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fast this process will eventually produce a pick from the distribution as a markov chain edit 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 32 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 voter model for opinion dynamics solutions edit one dimension edit the thermodynamic limit exists as long as the interaction decay is j i j i j α displaystyle j_ ij sim i j alpha with α 1 33 in the case of ferromagnetic interaction j i j i j α displaystyle j_ ij sim i j alpha with 1 α 2 dyson proved by comparison with the hierarchical case that there is phase transition at small enough temperature 34 in the case of ferromagnetic interaction j i j i j 2 displaystyle j_ ij sim i j 2 fröhlich and spencer proved that there is phase transition at small enough temperature in contrast with the hierarchical case 35 in the case of interaction j i j i j α displaystyle j_ ij sim i j alpha 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 free energy is analytic in the thermodynamic parameters 33 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 36 ising s exact solution edit in the nearest neighbor case with periodic or free boundary conditions an exact solution is available the hamiltonian of the one dimensional ising model on a lattice of l sites with free boundary conditions is h σ j i 1 l 1 σ i σ i 1 h i σ i displaystyle h sigma j sum _ i 1 ldots l 1 sigma _ i sigma _ i 1 h sum _ i sigma _ i where j and h can be any number since in this simplified case j is a constant representing the interaction strength between the nearest neighbors and h is the constant external magnetic field applied to lattice sites then the free energy is f β h lim l 1 β l ln z β 1 β ln e β j cosh β h e 2 β j sinh β h 2 e 2 β j displaystyle f beta h lim _ l to infty frac 1 beta l ln z beta frac 1 beta ln left e beta j cosh beta h sqrt e 2 beta j sinh beta h 2 e 2 beta j right and the spin spin correlation i e the covariance is σ i σ j σ i σ j c β e c β i j displaystyle langle sigma _ i sigma _ j rangle langle sigma _ i rangle langle sigma _ j rangle c beta e c beta i j where c β and c β are positive functions for t 0 for t 0 though the inverse correlation length c β vanishes proof edit the proof of this result is a simple computation if h 0 it is very easy to obtain the free energy in the case of free boundary condition i e when h σ j σ 1 σ 2 σ l 1 σ l displaystyle h sigma j left sigma _ 1 sigma _ 2 cdots sigma _ l 1 sigma _ l right then the model factorizes under the change of variables σ j σ j σ j 1 j 2 displaystyle sigma _ j sigma _ j sigma _ j 1 quad j geq 2 this gives z β σ 1 σ l e β j σ 1 σ 2 e β j σ 2 σ 3 e β j σ l 1 σ l 2 j 2 l σ j e β j σ j 2 e β j e β j l 1 displaystyle z beta sum _ sigma _ 1 ldots sigma _ l e beta j sigma _ 1 sigma _ 2 e beta j sigma _ 2 sigma _ 3 cdots e beta j sigma _ l 1 sigma _ l 2 prod _ j 2 l sum _ sigma _ j e beta j sigma _ j 2 left e beta j e beta j right l 1 therefore the free energy is f β 0 1 β ln e β j e β j displaystyle f beta 0 frac 1 beta ln left e beta j e beta j right with the same change of variables σ j σ j n e β j e β j e β j e β j n displaystyle langle sigma _ j sigma _ j n rangle left frac e beta j e beta j e beta j e beta j right n hence it decays exponentially as soon as t 0 but for t 0 i e in the limit β there is no decay if h 0 we need the transfer matrix method for the periodic boundary conditions case is the following the partition function is z β σ 1 σ l e β h σ 1 e β j σ 1 σ 2 e β h σ 2 e β j σ 2 σ 3 e β h σ l e β j σ l σ 1 σ 1 σ l v σ 1 σ 2 v σ 2 σ 3 v σ l σ 1 displaystyle z beta sum _ sigma _ 1 ldots sigma _ l e beta h sigma _ 1 e beta j sigma _ 1 sigma _ 2 e beta h sigma _ 2 e beta j sigma _ 2 sigma _ 3 cdots e beta h sigma _ l e beta j sigma _ l sigma _ 1 sum _ sigma _ 1 ldots sigma _ l v_ sigma _ 1 sigma _ 2 v_ sigma _ 2 sigma _ 3 cdots v_ sigma _ l sigma _ 1 the coefficients v σ σ displaystyle v_ sigma sigma can be seen as the entries of a matrix there are different possible choices a convenient one because the matrix is symmetric is v σ σ e β h 2 σ e β j σ σ e β h 2 σ displaystyle v_ sigma sigma e frac beta h 2 sigma e beta j sigma sigma e frac beta h 2 sigma or v e β h j e β j e β j e β h j displaystyle v begin bmatrix e beta h j e beta j e beta j e beta h j end bmatrix in matrix formalism z β tr v l λ 1 l λ 2 l λ 1 l 1 λ 2 λ 1 l displaystyle 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 where λ 1 is the highest eigenvalue of v while λ 2 is the other eigenvalue λ 1 e β j cosh β h e 2 β j cosh β h 2 2 sinh 2 β j e β j cosh β h e 2 β j sinh β h 2 e 2 β j displaystyle lambda _ 1 e beta j cosh beta h sqrt e 2 beta j cosh beta h 2 2 sinh 2 beta j e beta j cosh beta h sqrt e 2 beta j sinh beta h 2 e 2 beta j and λ 2 λ 1 this gives the formula of the free energy above in the thermodynamics limit for the non interaction case j 0 we got z n λ 1 n 2 cosh β h n displaystyle z_ n to lambda _ 1 n 2 cosh beta h n as the answer for the open boundary ising model comments edit 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 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 p 1 p e 2 β j displaystyle frac p 1 p e 2 beta j the problem is reduced to independent biased coin tosses this essentially completes the mathematical description from the description in terms of independent tosses the statistics of the model for long lines can be understood the line splits into domains each domain is of average length 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 s i s j e p i j displaystyle langle s_ i s_ j rangle propto e p i j 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 z configs e k s k k 1 p 1 p l displaystyle z sum _ text configs e sum _ k s_ k prod _ k 1 p 1 p l the logarithm divided by l is the free energy density β f log 1 p log 1 e 2 β j 1 e 2 β j displaystyle beta f log 1 p log left 1 frac e 2 beta j 1 e 2 beta j right which is analytic away from β a sign of a phase transition is a non analytic free energy so the one dimensional model does not have a phase transition one dimensional solution with transverse field edit 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 h σ j i 1 l σ i z σ i 1 z h i σ i x displaystyle h sigma j sum _ i 1 ldots l sigma _ i z sigma _ i 1 z h sum _ i sigma _ i x 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 σ n z i 1 n t i x displaystyle sigma _ n z prod _ i 1 n t_ i x σ n x t n z t n 1 z displaystyle sigma _ n x t_ n z t_ n 1 z upon rewriting the hamiltonian in terms of this change of basis matrices we obtain h σ h i 1 l t i z t i 1 z j i t i x displaystyle h sigma h sum _ i 1 ldots l t_ i z t_ i 1 z j sum _ i t_ i x since the roles of h and j are switched the hamiltonian undergoes a transition at j h 37 renormalization edit when there is no external field we can derive a functional equation that f β 0 f β displaystyle f beta 0 f beta satisfies using renormalization 38 specifically let z n β j displaystyle z_ n beta j be the partition function with n displaystyle n sites now we have z n β j σ e k σ 2 σ 1 σ 3 e k σ 4 σ 3 σ 5 displaystyle z_ n beta j sum _ sigma e k sigma _ 2 sigma _ 1 sigma _ 3 e k sigma _ 4 sigma _ 3 sigma _ 5 cdots where k β j displaystyle k beta j we sum over each of σ 2 σ 4 displaystyle sigma _ 2 sigma _ 4 cdots to obtain z n β j σ 2 cosh k σ 1 σ 3 2 cosh k σ 3 σ 5 displaystyle z_ n beta j sum _ sigma 2 cosh k sigma _ 1 sigma _ 3 cdot 2 cosh k sigma _ 3 sigma _ 5 cdots now since the cosh function is even we can solve a e k σ 1 σ 3 2 cosh k σ 1 σ 3 displaystyle ae k sigma _ 1 sigma _ 3 2 cosh k sigma _ 1 sigma _ 3 as a 2 cosh 2 k k 1 2 ln cosh 2 k textstyle a 2 sqrt cosh 2k k frac 1 2 ln cosh 2k now we have a self similarity relation 1 n ln z n k 1 2 ln 2 cosh 2 