statistical physics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...

, 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), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical models in physics, mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that r ...

. In the ANNNI model, competing

ferromagnetic Ferromagnetism is a property of certain materials (such as iron) that results in a significant, observable magnetic permeability, and in many cases, a significant magnetic coercivity, allowing the material to form a permanent magnet. Ferromagne ...

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 Spin (physics), spins (on different sublattices) pointing in oppos ...

exchange interaction In chemistry and physics, the exchange interaction is a quantum mechanical constraint on the states of indistinguishable particles. While sometimes called an exchange force, or, in the case of fermions, Pauli repulsion, its consequences cannot alw ...

s couple

spins The spins (as in having "the spins") is an adverse reaction of Substance intoxication, intoxication that causes a state of vertigo and nausea, causing one to feel as if "spinning out of control", especially when lying down. It is most commonly as ...

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, macros ...

s. To describe experimental results on magnetic orderings in

erbium Erbium is a chemical element; it has Symbol (chemistry), 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- ...

, the model was introduced in 1961 by Roger Elliott from the

University of Oxford The University of Oxford is a collegiate university, 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 List of oldest un ...

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

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 physics, chemistry, and other related fields like biology, 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 Sta ...

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 in most cases at least one is a metal, metallic element, although it is also sometimes used for mixtures of elements; herein only metallic alloys are described. Metallic alloys often have prop ...

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 ...

s, polytypes,

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 polari ...

, and other

solid Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...

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. It is the largest site of Neuron, neural integration in the central nervous system, and plays ...

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