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ma _ 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 non interacting fermionic strings by alexander polyakov and vladimir dotsenko this construction has been carried on the lattice and the continuum limit conjecturally describing the critical point is unknown in three as in two dimensions peierls argument shows that there is a phase transition this phase transition is rigorously known to be continuous in the sense that correlation length diverges and the magnetization goes to zero and is called the critical point it is believed that the critical point can be described by a renormalization group fixed point of the wilson kadanoff renormalization group transformation it is also believed that the phase transition can be described by a three dimensional unitary conformal field theory as evidenced by monte carlo simulations 41 42 exact diagonalization results in quantum models 43 and quantum field theoretical arguments 44 although it is an open problem to establish rigorously the renormalization group picture or the conformal field theory picture theoretical physicists have used these two methods to compute the critical exponents of the phase transition which agree with the experiments and with the monte carlo simulations this conformal field theory describing the three dimensional ising critical point is under active investigation using the method of the conformal bootstrap 45 46 47 48 this method currently yields the most precise information about the structure of the critical theory see ising critical exponents in 2000 sorin istrail of sandia national laboratories proved that the spin glass ising model on a nonplanar lattice is np complete that is assuming p np the general spin glass ising model is exactly solvable only in planar cases so solutions for dimensions higher than two are also intractable 49 istrail s result only concerns the spin glass model with spatially varying couplings and tells nothing about ising s original ferromagnetic model with equal couplings four dimensions and above edit main article high dimensional ising model 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 the accuracy of this approximation improves as the dimension becomes larger a deeper understanding of how the ising model behaves going beyond mean field approximations can be achieved using renormalization group methods see also edit annni model binder parameter boltzmann machine construction of an irreducible markov chain in the ising model geometrical frustration classical heisenberg model quantum heisenberg model quantum rotor model kuramoto model maximal evenness order operator spin model t j model classical xy model n vector model zn model footnotes edit see gallavotti 1999 chapters vi vii ernst ising contribution to the theory of ferromagnetism samuel stuart 1980 the use of anticommuting variable integrals in statistical mechanics i the computation of partition functions journal of mathematical physics 21 12 2806 2814 bibcode 1980jmp 21 2806s doi 10 1063 1 524404 see baierlein 1999 chapter 16 barahona francisco grötschel martin jünger michael reinelt gerhard 1988 an application of combinatorial optimization to statistical physics and circuit layout design operations research 36 3 493 513 doi 10 1287 opre 36 3 493 issn 0030 364x 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