Geometrical frustration

Geometric frustration ( also abbreviated as frustration ) is a phenomenon in condensed matter physics, in preventing the geometric properties of a crystal lattice or the presence of conflicting atomic forces the simultaneous minimization of all interaction energies at a given grid point. This can lead to highly degenerate ground states of 0 different entropy even at 0 K. In simpler terms, the substance can never be completely frozen, as the structure that makes them, does not have a single minimum energy state. Motion at the molecular level thus still takes place at zero temperature without energy supply.

The term frustration in the context of magnetic systems goes back to Gerard Toulouse (1977) and is particularly important in the so-called spin glasses. Magnetic systems with geometric frustration have been studied for many years. Early works include studies of an Ising model on a triangular lattice with antiferromagnetically coupled neighboring spins by GH Wannier, which was published in 1950. Later there were similar experiments on magnets with competing interactions, ie with different couplings, but each of which simple ( ferromagnetic or antiferromagnetic ) prefer a total different structures. In this case, incommensurable Spinanordungen may result (eg spiral structure ), how they were treated since 1959 by Akio Yoshimori, Thomas A. Kaplan, Roger J. Elliott and others. A renewed interest in such spin systems came about two decades later in the context of the previously mentioned spin glasses and spatially modulated magnetic superstructures. In spin glasses the geometrical frustration is compounded by stochastic disorder in the interactions. Well-known spin models with competing or frustrated interactions include the Sherrington - Kirkpatrick model with one that describes spin glasses, and represents the annni model that commensurate and incommensurate magnetic superstructures.

Magnetic order

Geometric frustration is a major phenomenon in magnetism of solid bodies, where it has to do with the topological arrangement of spins. A simple 2D example is shown in Figure 1. Three magnetic ions sit at the corners of a triangular lattice with antiferromagnetic interactions between them - because the locations of the ( otherwise identical ) particles are fixed on the grid, the particles are distinguishable by their location. The energy is minimal when each spin is opposite relative to its neighbors. Now, if the first two spins aligned antiparallel, so is the third frustrated because its two possible orientations, up and down, give the same energy. The third spin can not minimize with two other spins the same time its interaction energy. Since this occurs for each of the three spins of the six -fold degenerate ground state, only the two states where all spins up or down are have a higher energy.

In a similar manner (Figure 2) can be geometrically frustrated in three dimensions in a four tetrahedra arranged spins. When the spins interact antiferromagnetically, they can not align all anti-parallel. There are six nearest-neighbor interactions, of which four anti-parallel and therefore energetically are "favorable", but remain two energetically " adverse " interactions ( here 1-2, and 3-4 ).

Figure 2: Anti Ferromagnetic interacting spins in a tetrahedral arrangement

Geometric frustration is also possible when the pins are arranged non- collinearly. In a tetrahedron to the vertices of each spin is sitting, which is aligned along the respective axis through the center of the tetrahedron, the spins can be arranged so that they cancel each other out, so there is no net spin are ( Figure 3). This is equivalent to an antiferromagnetic spin interaction between each pair, and in this case there is no geometrical frustration before. With such geometrical axes frustration occurs when there is a ferromagnetic interaction between neighbors, so that the energy is minimized by parallel spins. The "best " arrangement is shown in Figure 4; there show two spins off towards the center and two from him. The resulting magnetic moment pointing upwards and maximizes the ferromagnetic interaction in this direction, the vector components in other directions cancel each other, i.e., they are arranged in anti- ferromagnetic. There are three different but equivalent arrangements in which two spins are facing outside and two inside, so that the base state is three -fold degenerate.

Figure 4: Spins with geometrical frustration along the plane passing through the center axes

Mathematical definition

The mathematical definition is quite simple ( and analogous to the so-called Wilson loop in quantum chromodynamics ): There are energy variables of the form

Considered, where G is the considered graph, while the so-called " energy exchange " between nearest - neighbors ( in predetermined units of energy ), the values ​​are to assume, while the inner products are scalar or vectorial spin variables. If the graph G is the ( square or triangular ) boundary surfaces P has the so-called " sticker tags", occur in the following so-called " loop products " of the form I1, I2, 2, 3, I3, I4, 4, 1 and I1, I2 2, 3 I3 1, which are also referred to as " Frustationsprodukt ". And that the sum of these products to form frustration, summed over all plaques. The result for the single plaque is either 1 or -1. In the negative case is the badge " geometrically frustrated ".

One can show that the result is gauge invariant, ie it does not change if one subjects the local values ​​of the exchange integrals and the spins at the same time the following gauge transformation: where the and any numbers But not only the " frustration products " but also measurable other variables, for example, do not change in such " Umeichungen ".