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This book covers all principal aspects of currently investigated frustrated systems, from exactly solved frustrated models to real experimental frustrated systems.

**Table of contents**

- Condensed Matter > Strongly Correlated Electrons
- Submission history
- Navigation menu
- Geometrical frustration
- Geometrical frustration - Wikipedia

Here our research focuses on questions as: What is the effect of Heisenberg-type perturbations in this model, or what are the consequences of magnetic fields and defects in the Kitaev-Heisenberg model? Further research topics include: Another research topic are one-dimensional Wigner lattices and their magnetic interactions. Finally it should be stressed, that there is a close connection to the research on generalized spin-orbital models, which are generically frustrated.

Frustrated quantum magnetism belongs to the very active research areas in condensed matter theory. This model has motivations both from the field of strongly correlated systems with orbital degeneracy and from that of solid-state based devices proposed for quantum computing. We find that the high degeneracy of ground states of the compass model is fragile and changes into twofold degenerate ground states for any finite amplitude of Heisenberg coupling. By computing the spin structure factors of finite clusters with Lanczos diagonalization, we evidence a rich variety of phases characterized by Z 2 symmetry, that are either ferromagnetic, C-type antiferromagnetic, or of Neel type, and analyze the effects of quantum fluctuations on phase boundaries.

In the ordered phases the anisotropy of compass interactions leads to a finite excitation gap to spin waves. Furthermore we have shown that for small nanoscale clusters with large anisotropy gap the lowest excitations are column-flip excitations that emerge due to Heisenberg perturbing interactions from the manifold of degenerate ground states of the compass model. The low energy column-flip or compass-type excitations are of particular interest, as they are robust against decoherence processes and are therefore well designed for storing information in quantum computing.

## Condensed Matter > Strongly Correlated Electrons

We have also pointed out that the dipolar interactions between nitrogen-vacancy centers forming a rectangular lattice in a diamond matrix may permit a solid-state realization of the anisotropic compass-Heisenberg model. Besides the mathematical beauty of the Kitaev model, recent studies were motivated by its possible relevance for orbitally degenerate systems with strong spin-orbit coupling such as layered iridates Na 2 IrO 3 and Li 2 IrO 3. These applications require to consider the extension, namely the Kitaev-Heisenberg model on honeycomb lattice, which reveals in its phase diagram apart of a liquid phase also several ordered e.

In particular we have explored the ground state properties of the Kitaev-Heisenberg model in a magnetic field, as well as the evolution of spin correlations in the presence of non-magnetic vacancies.

### Submission history

By means of exact diagonalizations, the phase diagram without vacancies was determined as a function of the magnetic field and the ratio between Kitaev and Heisenberg interactions. In the stripe phase, the combination of a vacancy and a small field breaks the six-fold symmetry of the model and stabilizes a particular stripe pattern.

Similar symmetry-breaking effects occur even at zero field due to interaction effects between vacancies. This selection mechanism and intrinsic randomness of vacancy positions may lead to spin-glass behavior.

The origin of the non-collinear arrangement of spins lies in the frustrated nature of magnetic superexchange interactions J 1 , J 2 and J 3 in these compounds. The remarkable magnetoelectric properties are another interesting feature of these compounds. Simultaneously with the helical spin arrangement in LiCuVO 4 appears a homogeneous electric polarization field that can be controlled by a magnetic field via the helix!

Such a control of electrial polarization by magnetic field or magnetization by an electric field is characteristic for multiferroic compounds. Effective hopping processes that contribute to superexchange in edgesharing Cu-O chains: In the ground state of the undoped chain all Cu ions are in the d 9 configuration. By an effective hopping process which involves intermediate O states a d 9 d 9 pair is virtually excited into pair formed by a d 10 configuration and a d 9 ligand-hole pair. The intermediate excitations are singlet configurations and therefore favor antiferromagnetism.

Importantly in the edgesharing compounds the 2nd neighbor process, via the Cu-O-O-Cu path, leads to a large AF superexchange integral J 2. Due to further processes that involve oxygen J 1 is negative. Typically these interactions lead to frustration and a helical spin arrangement. Three magnetic ions reside on the corners of a triangle with antiferromagnetic interactions between them; the energy is minimized when each spin is aligned opposite to neighbors.

Once the first two spins align antiparallel, the third one is frustrated because its two possible orientations, up and down, give the same energy. The third spin cannot simultaneously minimize its interactions with both of the other two. Since this effect occurs for each spin, the ground state is sixfold degenerate. Only the two states where all spins are up or down have more energy. Similarly in three dimensions, four spins arranged in a tetrahedron Figure 2 may experience geometric frustration. If there is an antiferromagnetic interaction between spins, then it is not possible to arrange the spins so that all interactions between spins are antiparallel.

There are six nearest-neighbor interactions, four of which are antiparallel and thus favourable, but two of which between 1 and 2, and between 3 and 4 are unfavourable. It is impossible to have all interactions favourable, and the system is frustrated. Geometrical frustration is also possible if the spins are arranged in a non- collinear way. If we consider a tetrahedron with a spin on each vertex pointing along the easy axis that is, directly towards or away from the centre of the tetrahedron , then it is possible to arrange the four spins so that there is no net spin Figure 3.

This is exactly equivalent to having an antiferromagnetic interaction between each pair of spins, so in this case there is no geometrical frustration. With these axes, geometric frustration arises if there is a ferromagnetic interaction between neighbours, where energy is minimized by parallel spins. The best possible arrangement is shown in Figure 4, with two spins pointing towards the centre and two pointing away.

