue = 7| pages = 055| last1 = Billó| first1 = M.| last2 = Caselle| first2 = M.| last3 = Gaiotto| first3 = D.| last4 = Gliozzi| first4 = F.| last5 = Meineri| first5 = M.| last6 = others| title = Line defects in the 3d Ising model| journal = JHEP| date = 2013|bibcode = 2013JHEP...07..055B | arxiv = 1304.4110}}</ref><ref>{{Cite journal|last1 = Cosme|first1 = Catarina|last2 = Lopes|first2 = J. M. Viana Parente|last3 = Penedones|first3 = Joao|title = Conformal symmetry of the critical 3D Ising model inside a sphere|journal = Journal of High Energy Physics|volume = 2015|issue = 8|pages = 22|date = 2015|arxiv = 1503.02011|doi = 10.1007/JHEP08(2015)022|bibcode = 2015JHEP...08..022C}}</ref> and theoretical arguments.<ref>{{Cite journal|last1 = Delamotte|first1 = Bertrand|last2 = Tissier|first2 = Matthieu|last3 = Wschebor|first3 = Nicolás|title = Scale invariance implies conformal invariance for the three-dimensional Ising model|journal = Physical Review E|volume = 93|issue = 12144|pages = 012144|year = 2016|arxiv = 1501.01776|doi = 10.1103/PhysRevE.93.012144|pmid = 26871060|bibcode = 2016PhRvE..93a2144D}}</ref> This conformal field theory is under active investigation using the method of the [[conformal bootstrap]].<ref>{{Cite journal| doi = 10.1103/PhysRevD.86.025022| volume = D86| issue = 2| pages = 025022| last1 = El-Showk| first1 = Sheer| last2 = Paulos| first2 = Miguel F.| last3 = Poland| first3 = David| last4 = Rychkov| first4 = Slava| last5 = Simmons-Duffin| first5 = David| last6 = Vichi| first6 = Alessandro| title = Solving the 3D Ising Model with the Conformal Bootstrap| journal = Phys. Rev.| date = 2012|arxiv = 1203.6064 |bibcode = 2012PhRvD..86b5022E }}</ref><ref name="cmin">{{Cite journal| doi = 10.1007/s10955-014-1042-7| volume = 157| issue = 4–5| pages = 869–914| last1 = El-Showk| first1 = Sheer| last2 = Paulos| first2 = Miguel F.| last3 = Poland| first3 = David| last4 = Rychkov| first4 = Slava| last5 = Simmons-Duffin| first5 = David| last6 = Vichi| first6 = Alessandro| title = Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents| journal = Journal of Statistical Physics| date = 2014|arxiv = 1403.4545 |bibcode = 2014JSP...tmp..139E }}</ref><ref name="SDPB">{{Cite journal| doi = 10.1007/JHEP06(2015)174| issn = 1029-8479| volume = 2015| issue = 6| pages = 174| last = Simmons-Duffin| first = David| title = A semidefinite program solver for the conformal bootstrap| journal = Journal of High Energy Physics| date = 2015|arxiv = 1502.02033 |bibcode = 2015JHEP...06..174S }}</ref><ref name="Kadanoff">{{cite web

In three dimensions, the Ising model was shown to have a representation in terms of non-interacting fermionic lattice strings by Alexander Polyakov. The critical point of the three-dimensional Ising model is described by a conformal field theory, as evidenced by Monte Carlo simulations and theoretical arguments. This conformal field theory is under active investigation using the method of the conformal bootstrap.<ref name="Kadanoff">{{cite web

在三维空间中，亚历山大 · 波里亚科夫用非相互作用的费米格子弦表示伊辛模型。三维 Ising 模型的临界点由一个共形场论来描述，这一点可以通过 Monte Carlo 模拟和理论论证得到证实。本共形场论正在使用共形引导法进行积极的调查

| last = Kadanoff

| last = Kadanoff

最后的卡达诺夫

| first = Leo P.

| first = Leo P.

首先是 Leo p。

| title = Deep Understanding Achieved on the 3d Ising Model

| title = Deep Understanding Achieved on the 3d Ising Model

对3d Ising 模型的深入理解

| website = Journal Club for Condensed Matter Physics

| website = Journal Club for Condensed Matter Physics

2012年3月15日 | 凝聚态物理学日报俱乐部网站

| date = April 30, 2014

| date = April 30, 2014

2014年4月30日

| url = http://www.condmatjournalclub.org/?p=2384

| url = http://www.condmatjournalclub.org/?p=2384

Http://www.condmatjournalclub.org/?p=2384

| access-date = July 19, 2015

| access-date = July 19, 2015

| 访问日期: 2015年7月19日

| archive-url = https://web.archive.org/web/20150722062827/http://www.condmatjournalclub.org/?p=2384

| archive-url = https://web.archive.org/web/20150722062827/http://www.condmatjournalclub.org/?p=2384

| 档案-网址 https://web.archive.org/web/20150722062827/http://www.condmatjournalclub.org/?p=2384

| archive-date = July 22, 2015

| archive-date = July 22, 2015

2015年7月22日

| url-status = dead

| url-status = dead

状态死机

}}</ref> This method currently yields the most precise information about the structure of the critical theory (see [[Ising critical exponents]]).

}}</ref> This method currently yields the most precise information about the structure of the critical theory (see Ising critical exponents).

} / ref 此方法目前可以得到关于临界理论结构的最精确的信息(请参阅 Ising 临界指数)。

In dimensions near four, the critical behavior of the model is understood to correspond to the [[renormalization group|renormalization]] behavior of the [[Quartic interaction|scalar phi-4 theory]] (see [[Kenneth G. Wilson|Kenneth Wilson]]).

In dimensions near four, the critical behavior of the model is understood to correspond to the renormalization behavior of the scalar phi-4 theory (see Kenneth Wilson).

在四维附近，模型的临界行为被理解为对应于标量 phi-4理论的重整化行为(见 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.

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.

在任何维度上，伊辛模型都可以用局部变化的平均场有效地描述。磁场定义为一个大区域上的平均自旋值，但不要太大以至于包括整个系统。该领域仍然有点到点的缓慢变化，作为平均体积移动。这些场的涨落用无限系统极限下的连续场理论来描述。

===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''.

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.

场 h 被定义为自旋变量的长波长傅里叶分量，在波长很长的限制下。取长波长平均值的方法有很多，这取决于如何截断高波长的细节。细节并不太重要，因为我们的目标是找到 h 的统计数据，而不是旋转。一旦 h 中的关联已知，自旋之间的长距离关联将与 h 中的长距离关联成正比。

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.

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.

对于任何缓变场 h 的值，自由能(对数概率)是 h 及其梯度的局部解析函数。自由能 f (h)定义为与长波场一致的所有伊辛构型的和。由于 h 是一个粗略的描述，所以有许多与 h 的每个值一致的 Ising 配置，只要匹配不需要太多的精确度。

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.

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.

由于任何区域的自旋允许值范围仅取决于该区域一个平均体积内的 h 值，因此每个区域的自由能贡献仅取决于该区域和邻近区域的 h 值。所以 f 是局部贡献的所有区域的和，它只依赖于 h 和它的导数。

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:

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:

由于 h 的对称性，只有偶数的幂才有贡献。通过在正方形晶格上的反射对称，只有梯度的偶数次方有所贡献。写出自由能量的前几项:

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

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

[ a h ^ 2 + sum { i ^ d i (部分 i h) ^ 2 + λ h ^ 4 + cdots right ] . / math

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

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

在正方格上，对称性保证导数项的系数 z 子 i / 子都是相等的。但是即使对于各向异性的 Ising 模型来说，z 子 i / 子在不同的方向上是不同的，h 中的涨落在一个坐标系中是各向同性的，不同的空间方向被重新调整。

On any lattice, the derivative term

On any lattice, the derivative term

在任何格上，导数项

:<math>Z_{ij} \, \partial_i H \, \partial_j H </math>

<math>Z_{ij} \, \partial_i H \, \partial_j H </math>

数学 z { ij } ，部分 i h ，部分 j h / 数学

is a positive definite [[quadratic form]], and can be used to ''define'' the metric for space. So any translationally invariant Ising model is rotationally invariant at long distances, in coordinates that make ''Zij'' = δ''ij''. Rotational symmetry emerges spontaneously at large distances just because there aren't very many low order terms. At higher order multicritical points, this [[accidental symmetry]] is lost.

is a positive definite quadratic form, and can be used to define the metric for space. So any translationally invariant Ising model is rotationally invariant at long distances, in coordinates that make Zij = δij. Rotational symmetry emerges spontaneously at large distances just because there aren't very many low order terms. At higher order multicritical points, this accidental symmetry is lost.