k 1 2 1 n 2 ln z n 2 k displaystyle frac 1 n ln z_ n k frac 1 2 ln left 2 sqrt cosh 2k right frac 1 2 frac 1 n 2 ln z_ n 2 k taking the limit we obtain f β 1 2 ln 2 cosh 2 k 1 2 f β displaystyle f beta frac 1 2 ln left 2 sqrt cosh 2k right frac 1 2 f beta where β j 1 2 ln cosh 2 β j displaystyle beta j frac 1 2 ln cosh 2 beta j when β displaystyle beta is small we have f β ln 2 displaystyle f beta approx ln 2 so we can numerically evaluate f β displaystyle f beta by iterating the functional equation until k displaystyle k is small two dimensions edit 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 the spin spin correlation functions were computed by mccoy and wu onsager s exact solution edit main article square lattice ising model onsager 1944 obtained the following analytical expression for the free energy of the ising model on the anisotropic square lattice when the magnetic field h 0 displaystyle h 0 in the thermodynamic limit as a function of temperature and the horizontal and vertical interaction energies j 1 displaystyle j_ 1 and j 2 displaystyle j_ 2 respectively β f ln 2 1 8 π 2 0 2 π d θ 1 0 2 π d θ 2 ln cosh 2 β j 1 cosh 2 β j 2 sinh 2 β j 1 cos θ 1 sinh 2 β j 2 cos θ 2 displaystyle 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 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 t c displaystyle t_ c which solves the equation sinh 2 j 1 k t c sinh 2 j 2 k t c 1 displaystyle sinh left frac 2j_ 1 kt_ c right sinh left frac 2j_ 2 kt_ c right 1 in the isotropic case when the horizontal and vertical interaction energies are equal j 1 j 2 j displaystyle j_ 1 j_ 2 j the critical temperature t c displaystyle t_ c occurs at the following point t c 2 j k ln 1 2 2 269185 j k displaystyle t_ c frac 2j k ln 1 sqrt 2 2 269185 cdots frac j k when the interaction energies j 1 displaystyle j_ 1 j 2 displaystyle j_ 2 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 h 0 displaystyle h 0 so the free energy and critical temperature are the same for the antiferromagnetic case for the triangular lattice which is not bi partite the ferromagnetic and antiferromagnetic ising model behave notably differently specifically around a triangle it is impossible to make all 3 spin pairs antiparallel so the antiferromagnetic ising model cannot reach the minimal energy state this is an example of geometric frustration onsager s formula for spontaneous magnetization edit 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 8 m 1 sinh 2 β j 1 sinh 2 β j 2 2 1 8 displaystyle m left 1 left sinh 2 beta j_ 1 sinh 2 beta j_ 2 right 2 right frac 1 8 where j 1 displaystyle j_ 1 and j 2 displaystyle j_ 2 are horizontal and vertical interaction energies a complete derivation was only given in 1951 by yang 1952 using a limiting process of transfer matrix eigenvalues the proof was subsequently greatly simplified in 1963 by montroll potts and ward 8 using szegő s limit formula for toeplitz determinants by treating the magnetization as the limit of correlation functions minimal model edit main article two dimensional critical ising model at the critical point the two dimensional ising model is a two dimensional conformal field theory the spin and energy correlation functions are described by a minimal model which has been exactly solved three dimensions edit in three as in two dimensions the most studied case of the ising model is the translation invariant model on a cubic lattice with nearest neighbor coupling in the zero magnetic field many theoreticians searched for an analytical three dimensional solution for many decades which would be analogous to onsager s solution in the two dimensional case 39 40 such a solution has not been found until now although there is no proof that it may not exist in three dimensions the ising model was shown to have a representation in terms of n...
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