statistical physics Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approxi ...

, the axial (or anisotropic) next-nearest neighbor Ising model, usually known as the ANNNI model, is a variant of the

Ising model The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...

in which competing

ferromagnetic Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...

and antiferromagnetic

exchange interaction In chemistry and physics, the exchange interaction (with an exchange energy and exchange term) is a quantum mechanical effect that only occurs between identical particles. Despite sometimes being called an exchange force in an analogy to classi ...

s 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

superstructure A superstructure is an upward extension of an existing structure above a baseline. This term is applied to various kinds of physical structures such as buildings, bridges, or ships. Aboard ships and large boats On water craft, the superstruct ...

s in

crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macr ...

s. To describe experimental results on magnetic orderings in erbium, the model was introduced in 1961 by Roger Elliott from the

University of Oxford The University of Oxford is a collegiate research university in Oxford, England. There is evidence of teaching as early as 1096, making it the oldest university in the English-speaking world and the world's second-oldest university in contin ...

, . The model has given its name in 1980 by

Michael E. Fisher Michael Ellis Fisher (3 September 1931 – 26 November 2021) was an English physicist, as well as chemist and mathematician, known for his many seminal contributions to statistical physics, including but not restricted to the theory of phase t ...

and Walter Selke., who analysed it first by

Monte Carlo method Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deter ...

s, and then by low temperature series expansions, showing the fascinating complexity of its phase diagram, including devil's staircases and a Lifshitz point. Indeed, it provides, for two- and three-dimensional systems, a theoretical basis for understanding numerous experimental observations on commensurate and incommensurate structures, as well as accompanying

phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states ...

s, in various

magnet A magnet is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, steel, nic ...

alloy An alloy is a mixture of chemical elements of which at least one is a metal. Unlike chemical compounds with metallic bases, an alloy will retain all the properties of a metal in the resulting material, such as electrical conductivity, ductilit ...

s, adsorbates,

polytypes In materials science, polymorphism describes the existence of a solid material in more than one form or crystal structure. Polymorphism is a form of isomerism. Any crystalline material can exhibit the phenomenon. Allotropy refers to polymorphis ...

, multiferroics, and other

solid Solid is one of the four fundamental states of matter (the others being liquid, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structura ...

s. Further possible applications range from modeling of

cerebral cortex The cerebral cortex, also known as the cerebral mantle, is the outer layer of neural tissue of the cerebrum of the brain in humans and other mammals. The cerebral cortex mostly consists of the six-layered neocortex, with just 10% consisting o ...

quantum information Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both t ...

References

{{reflist Statistical mechanics Lattice models Spin models