是一个积极的确定双线性形式，可以用来定义空间的度量。所以任何平移不变的 Ising 模型在很长的距离上都是旋转不变的，在坐标系中使 z 小于 ij / sub 小于 ij / sub。旋转对称在很远的距离自发地出现，只是因为没有很多低阶项。在高阶多临界点，这种偶然的对称性丢失。

Since β''F'' is a function of a slowly spatially varying field, the probability of any field configuration is:

Since βF is a function of a slowly spatially varying field, the probability of any field configuration is:

由于 f 是一个缓慢空间变化的场的函数，任何场构型的概率是:

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

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

数学 p (h) propto e ^ { int d ^ dx left [ AH ^ 2 + z | nabla h | ^ 2 + λ h ^ 4 right ]} . / math

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

The statistical average of any product of H terms is equal to:

任何 h 项乘积的统计平均值等于:

:<math>\langle H(x_1) H(x_2)\cdots H(x_n) \rangle = { \int DH \, P(H) H(x_1) H(x_2) \cdots H(x_n) \over \int DH \, P(H) }.</math>

<math>\langle H(x_1) H(x_2)\cdots H(x_n) \rangle = { \int DH \, P(H) H(x_1) H(x_2) \cdots H(x_n) \over \int DH \, P(H) }.</math>

H (x1) h (x2) cdots h (xn) rangle DH ，p (h) h (x1) h (x2) cdots h (xn) over int DH ，p (h)}. 数学

The denominator in this expression is called the ''partition function'', and the integral over all possible values of ''H'' is a statistical path integral. It integrates exp(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'', the only difference between this and the [[quantum field theory]] of a scalar field being that all the derivative terms enter with a positive sign, and there is no overall factor of ''i''.

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

这个表达式的分母叫做配分函数积分，h 的所有可能值的积分是一个统计路径积分。它将 exp (f)与 h 的所有值，与自旋的所有长波傅里叶成分相结合。F 是场 h 的一个欧氏拉格朗日函数，它与标量场的量子场论唯一的区别在于，所有的导数项都以正号进入，并且没有因子。

:<math>Z = \int DH \, e^{ - \int d^dx \left[ A H^2 + Z |\nabla H|^2 + \lambda H^4 \right]}</math>

<math>Z = \int DH \, e^{ - \int d^dx \left[ A H^2 + Z |\nabla H|^2 + \lambda H^4 \right]}</math>

数学 z int DH，e ^ { int d ^ dx left [ a h ^ 2 + z | nabla h | ^ 2 + λ h ^ 4 right ]} / math

===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.

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.

的形式可以用来预测哪些项是最重要的量纲分析。量纲分析并不是完全简单的，因为 h 的比例需要确定。

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

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

在一般情况下，选择 h 的标度律很容易，因为唯一有贡献的项是第一个项,

:<math>F = \int d^dx \, A H^2.</math>

<math>F = \int d^dx \, A H^2.</math>

数学 f int d ^ dx，a h ^ 2. / math

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.

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.

这个术语是最重要的，但它给琐碎的行为。这种形式的自由能是超局域的，这意味着它是来自每个点的独立贡献的和。这就像一维伊辛模型中的自旋翻转。H 在任何点的每个值都会完全独立于其他点的值而波动。

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.

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.

该领域的规模可以重新定义，以吸收系数 a，然后很明显，a 只决定了波动的总体规模。超局域模型描述了伊辛模型的长波长高温行为，因为在这个极限下，涨落平均值是独立于点与点之间的。

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:

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:

为了找到临界点，降低温度。随着温度下降，h 的波动上升，因为波动之间的相关性更强。这意味着大量自旋的平均值并不会像它们之间没有关联那样迅速变小，因为它们往往是一样的。这相当于在 h 不吸收 a 的单位系统中减少 a。相变只有在 f 的次导项能够贡献的情况下才能发生，但是由于第一项在长距离上占主导地位，因此系数 a 必须调整为零。这是临界点的位置:

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

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

数学 f int d ^ dx left [ t h ^ 2 + lambda h ^ 4 + z ( nabla h) ^ 2 right ] ，/ math

where ''t'' is a parameter which goes through zero at the transition.

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

其中 t 是一个参数，在跃迁时经过零。

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.

Since t is vanishing, fixing the scale of the field using this term makes the other terms blow up. Once t is small, the scale of the field can either be set to fix the coefficient of the H4 term or the (∇H)2 term to 1.

由于 t 正在消失，使用这个术语来确定字段的大小会导致其他术语爆炸。一旦 t 小，字段的尺度可以设置为固定 h sup 4 / sup 项的系数，或者(something h) sup 2 / sup 项为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''ddx'' is dimensionless, and ''Z'' has dimension 2 − ''d''/2. In this scaling, the gradient term is only important at long distances for ''d'' ≤ 4. Above four dimensions, at long wavelengths, the overall magnetization is only affected by the ultralocal terms.

To find the magnetization, fix the scaling of H so that λ is one. Now the field H has dimension −d/4, so that H4ddx is dimensionless, and Z has dimension 2 − d/2. In this scaling, the gradient term is only important at long distances for d ≤ 4. Above four dimensions, at long wavelengths, the overall magnetization is only affected by the ultralocal terms.

要找到磁化强度，固定 h 的比例，这样就是一。现在字段 h 的维数是 d / 4，因此 h sup 4 / sup d / sup x 是无量纲的，z 的维数是2-d / 2。在这个标度中，梯度项只对 d ≤4的长距离重要。在四维以上的长波长范围内，整体的磁化强度只受超局域项的影响。

There is one subtle point. The field ''H'' is fluctuating statistically, and the fluctuations can shift the zero point of ''t''. To see how, consider ''H''4 split in the following way:

There is one subtle point. The field H is fluctuating statistically, and the fluctuations can shift the zero point of t. To see how, consider H4 split in the following way:

有一点很微妙。磁场 h 在统计学上是波动的，波动可以改变 t 的零点:

:<math>H(x)^4 = -\langle H(x)^2\rangle^2 + 2\langle H(x)^2\rangle H(x)^2 + \left(H(x)^2 - \langle H(x)^2\rangle\right)^2</math>

<math>H(x)^4 = -\langle H(x)^2\rangle^2 + 2\langle H(x)^2\rangle H(x)^2 + \left(H(x)^2 - \langle H(x)^2\rangle\right)^2</math>

数学 h (x) ^ 4- langle h (x) ^ 2 rangle ^ 2 + 2 langle h (x) ^ 2 rangle h (x) ^ 2 + left (h (x) ^ 2- langle h (x) ^ 2 rangle right) ^ 2 / math

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 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.

第一项是对自由能的常数贡献，可以忽略不计。第二项是 t 中的有限位移。第三项是一个在长距离上可以缩放为零的量。这意味着在分析量纲分析 t 的标度时，位移 t 是很重要的。这在历史上是非常令人困惑的，因为在任何有限条件下 t 的移动是有限的，但是在过渡条件下 t 的移动是很小的。T 的分数变化很大，在单位 t 固定的情况下，移动看起来是无限的。

The magnetization is at the minimum of the free energy, and this is an analytic equation. In terms of the shifted ''t'',

The magnetization is at the minimum of the free energy, and this is an analytic equation. In terms of the shifted t,

磁化强度处于自由能的最小值，这是一个解析方程。在移动 t 方面,

:<math>{\partial \over \partial H } \left( t H^2 + \lambda H^4 \right ) = 2t H + 4\lambda H^3 = 0</math>

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

2 t h + 4 λ h ^ 30 / math

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

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

对于 t0，极小值在 h 处与 t 的平方根成正比。因此，兰道的灾难论点在大于5的维度上是正确的。大于5维的磁化强度指数等于平均场值。

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.

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.

当 t 为负值时，用一个新的正二次系数来描述新极小值的波动。因为这个词总是占主导地位，所以在低于转变温度的情况下，长距离的超局限性再次出现。

===Fluctuations===

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

To find the behavior of fluctuations, rescale the field to fix the gradient term. Then the length scaling dimension of the field is 1 − d/2. Now the field has constant quadratic spatial fluctuations at all temperatures. The scale dimension of the H2 term is 2, while the scale dimension of the H4 term is 4 − d. For d < 4, the H4 term has positive scale dimension. In dimensions higher than 4 it has negative scale dimensions.

为了找到涨落的行为，重新调整场来确定梯度项。该域的长度标度维数为1-d / 2。现在磁场在所有温度下都有常数的二次空间涨落。对于 d 4，h sup 4 / sup 项具有正的尺度维数，h sup 2 / sup 项的尺度维数为2，h sup 4 / sup 项的尺度维数为4。在高于4的维度中，它有负的尺度维度。

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

This is an essential difference. In dimensions higher than 4, fixing the scale of the gradient term means that the coefficient of the H4 term is less and less important at longer and longer wavelengths. The dimension at which nonquadratic contributions begin to contribute is known as the critical dimension. In the Ising model, the critical dimension is 4.

