page_title: Ising model
section_title: Basic properties and history
hierarchical_section_title: Ising model / Basic properties and history
caption_attribution_description: English: This tartan-like graph shows the Ising model probability density P ( &#x03C3; ) {\displaystyle P(\sigma )} for the two-sided lattice using the dyadic mapping. That is, a lattice configuration of length 2 N {\displaystyle 2N} &#x03C3; = ( &#x03C3; &#x2212; N &#x2212; 1 , &#x22EF; , &#x03C3; &#x2212; 2 , &#x03C3; &#x2212; 1 , &#x03C3; 0 , &#x03C3; 1 , &#x03C3; 2 , &#x22EF; , &#x03C3; N ) {\displaystyle \sigma =(\sigma _{-N-1},\cdots ,\sigma _{-2},\sigma _{-1},\sigma _{0},\sigma _{1},\sigma _{2},\cdots ,\sigma _{N})} is understood to consist of a sequence of "spins" &#x03C3; k = &#x00B1; 1 {\displaystyle \sigma _{k}=\pm 1} . This sequence may be represented by two real numbers 0 &#x2264; x , y &#x2264; 1 {\displaystyle 0\leq x,y\leq 1} with x ( &#x03C3; ) = &#x2211; k = 0 N ( &#x03C3; k + 1 2 ) 2 &#x2212; ( k + 1 ) {\displaystyle x(\sigma )=\sum _{k=0}^{N}\left({\frac {\sigma _{k}+1}{2}}\right)2^{-(k+1)}} and y ( &#x03C3; ) = &#x2211; k = 0 N ( &#x03C3; &#x2212; k &#x2212; 1 + 1 2 ) 2 &#x2212; ( k + 1 ) {\displaystyle y(\sigma )=\sum _{k=0}^{N}\left({\frac {\sigma _{-k-1}+1}{2}}\right)2^{-(k+1)}} The energy of a given configuration &#x03C3; {\displaystyle \sigma } is computed using the classical Hamiltonian, H ( &#x03C3; ) = &#x2211; k = &#x2212; N N V ( &#x03C4; k &#x03C3; ) {\displaystyle H(\sigma )=\sum _{k=-N}^{N}V(\tau ^{k}\sigma )} Here, &#x03C4; {\displaystyle \tau } is the shift operator, acting on the lattice by shifting all spins over by one position: &#x03C4; ( &#x22EF; , &#x03C3; 0 , &#x03C3; 1 , &#x03C3; 2 , &#x22EF; ) = ( &#x22EF; , &#x03C3; &#x2212; 1 , &#x03C3; 0 , &#x03C3; 1 , &#x22EF; ) {\displaystyle \tau (\cdots ,\sigma _{0},\sigma _{1},\sigma _{2},\cdots )=(\cdots ,\sigma _{-1},\sigma _{0},\sigma _{1},\cdots )} The interaction potential V {\displaystyle V} is given by the Ising model interaction V ( &#x03C3; ) = J &#x03C3; 0 &#x03C3; 1 + B &#x03C3; 0 {\displaystyle V(\sigma )=J\sigma _{0}\sigma _{1}+B\sigma _{0}} Here, the constant J {\displaystyle J} is the interaction strength between two neighboring spins &#x03C3; 0 {\displaystyle \sigma _{0}} and &#x03C3; 1 {\displaystyle \sigma _{1}} , while the constant B {\displaystyle B} may be interpreted as the strength of the interaction between the magnetic field and the magnetic moment of the spin. The set of all possible configurations &#x03A9; = { &#x03C3; } {\displaystyle \Omega =\{\sigma \}} form a canonical ensemble, with each different configuration occurring with a probability P ( &#x03C3; ) {\displaystyle P(\sigma )} given by the Boltzmann distribution P ( &#x03C3; ) = 1 Z ( T ) e &#x2212; H ( &#x03C3; ) / k B T {\displaystyle P(\sigma )={\frac {1}{Z(T)}}e^{-H(\sigma )/k_{B}T}} where k B {\displaystyle k_{B}} is Boltzmann's constant, T {\displaystyle T} is the temperature, and Z ( T ) {\displaystyle Z(T)} is the partition function. The partition function is defined to be such that the sum over all probabilities adds up to one; that is, so that Z ( T ) = &#x2211; &#x03C3; &#x2208; &#x03A9; e &#x2212; H ( &#x03C3; ) / k B T {\displaystyle Z(T)=\sum _{\sigma \in \Omega }e^{-H(\sigma )/k_{B}T}} Contents 1 Image details 2 As an invariant measure 3 Licensing 4 Original upload log Image details The image here shows P ( &#x03C3; ) = P ( x , y ) {\displaystyle P(\sigma )=P(x,y)} for the Ising model, with J = 0.3 {\displaystyle J=0.3} , B = 0 {\displaystyle B=0} and temperature T = 1 / k B {\displaystyle T=1/k_{B}} . The lattice is finite sized, with N = 10 {\displaystyle N=10} , so that all 1024 &#x00D7; 1024 = 2 N &#x00D7; 2 N {\displaystyle 1024\times 1024=2^{N}\times 2^{N}} lattice configurations are represented, each configuration denoted by one pixel. The color choices here are such that black represents values where P ( &#x03C3; ) = P ( x , y ) {\displaystyle P(\sigma )=P(x,y)} are zero, blue are small values, with yellow