Geometrical frustration

In condensed matter physics, the term geometrical frustration (or in short: frustration^[1]) refers to a phenomenon where atoms tend to stick to non-trivial positions^{[citation needed]} or where, on a regular crystal lattice, conflicting inter-atomic forces (each one favoring rather simple, but different structures) lead to quite complex structures. As a consequence of the frustration in the geometry or in the forces, a plenitude of distinct ground states may result at zero temperature, and usual thermal ordering may be suppressed at higher temperatures. Much studied examples are amorphous materials, glasses, or dilute magnets.

The term frustration, in the context of magnetic systems, has been introduced by Gerard Toulouse in 1977.^[2][3] Frustrated magnetic systems had been studied even before. Early work includes a study of the Ising model on a triangular lattice with nearest-neighbor spins coupled antiferromagnetically, by G. H. Wannier, published in 1950.^[4] Related features occur in magnets with competing interactions, where both ferromagnetic as well as antiferromagnetic couplings between pairs of spins or magnetic moments are present, with the type of interaction depending on the separation distance of the spins. In that case commensurability, such as helical spin arrangements may result, as had been discussed originally, especially, by A. Yoshimori,^[5] T. A. Kaplan,^[6] R. J. Elliott,^[7] and others, starting in 1959, to describe experimental findings on rare-earth metals. A renewed interest in such spin systems with frustrated or competing interactions arose about two decades later, beginning in the 1970s, in the context of spin glasses and spatially modulated magnetic superstructures. In spin glasses, frustration is augmented by stochastic disorder in the interactions, as may occur experimentally in non-stoichiometric magnetic alloys. Carefully analyzed spin models with frustration include the Sherrington–Kirkpatrick model,^[8] describing spin glasses, and the ANNNI model,^[9] describing commensurability magnetic superstructures. Recently, the concept of frustration has been used in brain network analysis to identify the non-trivial assemblage of neural connections and highlight the adjustable elements of the brain.^[10]