这是一个本质的区别。在大于4的维数中，梯度项尺度的确定意味着 h sup 4 / sup 项的系数在波长越来越长的情况下越来越不重要。非二次贡献开始贡献的维数称为临界维数。在 Ising 模型中，临界维数为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 distribution|Gaussian]] averages:

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:

在4以上的维度中，临界波动用一个长波长的纯二次自由能来描述。这意味着相关函数都可以用高斯平均值来计算:

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

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

S (x) s (y) rangle propto langle h (x) h (y) rangle g (x-y) dk over (2pi) ^ d { e ^ ik (x-y)} over k ^ 2 + t } / math

valid when ''x'' − ''y'' is large. The function ''G''(''x'' − ''y'') is the analytic continuation to imaginary time of the [[propagator|Feynman propagator]], since the free energy is the analytic continuation of the quantum field action for a free scalar field. For dimensions 5 and higher, all the other correlation functions at long distances are then determined by [[S-matrix#Wick's theorem|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.

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

当 x-y 很大时有效。函数 g (x-y)是费曼传播子虚时间的解析延拓，因为自由能是自由标量场作用的解析延拓。对于维数为5及以上的情况，所有其他长距离的相关函数都是由 Wick 定理确定的。由于对称性，所有的奇数时刻都是零。偶数阶矩是整个分区对每一对 g (x-y)乘积的和。

:<math>\langle S(x_1) S(x_2) \cdots S(x_{2n})\rangle = C^n \sum G(x_{i1},x_{j1}) G(x_{i2},x_{j2}) \ldots G(x_{in},x_{jn})</math>

<math>\langle S(x_1) S(x_2) \cdots S(x_{2n})\rangle = C^n \sum G(x_{i1},x_{j1}) G(x_{i2},x_{j2}) \ldots G(x_{in},x_{jn})</math>

S (x1) s (x2) cdots s (x2n }) rangle c ^ n sum g (x { i1} ，x { j1}) g (x { i2} ，x { j2}) ldots g (x { in } ，x { jn }) / math

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

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

其中 c 是比例常数。所以知道 g 就够了。它决定了场的所有多点相关性。

===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:

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:

为了确定 g 的形式，考虑路径积分中的场服从通过改变自由能导出的经典运动方程:

:<math>\begin{align}

<math>\begin{align}

数学 begin { align }

&&\left(-\nabla_x^2 + t\right) \langle H(x)H(y) \rangle &= 0 \\

&&\left(-\nabla_x^2 + t\right) \langle H(x)H(y) \rangle &= 0 \\

& 左(- nabla x ^ 2 + t 右) langle h (x) h (y) rangle & 0

\rightarrow {} && \nabla^2 G(x) + tG(x) &= 0

\rightarrow {} && \nabla^2 G(x) + tG(x) &= 0

2 g (x) + tG (x) & 0

\end{align}</math>

\end{align}</math>

End { align } / math

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.

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.

这仅在非重合点上有效，因为当点碰撞时 h 的相关性是奇异的。服从经典运动方程的原因和量子力学操作符服从经典波动的原因一样---- 它的涨落是由路径积分定义的。

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

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

在临界点 t 0，这是拉普拉斯方程，可以用高斯的静电学方法求解。定义一个电场模拟

:<math>E = \nabla G</math>

<math>E = \nabla G</math>

数学 e nabla g / math

Away from the origin:

Away from the origin:

远离原点:

:<math>\nabla \cdot E = 0</math>

<math>\nabla \cdot E = 0</math>

Math nabla cdot e 0 / math

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

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

因为 g 在 d 维上是球对称的，e 是 g 在 d 维上的径向梯度积分,

:<math>\int d^{d-1}S E_r = \mathrm{constant}</math>

<math>\int d^{d-1}S E_r = \mathrm{constant}</math>

数学 d ^ { d-1} s e r { constant } / math

This gives:

This gives:

这就给出了:

:<math>E = {C \over r^{d-1} }</math>

<math>E = {C \over r^{d-1} }</math>

数学 e { c over r ^ { d-1} / math

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

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

通过对 r 积分可以得到 g。

:<math>G(r) = {C \over r^{d-2} }</math>

<math>G(r) = {C \over r^{d-2} }</math>

数学 g (r){ c over r ^ { d-2}} / math

The constant ''C'' fixes the overall normalization of the field.

The constant C fixes the overall normalization of the field.

常量 c 修复了字段的整体标准化。

===''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:

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:

当 t 不等于零时，h 在温度稍微偏离临界点时波动，两点函数在很长距离衰减。它遵循的方程式被改变了:

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

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

数学 ^ 2 g + t g0 to {1 over r ^ d-1}{ d o (r ^ d-1} dG o (r ^ d-1) o (r)右) + t g (r)0 / math

For ''r'' small compared with <math>\sqrt{t}</math>, the solution diverges exactly the same way as in the critical case, but the long distance behavior is modified.

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

对于 r 小于 math sqrt { t } / math，解的发散方式与临界情况完全相同，但是长距离行为被修改。

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

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

为了说明如何做到这一点，我们可以很方便地将 Schwinger 在量子场论中提出的两点函数表示为积分:

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

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

Math g (x) int d tau {1 over ( sqrt {2 pi tau } right) ^ d } e ^ {-{ x ^ 2 over 4 tau }-t tau } / math

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 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.

这就是 g，因为这个积分的傅里叶变换很简单。每个固定贡献在 x 中是一个高斯分量，其傅里叶变换在 k 中是另一个倒数宽度的高斯分量。

:<math>G(k) = \int d\tau e^{-(k^2 - t)\tau} = {1 \over k^2 - t}</math>

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

数学 g (k) int d tau e ^ {-(k ^ 2-t) tau }{1 over k ^ 2-t } / math

This is the inverse of the operator ∇2 − ''t'' in ''k''-space, acting on the unit function in ''k''-space, which is the Fourier transform of a delta function source localized at the origin. So it satisfies the same equation as ''G'' with the same boundary conditions that determine the strength of the divergence at 0.

This is the inverse of the operator ∇2 − t in k-space, acting on the unit function in k-space, which is the Fourier transform of a delta function source localized at the origin. So it satisfies the same equation as G with the same boundary conditions that determine the strength of the divergence at 0.

这是 k 空间中操作符 sup 2 / sup-t 的逆，它作用于 k 空间中的单位函数，即定位于原点的 delta 函数源的傅里叶变换。因此，它满足与 g 相同的方程，具有相同的边界条件，决定了散度的强度在0。

The interpretation of the integral representation over the ''proper time'' τ is that the two point function is the sum over all random walk paths that link position 0 to position ''x'' over time τ. The density of these paths at time τ at position ''x'' is Gaussian, but the random walkers disappear at a steady rate proportional to ''t'' so that the Gaussian at time τ is diminished in height by a factor that decreases steadily exponentially. In the quantum field theory context, these are the paths of relativistically localized quanta in a formalism that follows the paths of individual particles. In the pure statistical context, these paths still appear by the mathematical correspondence with quantum fields, but their interpretation is less directly physical.

The interpretation of the integral representation over the proper time τ is that the two point function is the sum over all random walk paths that link position 0 to position x over time τ. The density of these paths at time τ at position x is Gaussian, but the random walkers disappear at a steady rate proportional to t so that the Gaussian at time τ is diminished in height by a factor that decreases steadily exponentially. In the quantum field theory context, these are the paths of relativistically localized quanta in a formalism that follows the paths of individual particles. In the pure statistical context, these paths still appear by the mathematical correspondence with quantum fields, but their interpretation is less directly physical.

在适当时间上的积分表示的解释是，两点函数是所有随机游走路径上的和，这些路径随时间将位置0与位置 x 连接起来。这些路径在 x 位置时的密度是高斯分布的，但是随机行走者消失的速率与 t 成正比，因此高斯分布在同一时刻的高度会随着一个因子以指数方式减小。在量子场论的背景下，这些是相对论定域量子的路径，形式主义遵循单个粒子的路径。在纯粹的统计上下文中，这些路径仍然是通过与量子场的数学对应出现的，但是它们的解释不是直接的物理解释。

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 <math>e^{-t\tau} = e^{-tr^2}</math>. The decay factor appropriate for position ''r'' is therefore <math>e^{-\sqrt t r}</math>.

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 <math>e^{-t\tau} = e^{-tr^2}</math>. The decay factor appropriate for position r is therefore <math>e^{-\sqrt t r}</math>.

积分表示立即表明 g (r)是正的，因为它被表示为正高斯函数的加权和。由于随机游动到达位置的适当时间为 r sup 2 / sup，在此时，高斯高度已经通过数学 e ^ {-t t } e ^ {-tr ^ 2} / math 衰减。因此，位置 r 的衰减因子是 math e ^ { sqrt t r } / math。

A heuristic approximation for ''G''(''r'') is:

A heuristic approximation for G(r) is:

G (r)的一个启发式近似是:

:<math>G(r) \approx { e^{-\sqrt t r} \over r^{d-2}}</math>

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

对于 r ^ { d-2} / math，大约 g (r) ^ { e-- sqrt t r }

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]].

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.

这并不是一个精确的形式，除了在三个维度中，路径之间的相互作用变得非常重要。高维中的精确形式是贝塞尔函数的变体。

===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 interpretation of the correlations as fixed size quanta travelling along random walks gives a way of understanding why the critical dimension of the H4 interaction is 4. 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 do not intersect.

