Ising Model  

In General > s.a. spin models; 2D gravity; lattice field theory [random].
* Idea: A crude model for ferromagnetic domains, based on a lattice of N fixed atoms of spin-1/2, with constant-coefficient Hamiltonian

H = – <ij> Jij si sj –  i=1N si B ,

where si = 1, B is the z-component of the magnetic field and the interaction energy is usually of the isotropic form Jij = J (J > 0 for ferromagnetism, J < 0 for antiferromagnetism); Without self-interactions, J = 0, the model is trivially solvable and does not depend on dimensions or type of lattice.
* History: The model was invented by the German physicist Wilhelm Lenz and investigated by his student Ernst Ising in the 1920s; Ising analyzed a 1D version of the model, and found no phase transitiion in the magnetization; A decade later other physicists found hints of magnetization in 2D, and in 1944 Lars Onsager confirmed the existsnce of a phase transition with an exact solution of the 2D Ising model; For three dimensions no exact solution has ever been found, but computer simulations give unmistakable evidence of an abrupt phase transition.
* Motivation: The 2D model is the only non-trivial exactly solvable model of phase transition.

Cases and Techniques > s.a. 2D ising model; Master Equation; Mean-Field Method; path integrals; stochastic quantization.
* In general: One can apply the mean-field approximation, which totally fails in 1D and gets better in higher dimensions, and the Bethe-Peierls approximation, which can be regarded as the lowest level of a Cluster Variation Method.
* 1D: The model is simple to solve and there is no spontaneous magnetization at any T, no B = 0 phase transition.
@ 1D: Reyes & Tsvelik NPB(06), cm/06 [correlation functions].
@ 1D, variations: Cassandro et al CMP(09) [with long-range interaction, random field]; Yuan et al PhyA(09) [with next-nearest-neighbor interactions].
@ 3D: Imbrie CMP(85) [random field, ground state]; Nigmatullin & Toboev TMP(89) [and 2D, thermodynamics]; Dotsenko et al PRL(93) [cluster boundaries]; Regge & Zecchina JPA(00)cm/99 [different lattices]; Ron et al PhyA(05) [fixed point]; Kozlovskii et al NPB(06) [free energy and equation of state]; Chung PLA(06) [magnetization and specific heat]; Canfora PLB(07)cm [Kallen-Lehman approach]; Caselle et al JHEP(07) [Monte Carlo, free energy of interfaces]; Canpolat et al PS(07) [effective-field approximation]; Nigro JSM(08)-a0710; Belletti et al JSP(09); Bittner et al NPB(09) [anisotropy of the interface tension].
@ 3D, random lattice: Ivaneyko et al PhyA(06); Lima et al PhyA(08).
@ 3D, with long-range-correlated disorder: Weinrib & Halperin PRB(83);
@ 3D, other variations: van Enter JSP(05) [random boundary conditions]; Kondratiev & Zhizhina JSP(07) [with birth and death dynamics]; Basuev TMP(07) [in half-space].
@ Higher dimensions: Aktekin PhyA(96) [4D, simulations]; Yokota PhyA(06) [replica symmetry breaking]; Sakai CMP(07) [lace expansion]; Klein & March PLA(08) [critical exponents]; Coupier AAP(08)-m/06 [conditions for Poisson approximations].
@ Other types: Bahmad et al PhyA(07) [mixed spin-1/2 and spin-1].
@ Spin-3/2: Canko & Keskin PLA(03) [ground state]; Keskin & Canko PLA(05) [relaxation phenomena near second-order phase transition]; Canko & Keskin PhyA(06).

References > s.a. [graphs; networks]; Potts Model; regge calculus.
@ General: Ising ZP(25); Imbrie PRL(84) [critical dimension]; Hayes AS(00)sep [I].
@ With magnetic field: Delfino JPA(04) [rev].
@ Phase transitions: Prüßner et al PhyA(00) [2D and 3D, critical exponents]; Liu & Gitterman AJP(03)aug [2D and 3D, critical T]; Zurek et al PRL(05)cm [dynamics]; Romá et al PhyA(06) [new order parameter]; Shimizu & Kawaguchi PLA(06) [and entanglement]; Aguirre-Contreras et al PLA(06) [critical T, diluted model]; Pérez Gaviro et al JPA(06); Dziarmaga PRB(06)cm [random lattice]; Pishtchev PLA(07) [critical exponents]; Machta et al JSP(08) [percolation signature]; Björnberg & Grimmett JSP(09)-a0901 [sharpness, hypercubic lattice]; Barré et al PhyA(09) [finite-size effects, random graphs]; > s.a. quantum phase transitions.
@ Entanglement: Novotny et al JPA(05) [one- and two-particle states]; Grimmett et al JSP(08)-a0704 [asymptotic scaling]; Furman et al a0805 [1D].
@ Continuum limit: Manrique et al CQG(06)ht/05 [Ising field theory, loop quantization techniques].
@ Other formulations: da Costa & Maciel RBEF(03)mp [combinatorial]; Diego ht/05 [integral representation].
@ Other variations and generalizations: Meyer pr(92) [spacetime Ising models]; O'Connor et al JPA(07) [Ising-like models, equation of state]; Bazhanov et al PLA(08)-a0706 [Faddeev-Volkov model].
@ Related topics: Issigoni & Paraskevaidis PhyA(05) [roughening T]; Suzuki JSM(09)-a0807 [dynamics of temperature quenching]; Gu et al PhyD(09)-a0809 [emergent properties]; > s.a. conformal invariance; Dimers.
> Online resources: Wikipedia page.

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