and red being progressively larger values. As an invariant measure This fractal tartan is invariant under the Baker's map. The shift operator &#x03C4; {\displaystyle \tau } on the lattice has an action on the unit square with the following representation: &#x03C4; ( x , y ) = ( x + &#x230A; 2 y &#x230B; 2 , 2 y &#x2212; &#x230A; 2 y &#x230B; ) {\displaystyle \tau (x,y)=\left({\frac {x+\left\lfloor 2y\right\rfloor }{2}}\,,\,2y-\left\lfloor 2y\right\rfloor \right)} This map (up to a reflection/rotation around the 45-degree axis) is essentially the Baker's map or equivalently the Horseshoe map. As the article on the Horseshoe map explains, the invariant sets have such a tartan pattern (an appropriately deformed Sierpinski carpet). In this case, the invariance arises from the translation invariance of the Gibbs states of the Ising model: that is, the energy H ( &#x03C3; ) {\displaystyle H(\sigma )} associated with the state &#x03C3; {\displaystyle \sigma } is invariant under the action of &#x03C4; {\displaystyle \tau } : H ( &#x03C4; k &#x03C3; ) = H ( &#x03C3; ) {\displaystyle H\left(\tau ^{k}\sigma \right)=H(\sigma )} for all integers k {\displaystyle k} . Similarly, the probability density is invariant as well: P ( &#x03C4; k &#x03C3; ) = P ( &#x03C3; ) {\displaystyle P\left(\tau ^{k}\sigma \right)=P(\sigma )} The naive classical treatment given here suffers from conceptual difficulties in the N &#x2192; &#x221E; {\displaystyle N\to \infty } limit. These problems can be remedied by using a more appropriate topology on the set of states that make up the configuration space. This topology is the cylinder set topology, and using it allows one to construct a sigma algebra and thus a measure on the set of states. With this topology, the probability density can be understood to be a translation-invariant measure on the topology. Indeed, there is a certain sense in which the seemingly fractal patterns generated by the iterated Baker's map or horseshoe map can be understood with a conventional and well-behaved topology on a lattice model. Created by Linas Vepstas User:Linas on 24 September 2006
context_page_description: 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. The spins are arranged in a graph, usually a lattice, 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.
  The Ising model was invented by the physicist Wilhelm Lenz, who gave it as a problem to his student Ernst Ising. The one-dimensional Ising model was solved by Ising 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 Lars Onsager.
context_section_description: The most studied case of the Ising model is the translation-invariant ferromagnetic zero-field model on a d-dimensional lattice, namely, Λ = Zᵈ, Jᵢⱼ = 1, h = 0.
  In his 1924 PhD thesis, Ising solved the model for the 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|:
  and the system is disordered. On the basis of this result, he incorrectly concluded that this model does not exhibit phase behaviour in any dimension.
  The Ising model undergoes a phase transition between an ordered and a disordered phase in 2 dimensions or more. Namely, the system is disordered for small β, whereas for large β the system exhibits ferromagnetic order:
  This was first proven by Rudolf Peierls in 1936, using what is now called a Peierls argument.
  The Ising model on a two-dimensional square lattice with no magnetic field was analytically solved by Lars Onsager (1944). Onsager showed that the correlation functions and free energy of the Ising model are determined by a noninteracting lattice fermion. Onsager announced the formula for the spontaneous magnetization for the 2-dimensional model in 1949 but did not give a derivation. Yang (1952) gave the first published proof of this formula, using a limit formula for Fredholm determinants, proved in 1951 by Szegő in direct response to Onsager's work.
caption: Visualization of the translation-invariant probability measure of the one-dimensional Ising model
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