将关联式解释为沿随机游动的固定尺寸量子，给出了 h sup 4 / sup 相互作用的临界维数为4。术语 h sup 4 / sup 可以看作是随机行走者在任意点密度的平方。由于有限阶相关函数只在涨落环境中引入少量新的随机游动，为了使这一项改变有限阶相关函数，新的路径必须相交。否则，密度的平方正好与密度成正比，而且只偏移一个常数即 h 浆 / 浆系数。但随机游动的交概率取决于维数，维数大于4的随机游动不相交。

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.

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.

普通随机游动的分形维数是2。覆盖路径所需的尺寸球的数量增加为 sup-2 / sup。分形维数2的两个物体仅在维数4或更少的空间中以合理的概率相交，这与一般两个平面的条件相同。库尔特 · 赛曼济克认为，这意味着大于4维的临界伊辛涨落应该由自由场来描述。这个论证最终成为数学证明。

===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 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.

四维伊辛模型是用一个波动场来描述的，但现在波动是相互作用的。在聚合物表示中，随机游动的交点几乎是可能的。在量子场延拓中，量子相互作用。

The negative logarithm of the probability of any field configuration ''H'' is the [[Thermodynamic free energy|free energy]] function

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

任何场组态 h 的概率的负对数是自由能函数

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

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

数学 f ^ int d ^ 4 x | | nabla h | ^ 2 + { t 2} h ^ 2 + { lambda 4[ h ^ 4 right ] ，/ math

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 diagram|Feynman expansion]] as particles travelling along random walks, splitting and rejoining at vertices. The interaction strength is parametrized by the classically dimensionless quantity λ.

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

这些数字因素可以简化运动方程。我们的目标是理解统计学上的波动。与任何其他非二次路径积分一样，相关函数具有 Feynman 展开式，即粒子沿着随机游动、分裂和在顶点重新连接。相互作用的强度是参数化的经典的无量纲量。

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

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

尽管量纲分析表明 z 和 z 都是无量纲的，但这是误导。长波长统计涨落不具有尺度不变性，只有当相互作用强度消失时才具有尺度不变性。

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.

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.

原因是有一个截止值用来定义 h，而这个截止值定义了最短的波长。波长在截止点附近的波动可以影响较长波长的波动。如果该系统是按截止比例缩放，参数将按量纲分析缩放，但然后比较参数不比较行为，因为重新缩放的系统有更多的模式。如果对系统进行重新标度，使短波长截止值保持不变，则长波长涨落将得到修正。

====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.

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.

研究缩放的一个快速启发式方法是在某一点切断 h 波数。波数大于的 h 的傅里叶模是不允许波动的。使整个系统更小的长度重新标定增加了所有波数，并使一些波动超过了截止值。

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.

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.

为了恢复旧的截止时间，对所有过去被禁止但现在波动的波形进行部分积分。在费曼图中，通过积分波数 k 的波动模式，可以将相关函数(量子场论)中的动量 k 与反传播因子成对连接起来。

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 2''bt''. The other two coefficients are dimensionless and do not change at all.

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 do not change at all.

在重新标度下，当系统缩小一个因子(1 + b)时，t 系数扩大一个因子(1 + b) sup 2 / sup 量纲分析。无穷小 b 的 t 变化为2bt。另外两个系数是无量纲的，根本不变。

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

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

积分输出的最低阶效应可以由运动方程计算出来:

:<math>\nabla^2 H + t H = - {\lambda \over 6} H^3.</math>

<math>\nabla^2 H + t H = - {\lambda \over 6} H^3.</math>

Math nabla ^ 2 h + t h-{ λ over 6} h ^ 3. / math

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

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

这个方程是任意一个相关函数(量子场论)内的恒等式，远离其他插入。在将模式与 k (1 + b)集成之后，它将是一个略有不同的同一性。

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

Since the form of the equation will be preserved, to find the change in coefficients it is sufficient to analyze the change in the H3 term. In a Feynman diagram expansion, the H3 term in a correlation function inside a correlation has three dangling lines. Joining two of them at large wavenumber k gives a change H3 with one dangling line, so proportional to H:

由于方程的形式将被保留，要找到系数的变化就足以分析 h sup 3 / sup 项的变化。在费曼图的展开式中，相关系数内的相关函数(量子场论)中的 h sup 3 / sup 项有3条悬空线。将其中两个大波数 k 连接起来，使一根悬垂线改变 h sup 3 / sup，因此与 h 成正比:

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

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

(2-pi) ^ 4}{1 / (k ^ 2 + t)} / math

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

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

因子3来自于这样一个事实，即循环可以用三种不同的方式来闭合。

The integral should be split into two parts:

The integral should be split into two parts:

积分应该分为两部分:

:<math>\int dk {1 \over k^2} - t \int dk { 1\over k^2(k^2 + t)} = A\Lambda^2 b + B b t</math>

<math>\int dk {1 \over k^2} - t \int dk { 1\over k^2(k^2 + t)} = A\Lambda^2 b + B b t</math>

<math>\int dk {1 \over k^2} - t \int dk { 1\over k^2(k^2 + t)} = A\Lambda^2 b + B b t</math>

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

The first part is not proportional to t, and in the equation of motion it can be absorbed by a constant shift in t. It is caused by the fact that the H3 term has a linear part. Only the second term, which varies from t to t, contributes to the critical scaling.

第一部分与 t 不成比例，在运动方程中，它可以被 t 中的常数移动所吸收。它是由于 h sup 3 / sup 项具有线性部分这一事实引起的。只有第二项(从 t 到 t 不等)对临界尺度有贡献。

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 product expansion|operator products]]:

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

这个新的线性项与左边的第一项相加，变化量 t 与 t 成正比。 T 的总变化量是量纲分析和运营商产品这两个项的总和:

:<math>\delta t = \left(2 - {B\lambda \over 2} \right)b t</math>

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

数学 t 左(2-{ b lambda over 2}右)

So ''t'' is rescaled, but its dimension is [[anomalous dimension|anomalous]], it is changed by an amount proportional to the value of λ.

So t is rescaled, but its dimension is anomalous, it is changed by an amount proportional to the value of λ.

所以 t 是重新标度的，但是它的尺寸是反常的，它的变化量与值成正比。

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

But λ also changes. The change in λ requires considering the lines splitting and then quickly rejoining. The lowest order process is one where one of the three lines from H3 splits into three, which quickly joins with one of the other lines from the same vertex. The correction to the vertex is

但也有变化。年的更改需要考虑线路的分裂，然后迅速重新连接。最低阶过程是 h sup 3 / sup 的三条线中的一条分裂成三条，这三条线迅速地与来自同一顶点的另一条线连接起来。对顶点的修正为

:<math>\delta \lambda = - {3 \lambda^2 \over 2} \int_k dk {1 \over (k^2 + t)^2} = -{3\lambda^2 \over 2} b</math>

<math>\delta \lambda = - {3 \lambda^2 \over 2} \int_k dk {1 \over (k^2 + t)^2} = -{3\lambda^2 \over 2} b</math>

(k ^ 2 + t) ^ 2}-(3 λ ^ 2 ^ 2) ^ 2}-(2} b / math)

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

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

数值因素要大三倍，因为在选择三条新线中的哪一条收缩时，需要额外的三倍因素。所以

:<math>\delta \lambda = - 3 B \lambda^2 b</math>

<math>\delta \lambda = - 3 B \lambda^2 b</math>

数学 delta lambda-3 b lambda ^ 2 b / math

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

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

这两个方程共同定义了重整化群方程的4个维度:

:<math>\begin{align}

<math>\begin{align}

数学 begin { align }

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

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

{ dt over t & 左(2-{ b lambda over 2}右) b

{d\lambda \over \lambda} &= {-3 B \lambda \over 2} b

{d\lambda \over \lambda} &= {-3 B \lambda \over 2} b

{ d lambda over lambda & {-3 b lambda over 2} b

\end{align}</math>

\end{align}</math>

End { align } / math

The coefficient ''B'' is determined by the formula

The coefficient B is determined by the formula

系数 b 由公式确定

:<math>B b = \int_{\Lambda<|k|<(1+b)\Lambda} {d^4k\over (2\pi)^4} {1 \over k^4}</math>

<math>B b = \int_{\Lambda<|k|<(1+b)\Lambda} {d^4k\over (2\pi)^4} {1 \over k^4}</math>

数学 b int { λ | k | (1 + b) Lambda }{ d ^ 4k over (2 pi) ^ 4}{1 over k ^ 4} / math

and is proportional to the area of a three-dimensional sphere of radius λ, times the width of the integration region ''b''Λ divided by Λ4:

and is proportional to the area of a three-dimensional sphere of radius λ, times the width of the integration region bΛ divided by Λ4:

与三维球面积成正比，乘以积分区域 b 的宽度除以 sup 4 / sup:

:<math>B= (2 \pi^2 \Lambda^3) {1\over (2\pi)^4} { b \Lambda} {1 \over b\Lambda^4} = {1\over 8\pi^2} </math>

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

数学 b (2 π ^ 2 λ ^ 3){1 / (2 π) ^ 4}{ b Lambda }{1 / (8 π ^ 2)} / math

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.

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.