The net magnetic moment points upwards, maximising ferromagnetic interactions in this direction, but left and right vectors cancel out i. There are three different equivalent arrangements with two spins out and two in, so the ground state is three-fold degenerate.

### Navigation menu

The mathematical definition is simple and analogous to the so-called Wilson loop in Quantum chromodynamics: One considers for example expressions "total energies" or "Hamiltonians" of the form. If the graph G has quadratic or triangular faces P , the so-called "plaquette variables" P W , "loop-products" of the following kind, appear:. One has to perform a sum over these products, summed over all plaquettes. In the last-mentioned case the plaquette is "geometrically frustrated". It can be shown that the result has a simple gauge invariance: Although most previous and current research on frustration focuses on spin systems, the phenomenon was first studied in ordinary ice.

This result was then explained by Linus Pauling [11] to an excellent approximation, who showed that ice possesses a finite entropy estimated as 0. In the hexagonal or cubic ice phase the oxygen ions form a tetrahedral structure with an O—O bond length 2. Every oxygen white ion is surrounded by four hydrogen ions black and each hydrogen ion is surrounded by 2 oxygen ions, as shown in Figure 5.

## Geometrical frustration

Maintaining the internal H 2 O molecule structure, the minimum energy position of a proton is not half-way between two adjacent oxygen ions. There are two equivalent positions a hydrogen may occupy on the line of the O—O bond, a far and a near position. Thus a rule leads to the frustration of positions of the proton for a ground state configuration: Pauling proposed that the open tetrahedral structure of ice affords many equivalent states satisfying the ice rules.

Pauling went on to compute the configurational entropy in the following way: Each O—O bond has two positions for a proton, leading to 2 2 N possible configurations. However, among the 16 possible configurations associated with each oxygen, only 6 are energetically favorable, maintaining the H 2 O molecule constraint. A mathematically analogous situation to the degeneracy in water ice is found in the spin ices.

A common spin ice structure is shown in Figure 6 in the cubic pyrochlore structure with one magnetic atom or ion residing on each of the four corners. Due to the strong crystal field in the material, each of the magnetic ions can be represented by an Ising ground state doublet with a large moment. Every tetrahedral cell must have two spins pointing in and two pointing out in order to minimize the energy. Currently the spin ice model has been approximately realized by real materials, most notably the rare earth pyrochlores Ho 2 Ti 2 O 7 , Dy 2 Ti 2 O 7 , and Ho 2 Sn 2 O 7. These materials all show nonzero residual entropy at low temperature.

The spin ice model is only one subdivision of frustrated systems. In general frustration is caused either by competing interactions due to site disorder see also the Villain model [12] or by lattice structure such as in the triangular , face-centered cubic fcc , hexagonal-close-packed , tetrahedron , pyrochlore and kagome lattices with antiferromagnetic interaction. So frustration is divided into two categories: The frustration of a spin glass is understood within the framework of the RKKY model, in which the interaction property, either ferromagnetic or anti-ferromagnetic, is dependent on the distance of the two magnetic ions.

Due to the lattice disorder in the spin glass, one spin of interest and its nearest neighbors could be at different distances and have a different interaction property, which thus leads to different preferred alignment of the spin. With the help of lithography techniques, it is possible to fabricate sub-micrometer size magnetic islands whose geometric arrangement reproduces the frustration found in naturally occurring spin ice materials. These islands are manually arranged to create a two-dimensional analog to spin ice.

In their previous work on a square lattice of frustrated magnets, they observed both ice-like short-range correlations and the absence of long-range correlations, just like in the spin ice at low temperature.

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These results solidify the uncharted ground on which the real physics of frustration can be visualized and modeled by these artificial geometrically frustrated magnets, and inspires further research activity. These artificially frustrated ferromagnets can exhibit unique magnetic properties when studying their global response to an external field using Magneto-Optical Kerr Effect. Another type of geometrical frustration arises from the propagation of a local order. A main question that a condensed matter physicist faces is to explain the stability of a solid.

It is sometimes possible to establish some local rules, of chemical nature, which lead to low energy configurations and therefore govern structural and chemical order.

This is not generally the case and often the local order defined by local interactions cannot propagate freely, leading to geometric frustration. A common feature of all these systems is that, even with simple local rules, they present a large set of, often complex, structural realizations. Geometric frustration plays a role in fields of condensed matter, ranging from clusters and amorphous solids to complex fluids.

The general method of approach to resolve these complications follows two steps. First, the constraint of perfect space-filling is relaxed by allowing for space curvature.

## Geometrical frustration - Wikipedia

An ideal, unfrustrated, structure is defined in this curved space. Then, specific distortions are applied to this ideal template in order to embed it into three dimensional Euclidean space. The final structure is a mixture of ordered regions, where the local order is similar to that of the template, and defects arising from the embedding.

Among the possible defects, disclinations play an important role. Two-dimensional examples are helpful in order to get some understanding about the origin of the competition between local rules and geometry in the large. Consider first an arrangement of identical discs a model for a hypothetical two-dimensional metal on a plane; we suppose that the interaction between discs is isotropic and locally tends to arrange the disks in the densest way as possible.

The best arrangement for three disks is trivially an equilateral triangle with the disk centers located at the triangle vertices. The study of the long range structure can therefore be reduced to that of plane tilings with equilateral triangles. A well known solution is provided by the triangular tiling with a total compatibility between the local and global rules: