2-Dimensional Ising Models| 2-Dimensional Ising Models |
In General > s.a. ising models; spin
models; lattice field theory [random].
* Motivation: The 2D Ising
model is the only non-trivial exactly solvable model of phase transition;
2016, Proof that all classical spin models are equivalent to 2D Ising models,
with possibly position-dependent couplings and external fields.
* Summary: Without a magnetic
field, one gets the Onsager solution (1940s) with a phase transition, while
with a magnetic field exact results were obtained in the 1980s by
Zamolodchikov, at the critical temperature.
@ As universal models:
De las Cuevas & Cubitt sci(16)mar
+ summary sci(16)mar.
@ General references:
Kac & Ward PR(52) [combinatorial];
Maddox Nat(92)oct [Onsager solution];
de Oliveira et al JPA(06) [Monte Carlo evolution];
García-Pelayo JMP(09) [isomorphism with persistent random walk];
Bostan et al a0904;
Strack & Jakubczyk PRB(09)-a0906;
Huber & Law a0907 [canonical paths];
Parisen Toldin et al JSP(09) [low-T paramagnetic-ferromagnetic transition];
Mangazeev et al PRE(10)-a1005 [scaling and universality, with magnetic field];
McCoy & Maillard PTP(12)-a1203 [rev];
Camia MPRF-a1205 [continuum scaling limit];
Kager et al JSP(13)-a1208 [signed-loop approach];
Siudem et al a1410
[infinite square lattice partition function, low-temperature expansion];
Chelkak et al AIHP(17)-a1507 [combinatorics];
Krieger a2009,
a2009 [partition function].
@ Susceptibility: Boukraa et al JPA(08)-a0808 [many terms in a series];
McCoy et al a1003 [rev];
Chan et al JSP(11)-a1012 [zero-field, many terms];
Tracy & Widom JMP(13) [diagonal susceptibility].
@ Correlation functions: Bell PR(66);
Wang PhyA(09);
Perk & Au-Yang JSP(09) [pair-correlation functions];
Iorgov & Lisovyy JSP(11)-a1012;
Chelkak et al a1202 [n-point spin correlations].
@ Critical behavior:
Lubetzky & Sly CMP(12) [mixing time];
Beffara & Duminil-Copin AP(12)-a1010;
Li CMP(12) [periodic models];
Witczak-Krempa PRL(15)-a1501;
Assis et al JPA(17)-a1705 [analyticity properties];
Caselle & Sorba PRD(20)-a2003.
@ Other specific concepts: Beale PRL(96) [exact energy distribution function];
Schülke & Zheng PLA(97) [global persistence exponent];
Kitatani et al JPA(03) [specific heat, ± J];
Mangazeev et al JPA(09) [scaling function, in magnetic field];
Camia et al a1205 [magnetization exponent];
Baxter JPA(16)-a1606 [square lattice, boundary free energies];
Freed & Teleman a1806 [topological dualities].
@ Numerical techniques:
Nakamura PRL(08) [Monte Carlo, quasi-1D];
Preis et al JCP(09) [GPU-accelerated Monte Carlo].
@ Random field:
Moss De Oliveira et al PhyA(90) [and 1D];
> s.a. renormalization.
> Related topics:
see entanglement entropy.
Different Types
@ Random lattice: Boulatov & Kazakpv PLB(87) [critical exponents];
Janke et al NPPS(94);
Lima et al PhyA(00);
De Sanctis a0811;
Dommers et al JSP(10)-a1005 [with power-law degree distribution];
Giardinà et al JSP(15)-a1412 [central-limit theorems];
Sasakura & Sato PTEP(14)-a1401;
Chen & Turunen CMP(20)-a1806 [critical temperature],
a2003 [phase transition].
@ Other lattices: Repetowicz et al JPA(99),
Repetowicz JPA(02) [quasiperiodic, Penrose tiling];
Oitmaa & Keppert JPA(02) [on a 4-6 lattice];
Bugrij & Lisovyy PLA(03)-a0708 [finite lattice, spin matrix elements],
TMP(04)-a0708 [anisotropic lattice, correlation functions];
Wan ht/05 [with non-local links];
Balint et al a0806 [triangular lattice];
Björnberg JSP(09) [on star-like graphs];
Viana et al PLA(09) [anisotropic lattice, antiferromagnetic, longitudinal field];
Codello JPA(10) [Archimedean and Laves lattices];
Mellor & Hibberd a1106 [Union Jack lattice];
Gandolfo et al JSP(12)-a1207 [Cayley tree, Gibbs states];
Yoshida & Kubica a1404 [on a fractal (Sierpiński) lattice].
@ On dynamical triangulation: Benedetti & Loll GRG(07)gq/06;
Sato & Tanaka PRD(18)-a1710 [criticality at absolute zero].
@ On causal triangulations: Napolitano & Turova JSP(16)-a1504 [random planar triangulations].
@ Different global topologies: Burda & Jurkiewicz PLB(88) [on T2];
Nigro PhyA(13)-a1010 [cylinder, spatially periodic boundary conditions];
Assis & McCoy JPA(11)-a1011 [half-plane lattice];
Lu & Wu PRE(01)cm/00 [non-orientable surface];
Greenblatt a1409 [cylinder, finite-size corrections];
Matsuura & Sakai PTEP(15)-a1507 [with twisted boundary conditions];
Mohammed & Mahapatra IJMPC(18)-a1601 [different boundary conditions].
@ Different interactions: Van den Nest et al PRL(08)-a0708 [arbitrary
graph with inhomogeneous pairwise interactions equivalent to 2D square lattice with suitable couplings];
Picco a1207,
Blanchard et al EPL(13)-a1211 [long-range interactions, simulations of critical behavior].
@ Other variations:
Kutlu PhyA(97) [with more interactions];
Roder et al PhyA(99) [high-T analysis];
Bittner et al PhyA(00) [fluctuating];
Dorogovtsev et al PRE(02)cm [Tc];
Shiwa & Sakaniwa JPA(06)cm/05 [on a constant negative curvature surface];
Rojas & de Souza PLA(09) [exactly-solvable models];
Gandolfo et al CMP(15)-a1310 [on the Lobachevsky plane];
Aasen et al JPA(16)-a1601 [with topological defects];
> s.a. ising models.
@ Relationship with different models: Wetterich NPB(17)-a1612 [and free massless Dirac fermions in 2D Minkowski space].
> Gravity-related models:
see 2D gravity; spin networks.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
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