在其他维度中，常数 b 发生了变化，但在 t 流和耦合流中都出现了相同的常数。这是因为只有一个顶点的闭环对 t 的导数是一个有两个顶点的闭环。这意味着耦合的标度和 t 之间的唯一区别是连接和分裂的组合因素。

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

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.

从四维理论出发来研究三维问题是可能的，因为随机游动的交集概率不断地依赖于空间的维数。在费曼图的语言中，当维数改变时，耦合不会发生很大的变化。

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

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

如果没有关于如何做到这一点的规定，那么继续脱离维度4的过程就不能完全得到很好的定义。这个处方只在图上有很好的定义。它用4维中的 Schwinger 表示取代了4维中的 Schwinger 表示，4维定义为:

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

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

Math g (x-y) int d tau {1 over t ^ d over 2} e ^ { x ^ 2 over 2 tau } + t tau } / math

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

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

在尺寸4-中，耦合具有正的尺度尺寸，这必须加到流量中。

:<math>\begin{align}

<math>\begin{align}

数学 begin { align }

{d\lambda \over \lambda} &= \varepsilon - 3 B \lambda \\

{d\lambda \over \lambda} &= \varepsilon - 3 B \lambda \\

{ d lambda } & varepsilon-3 b lambda

{dt \over t} &= 2 - \lambda B

{dt \over t} &= 2 - \lambda B

{ dt over t } & 2- lambda b

\end{align}</math>

\end{align}</math>

End { align } / math

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

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

系数 b 是维数相关的，但是它会取消。年的固定点不再是零，而是在:

:<math>\lambda = {\varepsilon \over 3B} </math>

<math>\lambda = {\varepsilon \over 3B} </math>

数学，数学，数学

where the scale dimensions of ''t'' is altered by an amount λ''B'' = ε/3.

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

其中 t 的比例尺寸被 b / 3所改变。

The magnetization exponent is altered proportionately to:

The magnetization exponent is altered proportionately to:

按比例改变磁化指数:

:<math>\tfrac{1}{2} \left( 1 - {\varepsilon \over 3}\right)</math>

<math>\tfrac{1}{2} \left( 1 - {\varepsilon \over 3}\right)</math>

左(1-{ varepsilon over 3}右) / math

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.

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.

这是0.333在3维度(1)和0.166在2维度(2)。这与测量的指数. 308和昂萨格二维指数相差不大。

===Infinite dimensions – mean field===

{{main|Mean field theory}}

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

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

完全连通图上伊辛模型的行为可以用平均场理论来完全理解。这种类型的描述适用于非常高维的正方形格子，因为这样每个位置都有非常多的邻居。

The idea is that if each spin is connected to a large number of spins, only the average ratio of + spins to − spins is important, since the fluctuations about this mean will be small. 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 ''N'' → ∞ is smooth. In terms of the new ''J'', the energy cost for flipping a spin is ±2''JH''.

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

这个想法是，如果每个自旋都与大量的自旋相关联，那么只有 + 自旋与自旋的平均比例是重要的，因为这个平均值的涨落是很小的。平均场 h 是自旋的平均分数，是 + 减去自旋的平均分数，是-。在平均场 h 中翻转一个自旋的能量成本是2JNH。重新定义 j 可以方便地吸收因子 n，使极限 n →∞光滑。对于新的 j，翻转一个自旋的能量消耗是2JH。

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

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:

这个能量代价给出了自旋为 + 的概率 p 与自旋为-的概率1-p 的比值。这个比例就是玻尔兹曼因子:

:<math>{p\over 1-p} = e^{2\beta JH}</math>

<math>{p\over 1-p} = e^{2\beta JH}</math>

1-p ^ {2-beta JH } / math

so that

so that

所以

:<math>p = {1 \over 1 + e^{-2\beta JH} }</math>

<math>p = {1 \over 1 + e^{-2\beta JH} }</math>

数学 p {1 over 1 + e ^ {-2 beta JH } / math

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 2''p'' − 1. But this average is the same for all spins, and is therefore equal to ''H''.

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.

自旋的平均值是用权重 p 和1-p 的平均值1和-1给出的，因此平均值是2p-1。但是这个平均值对所有的自旋都是一样的，因此等于 h。

:<math> H = 2p - 1 = { 1 - e^{-2\beta JH} \over 1 + e^{-2\beta JH}} = \tanh (\beta JH)</math>

<math> H = 2p - 1 = { 1 - e^{-2\beta JH} \over 1 + e^{-2\beta JH}} = \tanh (\beta JH)</math>

2 p-1{1-e ^ {-2 beta JH } over 1 + e ^ {2 beta JH } tanh ( beta JH) / math

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

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

这个方程的解是可能的一致平均场。For βJ < 1 there is only the one solution at H = 0.对于较大的值，有三个解，在 h0处的解是不稳定的。

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 β''J'' = 1 the mean field Ising model undergoes a phase transition in the limit of large ''N''.

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 βJ = 1 the mean field Ising model undergoes a phase transition in the limit of large N.

这种不稳定性意味着平均场在零点以上增加一点点就会产生一个大于平均场值的自旋的统计分数。因此，在零值以上波动的平均场将产生一个更大的平均场，并最终在稳定解处稳定下来。这意味着对于低于临界值 j1的温度，平均场 Ising 模型在大 n 的极限下经历了一个相变。

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.

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.

在临界温度以上，由于平均场将涨落恢复到零场，h 中的涨落受到抑制。在临界温度以下，平均场被驱动到一个新的平衡值，即方程的正 h 解或负 h 解。

For β''J'' = 1 + ε, just below the critical temperature, the value of ''H'' can be calculated from the Taylor expansion of the hyperbolic tangent:

For βJ = 1 + ε, just below the critical temperature, the value of H can be calculated from the Taylor expansion of the hyperbolic tangent:

对于 j1 + ，刚好低于临界温度，h 的值可以通过双曲正切的泰勒展开式计算出来:

:<math>H = \tanh(\beta J H) = (1+\varepsilon)H - {(1+\varepsilon)^3H^3\over 3}</math>

<math>H = \tanh(\beta J H) = (1+\varepsilon)H - {(1+\varepsilon)^3H^3\over 3}</math>

数学 h ( beta j h)(1 + varepsilon) h-{(1 + varepsilon) ^ 3H ^ 3除以3} / math

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

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

除以 h 去掉 h0时的不稳定解，稳定解是:

:<math>H = \sqrt{3\varepsilon}</math>

<math>H = \sqrt{3\varepsilon}</math>

数学，数学，数学

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 [[Lev Landau|Landau]] to suspect that all Ising type phase transitions in all dimensions should follow this law.

The spontaneous magnetization H grows near the critical point as the square root of the change in temperature. This is true whenever H can be calculated from the solution of an analytic equation which is symmetric between positive and negative values, which led Landau to suspect that all Ising type phase transitions in all dimensions should follow this law.

自发磁化 h 在温度变化的平方根附近生长。当 h 可以通过一个解析方程的解来计算时，这是正确的，这个解析方程在正负值之间是对称的，这使朗道怀疑所有维度上的所有伊辛型相变都应该遵循这个规律。

The mean field exponent is [[Universality (dynamical systems)|universal]] because changes in the character of solutions of analytic equations are always described by [[catastrophe theory|catastrophes]] 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:

The mean field exponent is universal because changes in the character of solutions of analytic equations are always described by catastrophes 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:

平均场指数具有普遍意义，因为解析方程解的性质的变化总是用多项式方程泰勒级数中的灾变来描述。根据对称性，h 的方程在右边必须只有 h 的奇数次方。改变应该只是平稳地改变系数。当右侧 h 的系数为1时，这种转变就发生了。过渡期附近:

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

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

(1 + a varepsilon) h + b h ^ 3 + cdots / math

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 β''F'' is either non-analytic or non-generic at the exact β where the transition occurs.

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

不管 a 和 b 是什么，只要它们都没有调到零，自发磁化强度就会随着。只有当自由能 f 在跃迁发生的确切位置是非解析的或非泛型的时候，这个论证才会失败。

But the spontaneous magnetization in magnetic systems and the density in gasses near the critical point are measured very accurately. The density and the magnetization in three dimensions have the same power-law dependence on the temperature near the critical point, but the behavior from experiments is:

But the spontaneous magnetization in magnetic systems and the density in gasses near the critical point are measured very accurately. The density and the magnetization in three dimensions have the same power-law dependence on the temperature near the critical point, but the behavior from experiments is:

但是磁系统中的自发磁化和临界点附近气体的密度被非常精确地测量出来了。三维空间的密度和磁化强度与临界点附近的温度具有同样的幂律关系，但实验表明:

:<math>H \propto \varepsilon^{0.308}</math>

<math>H \propto \varepsilon^{0.308}</math>

0.308} / math

The exponent is also universal, since it is the same in the Ising model as in the experimental magnet and gas, but it is not equal to the mean field value. This was a great surprise.

The exponent is also universal, since it is the same in the Ising model as in the experimental magnet and gas, but it is not equal to the mean field value. This was a great surprise.

