In statistical physics, the axial (or anisotropic) next-nearest neighbor Ising model, usually known as the ANNNI model, is a variant of the Ising model in which competing ferromagnetic and antiferromagnetic exchange interactions couple spins at nearest and next-nearest neighbor sites along one of the crystallographic axes of the lattice. The model is a prototype for complicated spatially modulated magnetic superstructures in crystals.
The model was introduced in 1961 by Roger Elliott from the University of Oxford, but only given this name in 1980 by Michael E. Fisher and Walter Selke. It provides a theoretical basis for understanding numerous experimental observations on commensurate and incommensurate structures, as well as accompanying phase transitions, in magnets, alloys, adsorbates, polytypes, multiferroics, and other solids; further applications range from modeling of cortex to quantum information.  
- "Phenomenological discussion of magnetic ordering in the heavy rare-earth metals". Phys. Rev. 124 (2): 346–353. 1961. doi:10.1103/PhysRev.124.346. Bibcode: 1961PhRv..124..346E.
- M.E. Fisher and W. Selke (1980). "Infinitely many commensurate phases in a simple Ising model". Phys. Rev. Lett. 44 (23): 1502–1505. doi:10.1103/PhysRevLett.44.1502. Bibcode: 1980PhRvL..44.1502F.
- P. Bak (1982). "Commensurate phases, incommensurate phases, and the devil's staircase". Reports on Progress in Physics 45 (6): 587–629. doi:10.1088/0034-4885/45/6/001. Bibcode: 1982RPPh...45..587B.
- W. Selke (1988). "The ANNNI model—Theoretical analysis and experimental application". Physics Reports 170 (4): 213–264. doi:10.1016/0370-1573(88)90140-8. Bibcode: 1988PhR...170..213S.
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