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 Mathematics, mathematical tools for dealing with large populations ...

, 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 In materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usually related to the spins of electrons, align in a regular pattern with neighboring spins (on different sublattices) pointing in opposite directions. ...

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 classical ...

s couple

spins The spins (as in having "the spins")Diane Marie Leiva. ''The Florida State University College of Education''Women's Voices on College Drinking: The First-Year College Experience"/ref> is an adverse reaction of intoxication that causes a state of v ...

at nearest and next-nearest neighbor sites along one of the crystallographic axes of the

lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...

. 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, macros ...

s. To describe experimental results on magnetic orderings in

erbium Erbium is a chemical element with the symbol Er and atomic number 68. A silvery-white solid metal when artificially isolated, natural erbium is always found in chemical combination with other elements. It is a lanthanide, a rare-earth element ...

, the model was introduced in 1961 by

Roger Elliott Major General Roger Elliott ( 1665 – 16 May 1714 ) was one of the earliest British Governors of Gibraltar. A member of the Eliot family, his son Granville Elliott became the first Count Elliott and his nephew George Augustus Eliott als ...

from the

University of Oxford , mottoeng = The Lord is my light , established = , endowment = £6.1 billion (including colleges) (2019) , budget = £2.145 billion (2019–20) , chancellor ...

, . The model has given its name in 1980 by Michael E. Fisher and

Walter Selke Walter Selke (born 1947) is a German retired professor for Theoretical Physics at the RWTH Aachen. After having received his doctoral degree at the Leibniz University Hannover, followed by postdoctoral positions at the Saarland University, Corn ...

., 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 determi ...

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 of ...

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, nickel, ...

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, ductility, ...

adsorbate Adsorption is the adhesion of atoms, ions or molecules from a gas, liquid or dissolved solid to a surface. This process creates a film of the ''adsorbate'' on the surface of the ''adsorbent''. This process differs from absorption, in which a f ...

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 polymorphism ...

multiferroics Multiferroics are defined as materials that exhibit more than one of the primary ferroic properties in the same phase: * ferromagnetism – a magnetisation that is switchable by an applied magnetic field * ferroelectricity – an electric polarisa ...

, and other

solid Solid is one of the State of matter#Four fundamental states, four fundamental states of matter (the others being liquid, gas, and Plasma (physics), plasma). The molecules in a solid are closely packed together and contain the least amount o ...

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 of ...

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 th ...

References

{{reflist Statistical mechanics Lattice models Spin models