这个指数也是通用的，因为它在伊辛模型中与实验磁体和气体中相同，但它不等于平均场值。这是一个巨大的惊喜。

This is also true in two dimensions, where

This is also true in two dimensions, where

在二维空间中也是如此

:<math>H \propto \varepsilon^{0.125}</math>

<math>H \propto \varepsilon^{0.125}</math>

0.125} / math

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

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

但这并不令人惊讶，因为昂萨格已经预测到了。

===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.

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.

在三维空间中，场论中的微扰级数是耦合常数中的一个展开式，它并不是特别小。在固定点耦合的有效尺寸是粒子路径分叉率的一倍，因此膨胀参数约为1 / 3。在二维情况下，扰动展开参数为2 / 3。

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

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

但是重整化也可以直接有效地应用于自旋，而不需要传递到平均场。从历史上看，这种方法应归功于 Leo Kadanoff，并且早于微扰展开。

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.

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.

我们的想法是迭代地整合出晶格自旋，从而产生耦合流。但是现在的耦合是晶格能系数。事实上，连续介绍存在保证这个迭代将收敛到一个固定点当温度被调整到临界。

====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:

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

写出二维伊辛模型，其中包含无限多个可能的高阶相互作用。为了保持自旋反射的对称性，只有力量才能做出贡献:

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

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

数学 e sum { ij } s i j + sum j { ijkl } s i s j k s l ldots. / math

By translation invariance,''Jij'' is only a function of i-j. By the accidental rotational symmetry, at large i and j its size only depends on the magnitude of the two-dimensional vector ''i'' − ''j''. The higher order coefficients are also similarly restricted.

By translation invariance,Jij is only a function of i-j. By the accidental rotational symmetry, at large i and j its size only depends on the magnitude of the two-dimensional vector i − j. The higher order coefficients are also similarly restricted.

利用平移不变性，j 次方 ij / sub 只是 i-j 的函数。通过偶然的旋转对称，在很大程度上 i 和 j 的大小只取决于二维向量 i-j 的大小。高阶系数也同样受到限制。

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 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.

重整化迭代将晶格分为偶自旋和奇自旋两部分。奇自旋存在于奇棋盘格子的位置，偶自旋存在于偶棋盘格子的位置。用位置(i，j)对自旋进行索引时，奇位点是 i + j 的奇位点，偶位点是 i + j 的偶位点，偶位点只与奇位点相连。

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 <math>\scriptstyle \sqrt{2}</math> larger.

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 <math>\scriptstyle \sqrt{2}</math> larger.

奇自旋的两个可能值将通过对两个可能值的求和得到综合。这将为剩下的偶数自旋产生一个新的自由能函数，带有新的调整过的耦合。这些均匀的自旋再次出现在晶格中，轴与原来的自旋倾斜了45度。不旋转系统将恢复旧的配置，但使用新的参数。这些参数描述了在距离 math scriptstyle sqrt {2} / math larger 处自旋间的相互作用。

Starting from the Ising model and repeating this iteration eventually changes all the couplings. When the temperature is higher than the critical temperature, 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.

Starting from the Ising model and repeating this iteration eventually changes all the couplings. When the temperature is higher than the critical temperature, 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.

从伊辛模型开始，重复这个迭代最终改变了所有的耦合。当温度高于临界温度时，由于距离较远的自旋是不相关的，耦合将收敛到零。但是当温度达到临界值时，所有的自旋之间都会存在非零系数。只考虑前几项就可以近似地计算流动。当包含更多的术语时，这种截断的流将产生越来越好的临界指数近似值。

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.

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.

最简单的近似方法是只保留通常的 j 项，抛弃其他所有项。这将在 j 中产生一个流，类似于在展开式中固定点 t 的流。

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:

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:

要查找 j 中的变化，请考虑一个奇怪站点的四个邻居。这是唯一与它相互作用的自旋。在奇数位置自旋的两个值之和对配分函数的乘法贡献是:

:<math> e^{J (N_+ - N_-)} + e^{J (N_- - N_+)} = 2 \cosh(J[N_+ - N_-])</math>

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

数学 e ^ { j (n +-n -)} + e ^ { j (n-n +)}2 cosh (j [ n +-n-]) / math

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

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

其中 n 子 / 子是邻居的数量。忽略2的因子，这个奇数位置的自由能贡献是:

:<math> F = \log(\cosh[J(N_+ - N_-)]).</math>

<math> F = \log(\cosh[J(N_+ - N_-)]).</math>

数学 f log ( cosh [ j (n +-n -)]) . / math

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:

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:

这包括最近邻相互作用和次近邻相互作用，正如预期的那样，还包括一个将被丢弃的四自旋相互作用。要截断到最近邻相互作用，考虑所有自旋相同和相等数目 + 和-之间的能量差是:

:<math> \Delta F = \ln(\cosh[4J]).</math>

<math> \Delta F = \ln(\cosh[4J]).</math>

Math Delta f ln ( cosh [4J ]) . / math

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

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:

从最近邻耦合出发，所有自旋相等和交错自旋之间的能量差为8J。所有自旋相等和非交错但净零自旋之间的能量差为4J。忽略四自旋相互作用，一个合理的截断是这两个能量或6J 的平均值。由于每个链接都会产生两个奇怪的旋转，因此与前一个链接比较的正确值是:

:<math>3J' = \ln(\cosh[4J]).</math>

<math>3J' = \ln(\cosh[4J]).</math>

Math 3J’ ln ( cosh [4J ]) . / math

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.

For small J, this quickly flows to zero coupling. Large Js flow to large couplings. The magnetization exponent is determined from the slope of the equation at the fixed point.

对于小 j 来说，这很快就变成了零耦合。大型 Js 流到大型耦合器。磁化指数由方程在定点处的斜率确定。

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

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

这种方法的变体在二维和三维中包含多项时，可以为临界指数提供良好的数值近似。

==Applications==

===Magnetism===

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

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

该模型的最初动机是铁磁现象。铁是有磁性的，一旦被磁化，与任何原子时间相比，它会保持长时间的磁化。

In the 19th century, it was thought that magnetic fields are due to currents in matter, and [[André-Marie Ampère|Ampère]] postulated that permanent magnets are caused by permanent atomic currents. The motion of classical charged particles could not explain permanent currents though, as shown by [[Joseph Larmor|Larmor]]. In order to have ferromagnetism, the atoms must have permanent [[magnetic moment]]s which are not due to the motion of classical charges.

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

在19世纪，人们认为磁场是由物质中的电流引起的，而 amp 重新假定永磁体是由永久原子电流引起的。经典带电粒子的运动不能解释永久电流，但是，如拉莫尔所示。为了获得铁磁性，原子必须具有不受经典电荷运动影响的永磁矩。

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.

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.

一旦电子的自旋被发现，很明显，磁性应该是由于大量的电子朝同一方向旋转。人们很自然地会问，电子是如何知道自旋的方向的，因为磁铁一侧的电子并不直接与另一侧的电子相互作用。他们只能影响他们的邻居。伊辛模型旨在研究是否可以使大部分的电子仅仅使用局部力就可以朝同一方向自旋。

===Lattice gas===

The Ising model can be reinterpreted as a statistical model for the motion of atoms. Since the kinetic energy depends only on momentum and not on position, while the statistics of the positions only depends on the potential energy, the thermodynamics of the gas only depends on the potential energy for each configuration of atoms.

The Ising model can be reinterpreted as a statistical model for the motion of atoms. Since the kinetic energy depends only on momentum and not on position, while the statistics of the positions only depends on the potential energy, the thermodynamics of the gas only depends on the potential energy for each configuration of atoms.

伊辛模型可以重新解释为原子运动的统计模型。由于动能只取决于动量而不取决于位置，而位置的统计只取决于势能，气体的热力学只取决于每种原子构型的势能。

A coarse model is to make space-time a lattice and imagine that each position either contains an atom or it doesn't. The space of configuration is that of independent bits ''Bi'', where each bit is either 0 or 1 depending on whether the position is occupied or not. An attractive interaction reduces the energy of two nearby atoms. If the attraction is only between nearest neighbors, the energy is reduced by −4''JB''''i''''B''''j'' for each occupied neighboring pair.

A coarse model is to make space-time a lattice and imagine that each position either contains an atom or it doesn't. The space of configuration is that of independent bits Bi, where each bit is either 0 or 1 depending on whether the position is occupied or not. An attractive interaction reduces the energy of two nearby atoms. If the attraction is only between nearest neighbors, the energy is reduced by −4JBiBj for each occupied neighboring pair.

一个粗糙的模型是把时空变成一个格子，并假设每个位置要么包含一个原子，要么不包含。其配置空间为独立位 b 子 i / 子，其中每个位是0还是1，取决于位置是否被占用。引力相互作用降低了附近两个原子的能量。如果吸引力仅存在于最近邻居之间，则每个占据的邻居对的能量减少为 -4JB 子 i / sub b 子 j / sub。

The density of the atoms can be controlled by adding a [[chemical potential]], which is a multiplicative probability cost for adding one more atom. A multiplicative factor in probability can be reinterpreted as an additive term in the logarithm – the energy. The extra energy of a configuration with ''N'' atoms is changed by ''μN''. The probability cost of one more atom is a factor of exp(−''βμ'').

