ANNNI model

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,[1] but only given this name in 1980 by Michael E. Fisher and Walter Selke.[2] It provides a theoretical basis for understanding numerous experimental observations on commensurate and incommensurate structures,[3] 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.[4]


  1. R. J. Elliott (1961). "Phenomenological discussion of magnetic ordering in the heavy rare-earth metals". Phys. Rev. 124 (2): 346–353. Bibcode:1961PhRv..124..346E. doi:10.1103/PhysRev.124.346.
  2. M.E. Fisher and W. Selke (1980). "Infinitely many commensurate phases in a simple Ising model". Phys. Rev. Lett. 44 (23): 1502–1505. Bibcode:1980PhRvL..44.1502F. doi:10.1103/PhysRevLett.44.1502.
  3. P. Bak (1982). "Commensurate phases, incommensurate phases, and the devil's staircase". Reports on Progress in Physics. 45 (6): 587–629. Bibcode:1982RPPh...45..587B. CiteSeerX doi:10.1088/0034-4885/45/6/001.
  4. W. Selke (1988). "The ANNNI model—Theoretical analysis and experimental application". Physics Reports. 170 (4): 213–264. Bibcode:1988PhR...170..213S. doi:10.1016/0370-1573(88)90140-8.
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