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.
main page – abbreviations – journals – comments – other
sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 24
oct
2009