The density of the atoms can be controlled by adding a chemical potential, which is a multiplicative probability cost for adding one more atom. A multiplicative factor in probability can be reinterpreted as an additive term in the logarithm – the energy. The extra energy of a configuration with N atoms is changed by μN. The probability cost of one more atom is a factor of exp(−βμ).

原子的密度可以通过增加一个化学势来控制，这是增加一个原子的乘积概率代价。概率中的乘法因子可以重新解释为对数中的加法项——能量。含有 n 个原子的组态的额外能量被 n 改变。多一个原子的概率成本是 exp (-)的一个因素。

So the energy of the lattice gas is:

So the energy of the lattice gas is:

所以晶格气体的能量是:

:<math>E = - \frac{1}{2} \sum_{\langle i,j \rangle} 4 J B_i B_j + \sum_i \mu B_i</math>

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

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

Rewriting the bits in terms of spins, <math>B_i = (S_i + 1)/2. </math>

Rewriting the bits in terms of spins, <math>B_i = (S_i + 1)/2. </math>

用自旋重写位元，数学 b i (s i + 1) / 2。数学

:<math>E = - \frac{1}{2} \sum_{\langle i,j \rangle} J S_i S_j - \frac{1}{2} \sum_i (4 J - \mu) S_i</math>

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

数学 e-frac {1}{2}{1}{1}{1}{2}{1}{1}{4 j-mu) s i / math

For lattices where every site has an equal number of neighbors, this is the Ising model with a magnetic field ''h'' = (''zJ'' − ''μ'')/2, where ''z'' is the number of neighbors.

For lattices where every site has an equal number of neighbors, this is the Ising model with a magnetic field h = (zJ − μ)/2, where z is the number of neighbors.

对于每个位置有相同数量邻居的晶格，这是磁场为 h (zJ -) / 2的 Ising 模型，其中 z 是邻居的数量。

In biological systems, modified versions of the lattice gas model have been used to understand a range of binding behaviors. These include the binding of ligands to receptors in the cell surface,<ref>{{Cite journal|last=Shi|first=Y.|last2=Duke|first2=T.|date=1998-11-01|title=Cooperative model of bacteril sensing|journal=Physical Review E|language=en|volume=58|issue=5|pages=6399–6406|doi=10.1103/PhysRevE.58.6399|arxiv=physics/9901052|bibcode=1998PhRvE..58.6399S}}</ref> the binding of chemotaxis proteins to the flagellar motor,<ref>{{Cite journal|last=Bai|first=Fan|last2=Branch|first2=Richard W.|last3=Nicolau|first3=Dan V.|last4=Pilizota|first4=Teuta|last5=Steel|first5=Bradley C.|last6=Maini|first6=Philip K.|last7=Berry|first7=Richard M.|date=2010-02-05|title=Conformational Spread as a Mechanism for Cooperativity in the Bacterial Flagellar Switch|journal=Science|language=en|volume=327|issue=5966|pages=685–689|doi=10.1126/science.1182105|issn=0036-8075|pmid=20133571|bibcode = 2010Sci...327..685B |url=https://semanticscholar.org/paper/680aa07b7425c7addc6e02ef49356d31cfb84d48}}</ref> and the condensation of DNA.<ref>{{Cite journal|last=Vtyurina|first=Natalia N.|last2=Dulin|first2=David|last3=Docter|first3=Margreet W.|last4=Meyer|first4=Anne S.|last5=Dekker|first5=Nynke H.|last6=Abbondanzieri|first6=Elio A.|date=2016-04-18|title=Hysteresis in DNA compaction by Dps is described by an Ising model|url=http://www.pnas.org/content/early/2016/04/14/1521241113|journal=Proceedings of the National Academy of Sciences|language=en|pages=4982–7|doi=10.1073/pnas.1521241113|issn=0027-8424|pmid=27091987|pmc=4983820|volume=113|issue=18|bibcode=2016PNAS..113.4982V}}</ref>

In biological systems, modified versions of the lattice gas model have been used to understand a range of binding behaviors. These include the binding of ligands to receptors in the cell surface, the binding of chemotaxis proteins to the flagellar motor, and the condensation of DNA.

在生物系统中，修改过的格子气模型已经被用来理解一系列的结合行为。这些作用包括配体与细胞表面受体的结合、趋化蛋白与鞭毛运动的结合以及 DNA 的凝聚。

===Application to neuroscience===

The activity of [[neuron]]s 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, [[J. J. Hopfield|Hopfield]] suggested that a dynamical Ising model would provide a [[Hopfield net|first approximation]] to a neural network which is capable of [[learning]].<ref>{{Citation| author= J. J. Hopfield| title = Neural networks and physical systems with emergent collective computational abilities| journal = Proceedings of the National Academy of Sciences of the USA| volume= 79 | pages= 2554–2558| year= 1982| doi = 10.1073/pnas.79.8.2554| pmid = 6953413| issue= 8| pmc= 346238| postscript= .|bibcode = 1982PNAS...79.2554H }}</ref>

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

大脑中神经元的活动可以用统计学的方法来模拟。每个神经元在任何时候都是活动的 + 或不活动的-。活跃神经元是那些在任何给定的时间窗内向轴突发送动作电位的神经元，而不活跃的神经元是那些不发送动作电位的神经元。因为神经活动在任何时候都是由独立的比特模拟的，霍普菲尔德建议，动态伊辛模型将提供一个能够学习的神经网络的第一近似值。

Following the general approach of Jaynes,<ref>{{Citation| author=Jaynes, E. T.| title= Information Theory and Statistical Mechanics | journal= Physical Review| volume = 106 | pages= 620–630 | year= 1957| doi=10.1103/PhysRev.106.620| postscript=.|bibcode = 1957PhRv..106..620J| issue=4 | url= https://semanticscholar.org/paper/08b67692bc037eada8d3d7ce76cc70994e7c8116 }}</ref><ref>{{Citation| author= Jaynes, Edwin T.| title = Information Theory and Statistical Mechanics II |journal = Physical Review |volume =108 | pages = 171–190 | year = 1957| doi= 10.1103/PhysRev.108.171| postscript= .|bibcode = 1957PhRv..108..171J| issue= 2 }}</ref> a recent interpretation of Schneidman, Berry, Segev and Bialek,<ref>{{Citation|author1=Elad Schneidman |author2=Michael J. Berry |author3=Ronen Segev |author4=William Bialek | title= Weak pairwise correlations imply strongly correlated network states in a neural population| journal=Nature| volume= 440 | pages= 1007–1012| year=2006| doi= 10.1038/nature04701| pmid= 16625187| issue= 7087| pmc= 1785327| postscript= .|arxiv = q-bio/0512013 |bibcode = 2006Natur.440.1007S |title-link=neural population }}</ref>

Following the general approach of Jaynes, a recent interpretation of Schneidman, Berry, Segev and Bialek,

遵循 Jaynes 的一般方法，最近对 Schneidman，Berry，Segev 和 Bialek 的解释,

is that the Ising model is useful for any model of neural function, because a statistical model for neural activity should be chosen using the [[principle of maximum entropy]]. Given a collection of neurons, a statistical model which can reproduce the average firing rate for each neuron introduces a [[Lagrange multiplier]] for each neuron:

is that the Ising model is useful for any model of neural function, because a statistical model for neural activity should be chosen using the principle of maximum entropy. Given a collection of neurons, a statistical model which can reproduce the average firing rate for each neuron introduces a Lagrange multiplier for each neuron:

伊辛模型对于任何神经功能模型都是有用的，因为神经活动的统计模型应该用最大熵原理来选择。给定一组神经元，一个能够重现每个神经元平均放电率的统计模型为每个神经元引入一个拉格朗日乘数:

:<math>E = - \sum_i h_i S_i</math>

<math>E = - \sum_i h_i S_i</math>

数学 e sum i h i s i / math

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:

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:

但是该模型中每个神经元的活动在统计上是独立的。为了考虑到双相关性，当一个神经元倾向于和另一个一起激活(或者不激活)时，引入双倍拉格朗日乘数:

:<math>E= - \tfrac{1}{2} \sum_{ij} J_{ij} S_i S_j - \sum_i h_i S_i</math>

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

数学 e- tfrac {2} sum { ij } s i s- sum i i s i / math

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

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

数学不仅仅局限于邻居。注意，伊辛模型的这种推广有时在统计学中被称为二次指数二元分布。

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.

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.

