Ising Models  

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 − μ ∑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 transition 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.
* Lee-Yang theorem: For any graph, the zeros of the partition function of the ferromagnetic Ising model lie on the unit circle in C; In fact, the union of the zeros of all graphs is dense on the unit circle.
@ References: Ising ZP(25); Imbrie PRL(84) [critical dimension]; Hayes AS(00)sep [I]; McCoy a1111-conf [rev]; Mosseri PRE(15)-a1409 [arbitrary graphs, energy spectrum from Hadamard transform]; Collevecchio et al a1409 [Prokofiev-Svistunov (worm) algorithm]; Ising et al a1706 [history]; Peters & Regts JLMS(19)-a1810 [arbitrary graphs, zeros of the partition function].

Cases and Techniques > s.a. 2D ising models.
* 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; See, however, the transverse-field case.
@ 1D: Pfeuty AP(70) [transverse field]; Reyes & Tsvelik NPB(06), PRB(06)cm [correlation functions]; Mbeng et al a2009 [quantum Ising chain, intro].
@ 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]; Litim & Zappalà PRD(11) [exponents, functional renormalization group approach]; Perk ChPB(13)-a1307; Aizenman et al CMP(15)-a1311; Gliozzi & Rago JHEP(14)-a1403 [critical exponents]; Cosme et al JHEP(15)-a1503 [conformal symmetry]; Talalaev a1805 [integrable structure]; Rychkov a2007 [conformal bootstrap approach]; > s.a. Scale Invariance.
@ 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: 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]; Temesvári NPB(10)-a0911 [free energy, perturbative]; Bonzom et al PLB(12)-a1108 [random lattice, no phase transition in the continuum limit]; Chatterjee CMP(15)-a1404 [no replica symmetry breaking in the random field Ising model]; Ott & Velenik a2007 [correlations].
@ Antiferromagnetic: Azcoiti et al NPB(14) [Monte Carlo algorithm].
@ Other types: Bahmad et al PhyA(07) [mixed spin-1/2 and spin-1]; Serva PhyA(11) [dilute model]; Affonso et al a2105 [long-range systems].
@ Spin-3/2: Canko & Keskin PLA(03) [ground state]; Keskin & Canko PLA(05) [relaxation phenomena near second-order phase transition]; Canko & Keskin PhyA(06).
@ Simulations: Aktekin PhyA(96) [4D]; Cervera-Lierta Quant(18)-a1807 [1D, on a quantum computer]; Crosson & Slezak a2002 [path integral Monte Carlo].
> Techniques: see ClusterExpansion; Master Equation; Mean-Field Method; path integrals; renormalization group; stochastic quantization.

References > s.a. graph theory in physics; networks / Potts Model; regge calculus.
@ 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]; Bissacot & Cioletti JSP(10)-a1001 [with non-uniform external fields]; Gessner et al EPL(14)-a1403 [monitoring the time evolution of a single spin]; Bonati EJP(14) [the Peierls argument in higher dimensions]; Duminil-Copin a1607-proc [random currents expansion]; > s.a. quantum phase transitions.
@ With magnetic field: Delfino JPA(04) [rev]; March PLA(14) [field of arbitrary strength]; Cioletti & Vila JSP(16)-a1506 [general lattices, graphical representation].
@ Entanglement: Novotný et al JPA(05) [one- and two-particle states]; Grimmett et al JSP(08)-a0704 [asymptotic scaling]; Furman et al PRA(08)-a0805 [1D]; Chang & Wu PRA(10)-a1001 [dynamics, and phase transition]; Foss-Feig et al NJP(13)-a1306 [dynamics of spin-spin correlation functions].
@ Continuum limit: Manrique et al CQG(06)ht/05 [Ising field theory, loop quantization techniques].
@ Other formulations: Rosengren JPA(86), 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]; Hernández et al JMP(13)-a1301, Hernández a1402 [coupled to 2D causal dynamical triangulations].
@ Related topics: Issigoni & Paraskevaidis PhyA(05) [roughening T]; Suzuki JSM(09)-a0807 [dynamics of temperature quenching]; Gu et al PhyD(09)-a0809 [emergent properties]; Streib et al JSP(14) [partition function, binomial approximation method]; Lubetzky & Sly a1401 [Glauber dynamics and information percolation]; Zintchenko et al PRB(15)-a1408 [ground states]; Bravyi & Hastings CMP-a1410 [complexity]; Navez et al PRB(17)-a1603 [propagation of quantum fluctuations]; Tee a2105 [emergent spacetime as the ground state]; > s.a. conformal invariance; Dimers; thermalization.
> Online resources: see Wikipedia page.

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