Ising Model  

In General > s.a. spin models; 2D gravity; lattice field theory [random].
* Motivation: The 2D model is the only non-trivial exactly solvable model of phase transition.
* 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.

Cases and Techniques > s.a. Master Equation; Mean-Field Method; path integrals; stochastic quantization.
* 1D: The model is simple to solve and there is no spontaneous magnetization at any T, no B = 0 phase transition.
* 2D: Without a magnetic field, one gets the Onsager solution (1940's) with a phase transition, while with a magnetic field exact results were obtained in the 1980's by Zamolodchikov, at the critical temperature.
* 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: Moss De Oliveira et al PhyA(90) [and 2D, random field]; Reyes & Tsvelik NPB(06), cm/06 [correlation functions].
@ 2D: Kac & Ward PR(52) [combinatorial]; Bell PR(66) [correlation functions, etc]; Maddox Nat(92)oct [Onsager solution]; Schülke & Zheng PLA(97) [global persistence exponent]; Kitatani et al JPA(03) [specific heat, J]; de Oliveira et al JPA(06) [Monte Carlo evolution].
@ 2D, random lattice: Boulatov & Kazakpv PLB(87) [critical exponents]; Janke et al NPPS(94); Lima et al PhyA(00).
@ 2D, other variations: Burda & Jurkiewicz PLB(88) [on T2]; Kutlu PhyA(97) [with more intreactions]; Repetowicz et al JPA(99), Repetowicz JPA(02) [quasiperiodic, Penrose tiling]; Roder et al PhyA(99) [high-T analysis]; Bittner et al PhyA(00) [fluctuating]; Lu & Wu PRE(01)cm/00 [non-orientable surface]; Oitmaa & Keppert JPA(02) [on a 4-6 lattice]; Dorogovtsev et al PRE(02)cm [Tc]; Bugrij & Lisovyy PLA(03)-a0708 [finite lattice, spin matrix elements], TMP(04)-a0708 [anisotropic lattice, correlation functions]; Wan ht/05 [with non-local links]; Shiwa & Sakaniwa JPA(06)cm/05 [on constant negative curvature surface]; Benedetti & Loll GRG(07)gq/06 [on dynamical triangulation]; Balint et al a0806 [triangular lattice].
@ 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 a0710.
@ 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].
@ Other types: Van den Nest et al PRL(08)-a0708 [arbitrary graph with inhomogeneous pairwise interactions equivalent to 2D square lattice with suitable couplings]; 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].
@ With magnetic field: Delfino JPA(04) [rev].
@ Phase transitions: Prüßner et al PhyA(00) [critical exponents]; Liu & Gitterman AJP(03) [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].
@ 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 a0706 [Faddeev-Volkov model].
@ Related topics: Issigoni & Paraskevaidis PhyA(05) [roughening T]; Suzuki a0807 [dynamics of temperature quenching]; > s.a. conformal invariance.
> Online resources: Wikipedia page.

Main page – Abbreviations – Journals – Comments – Other sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified 23 jul 2008