这个能量函数只引入了具有值的自旋和具有相同值的一对自旋的概率偏差。高阶相关性不受乘子的约束。从这个分布中取样的活动模式需要在计算机中以可以想象到的最有效的编码方案存储最大数量的比特，与具有相同平均活动和成对相关性的任何其他分布相比。这意味着伊辛模型适用于任何以尽可能随机的位来描述的系统，对于物理学和社会科学中经常出现的成对相关性和平均1的数量都有约束。

===Spin glasses===

With the Ising model the so-called [[spin glasses]] can also be described, by the usual Hamiltonian

With the Ising model the so-called spin glasses can also be described, by the usual Hamiltonian

在伊辛模型中，所谓的自旋玻璃也可以用通常的哈密顿量来描述

<math>\hat H=-\frac{1}{2}\,\sum J_{i,k}\,S_i\,S_k,</math>

<math>\hat H=-\frac{1}{2}\,\sum J_{i,k}\,S_i\,S_k,</math>

数学 h-frac {2} ，和 j { i，k } ，s i，s k，/ math

where the ''S''-variables describe the Ising spins, while the ''Ji,k'' are taken from a random distribution. For spin glasses a typical distribution chooses antiferromagnetic bonds with probability ''p'' and ferromagnetic bonds with probability 1 − ''p''. These bonds stay fixed or "quenched" even in the presence of thermal fluctuations. When ''p'' = 0 we have the original Ising model. This system deserves interest in its own; particularly one has "non-ergodic" properties leading to strange relaxation behaviour. Much attention has been also attracted by the related bond and site dilute Ising model, especially in two dimensions, leading to intriguing critical behavior.<ref>{{Citation|author= J-S Wang, [[Walter Selke|W Selke]], VB Andreichenko, and VS Dotsenko| title= The critical behaviour of the two-dimensional dilute model|journal= Physica A|volume= 164| issue= 2| pages= 221–239 |year= 1990|doi=10.1016/0378-4371(90)90196-Y|bibcode = 1990PhyA..164..221W }}</ref>

where the S-variables describe the Ising spins, while the Ji,k are taken from a random distribution. For spin glasses a typical distribution chooses antiferromagnetic bonds with probability p and ferromagnetic bonds with probability 1 − p. These bonds stay fixed or "quenched" even in the presence of thermal fluctuations. When p = 0 we have the original Ising model. This system deserves interest in its own; particularly one has "non-ergodic" properties leading to strange relaxation behaviour. Much attention has been also attracted by the related bond and site dilute Ising model, especially in two dimensions, leading to intriguing critical behavior.

其中 s 变量描述伊辛自旋，而 j 子 i，k / 子取自一个随机分布。自旋玻璃的典型分布是反铁磁键和铁磁键的概率分别为 p 和1-p。这些键保持固定或“淬灭” ，即使在热波动的存在。当 p0时，我们得到了原始的 Ising 模型。这个系统本身值得关注，特别是它具有“非遍历”性质，导致奇怪的松弛行为。相关的键和位置稀释伊辛模型也引起了人们的广泛关注，尤其是在二维情况下，由此产生了一些有趣的临界行为。

===Sea ice===

2D [[melt pond]] approximations can be created using the Ising model; sea ice topography data bears rather heavily on the results. The state variable is binary for a simple 2D approximation, being either water or ice.<ref>{{Citation|author= Yi-Ping Ma|author2= Ivan Sudakov|author3= Courtenay Strong|author4= Kenneth Golden|title= Ising model for melt ponds on Arctic sea ice|journal= |volume= |issue= |pages= |year= 2017|arxiv=1408.2487v3}}</ref>

2D melt pond approximations can be created using the Ising model; sea ice topography data bears rather heavily on the results. The state variable is binary for a simple 2D approximation, being either water or ice.

使用伊辛模型可以创建二维融化池的近似值; 海冰地形数据对结果有相当大的影响。状态变量是简单的二维近似的二进制变量，不是水就是冰。

==See also==

{{div col|colwidth=25em}}

* [[ANNNI model]]

* [[Binder parameter]]

* [[Boltzmann machine]]

* [[Conformal bootstrap]]

* [[Geometrically frustrated magnet]]

* [[Heisenberg model (classical)|Classical Heisenberg model]]

* [[Heisenberg model (quantum)|Quantum Heisenberg model]]

* [[Hopfield net]]

* [[Ising critical exponents]]

* [[John Clive Ward|J. C. Ward]]

* [[Kuramoto model]]

* [[Maximal evenness]]

* [[Order operator]]

* [[Potts model]] (common with [[Ashkin–Teller model]])

* [[Spin model]]s

* [[Square-lattice Ising model]]

* [[Swendsen–Wang algorithm]]

* [[t-J model]]

* [[Two-dimensional critical Ising model]]

* [[Wolff algorithm]]

* [[XY model]]

* [[Z N model]]

{{div col end}}

==Footnotes==





! -- 参见维基百科: 脚注解释如何使用 ref erences / tags 生成脚注 --

{{Reflist|30em}}

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* {{springer|author=[[Kurt Binder|K. Binder]]|title=Ising model|doi=I/i120080}}

* Stephen G. Brush (1967), [https://dx.doi.org/10.1103/RevModPhys.39.883 History of the Lenz-Ising Model]. ''Reviews of Modern Physics'' (American Physical Society) vol. 39, pp 883–893. ( 10.1103/RevModPhys.39.883)

* {{citation|first=R.|last=Baierlein|title=Thermal Physics|publisher=Cambridge University Press|location=Cambridge|year=1999|isbn=978-0-521-59082-2|url-access=registration|url=https://archive.org/details/thermalphysics00ralp}}

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* Ross Kindermann and J. Laurie Snell (1980), ''[http://www.ams.org/online_bks/conm1/ Markov Random Fields and Their Applications]''. American Mathematical Society. {{ISBN|0-8218-3381-2}}.

*[[Hagen Kleinert|Kleinert, H]] (1989), ''Gauge Fields in Condensed Matter'', Vol. I, "Superflow and Vortex Lines", pp. 1–742, Vol. II, "Stresses and Defects", pp. 743–1456, [https://web.archive.org/web/20100113041810/http://worldscibooks.com/physics/0356.html World Scientific (Singapore)]; Paperback {{ISBN|9971-5-0210-0}} '' (also available online: [http://www.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents1.html Vol. I] and [http://www.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents2.html Vol. II])''

*[[Hagen Kleinert|Kleinert, H]] and Schulte-Frohlinde, V (2001), ''Critical Properties of φ4-Theories'', [https://web.archive.org/web/20080226151023/http://www.worldscibooks.com/physics/4733.html World Scientific (Singapore)]; Paperback {{ISBN|981-02-4658-7}}'' (also available [http://users.physik.fu-berlin.de/~kleinert/kleinert/?p=booklist&details=6 online])''

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* Barry M. McCoy and Tai Tsun Wu (1973), ''The Two-Dimensional Ising Model''. Harvard University Press, Cambridge Massachusetts, {{ISBN|0-674-91440-6}}

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*{{Citation | last1=Istrail | first1=Sorin | title=Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing | chapter-url=http://www.cs.brown.edu/~sorin/pdfs/Ising-paper.pdf | publisher=ACM | mr=2114521 | year=2000 | chapter=Statistical mechanics, three-dimensionality and NP-completeness. I. Universality of intractability for the partition function of the Ising model across non-planar surfaces (extended abstract) | pages=87–96 | doi=10.1145/335305.335316| isbn=978-1581131840 }}

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{{refend}}

==External links==

* [http://web.mit.edu/redingtn/www/netadv/Xising.html Ising model at The Net Advance of Physics]

* [[Barry Arthur Cipra]], "The Ising model is [[NP-complete]]", [[SIAM News]], Vol. 33, No. 6; [https://web.archive.org/web/20070926223950/http://www.siam.org/pdf/news/654.pdf online edition (.pdf)]

* [http://scienceworld.wolfram.com/physics/IsingModel.html Science World article on the Ising Model]

* [http://physics.ucsc.edu/~peter/ising/ising.html A dynamical 2D Ising java applet by UCSC]

* [https://sites.google.com/view/chremos-group/applets/ising-model A dynamical 2D Ising java applet]

* [http://www.physics.uci.edu/~etolleru/IsingApplet/IsingApplet.html A larger/more complicated 2D Ising java applet]

* [http://demonstrations.wolfram.com/IsingModel/ Ising Model simulation] by Enrique Zeleny, the [[Wolfram Demonstrations Project]]

* [http://ibiblio.org/e-notes/Perc/contents.htm Phase transitions on lattices]

* [http://www.sandia.gov/media/NewsRel/NR2000/ising.htm Three-dimensional proof for Ising Model impossible, Sandia researcher claims]

* [http://isingspinwebgl.com Interactive Monte Carlo simulation of the Ising, XY and Heisenberg models with 3D graphics(requires WebGL compatible browser)]

* [https://github.com/AmazaspShumik/BayesianML-MCMC/blob/master/Gibbs%20Ising%20Model/GibbsIsingModel.m Ising Model code ], [https://github.com/AmazaspShumik/BayesianML-MCMC/blob/master/Gibbs%20Ising%20Model/imageDenoisingExample.m image denoising example with Ising Model]

* [http://www.damtp.cam.ac.uk/user/tong/statphys/five.pdf David Tong's Lecture Notes ] provide a good introduction

{{Stochastic processes}}

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[[Category:Spin models]]

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类别: 统计力学

[[Category:Lattice models]]

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类别: 格子模型

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